Existence of Asymptotically Almost Automorphic Mild...

Research ArticleExistence of Asymptotically Almost Automorphic Mild Solutionsof Semilinear Fractional Differential Equations

Junfei Cao1 Zaitang Huang 2 and GastonM NrsquoGueacutereacutekata 3

1Department of Mathematics Guangdong University of Education Guangzhou 510303 China2School of Mathematical Sciences Guangxi Teachers Education University Nanning 530023 China3Department of Mathematics Morgan State University Baltimore MD 21251 USA

Correspondence should be addressed to Gaston M NrsquoGuerekata nguerekataaolcom

Received 21 December 2017 Revised 18 April 2018 Accepted 10 May 2018 Published 1 August 2018

Academic Editor Patricia J Y Wong

Copyright copy 2018 Junfei Cao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinearfractional differential equations D120572119905 119909(119905) = 119860119909(119905) + D120572minus1119905 119865(119905 119909(119905) 119861119909(119905)) 119905 isin R where 1 lt 120572 lt 2 119860 is a linear densely definedoperator of sectorial type on a complex Banach space 119883 and 119861 is a bounded linear operator defined on 119883 119865 is an appropriatefunction defined on phase space and the fractional derivative is understood in the Riemann-Liouville sense Combining the fixedpoint theorem due to Krasnoselskii and a decomposition technique we prove the existence of asymptotically almost automorphicmild solutions to such problems Our results generalize and improve some previous results since the (locally) Lipschitz continuityon the nonlinearity 119865 is not requiredThe results obtained are utilized to study the existence of asymptotically almost automorphicmild solutions to a fractional relaxation-oscillation equation

1 Introduction

The almost periodic function introduced seminally by Bohrin 1925 plays an important role in describing the phe-nomena that are similar to the periodic oscillations whichcan be observed frequently in many fields such as celes-tial mechanics nonlinear vibration electromagnetic theoryplasma physics engineering and ecosphere The concept ofalmost automorphy which is an important generalizationof the classical almost periodicity was first introduced inthe literature [1ndash4] by Bochner in relation to some aspectsof differential geometry Since then this pioneer work hasattracted more and more attention and has been substan-tially extended in several different directions Many authorshave made important contributions to this theory (see forinstance [5ndash17] and the references therein) Especially in [56] the authors gave an important overview about the theoryof almost automorphic functions and their applications todifferential equations

As a natural extension of almost automorphy the con-cept of asymptotic almost automorphy which is the centralissue to be discussed in this paper was introduced in the

literature [18] by NrsquoGuerekata in the early eighties Sincethen this notion has found several developments and hasbeen generalized into different directions Until now theasymptotically almost automorphic functions as well as theasymptotically almost automorphic solutions for differentialsystems have been investigated by many mathematicianssee [19] by Bugajewski and NrsquoGuerekata [20] by DiaganaHernandez and dos Santos and [21] by Ding Xiao andLiang for the asymptotically almost automorphic solutionsto integrodifferential equations see [22] by Zhao Changand NrsquoGuerekata for the asymptotically almost automorphicsolutions to the nonlinear delay integral equations and see[23] by Chang and Tang and [24] by Zhao Chang andNieto for the asymptotically almost automorphic solutionsto stochastic differential equations and the existence ofasymptotically almost automorphic solutions has becomeoneof the most attractive topics in the qualitative theory ofdifferential equations due to its significance and applicationsin physics mathematical biology control theory and so onWe refer the reader to the monographs of NrsquoGuerekata [25]for the recently theory and applications of asymptoticallyalmost automorphic functions

HindawiInternational Journal of Differential EquationsVolume 2018 Article ID 8243180 23 pageshttpsdoiorg10115520188243180

2 International Journal of Differential Equations

With motivation coming from a wide range of engineer-ing and physical applications fractional differential equationshave recently attracted great attention of mathematiciansand scientists This kind of equations is a generalization ofordinary differential equations to arbitrary noninteger ordersFractional differential equations find numerous applicationsin the field of viscoelasticity feedback amplifiers electri-cal circuits electro analytical chemistry fractional multi-poles neuron modelling encompassing different branches ofphysics chemistry and biological sciences [26ndash32] Manyphysical processes appear to exhibit fractional order behaviorthat may vary with time or space In recent years therehas been a significant development in ordinary and partialdifferential equations involving fractional derivatives weonly enumerate here the monographs of Kilbas et al [26 27]Diethelm [28] Hilfer [29] Podlubny [30] Miller [31] andZhou [32] and the papers of Agarwal et al [33 34] Benchohraet al [35 36] El-Borai [37] Lakshmikantham et al [38ndash41]Mophou et al [42ndash45]NrsquoGuerekata [46] andZhou et al [47ndash50] and the reference therein

The study of almost periodic and almost automorphictype solutions to fractional differential equations was initi-ated by Araya and Lizama [11] In their work the authorsinvestigated the existence and uniqueness of an almostautomorphic mild solution of the semilinear fractional dif-ferential equation

D120572119905 119909 (119905) = 119860119909 (119905) + 119865 (119905 119909 (119905)) 119905 isin R 1 lt 120572 lt 2 (1)

when 119860 is a generator of an 120572-resolvent family and D120572119905 is theRiemann-Liouville fractional derivative In [51] Cuevas andLizama considered the fractional differential equation

D120572119905 119909 (119905) = 119860119909 (119905) +D120572minus1119905 119865 (119905 119909 (119905)) 119905 isin R 1 lt 120572 lt 2 (2)

where 119860 is a linear operator of sectorial negative type ona complex Banach space 119883 and the fractional derivative isunderstood in the Riemann-Liouville sense Under suitableconditions on 119865(119905 119909) the authors proved the existence anduniqueness of an almost automorphic mild solution to (2)Cuevas et al [52 53] studied respectively the pseudo almostperiodic and pseudo almost periodic class infinity mildsolutions to (2) assuming that 119865 R times 119883 997888rarr 119883 and(119905 119909) 997888rarr 119865(119905 119909) is a pseudo almost periodic and pseudoalmost periodic of class infinity function satisfying suitableconditions in 119909 isin 119883 Agarwal et al [54] studied the existenceand uniqueness of a weighted pseudo almost periodic mildsolution to equation (2) Ding et al [55] investigated theexistence and uniqueness of almost automorphic solution to(2) assuming that 119865 R times 119883 997888rarr 119883 and (119905 119909) 997888rarr 119865(119905 119909) isStepanov-like almost automorphic in 119905 isin R satisfying somekind of Lipschitz conditions Cuevas et al [56] studied theexistence of almost periodic (resp pseudo almost periodic)mild solutions to equation (2) assuming that 119865 Rtimes119883 997888rarr 119883and (119905 119909) 997888rarr 119865(119905 119909) is Stepanov almost (resp Stepanov-like pseudo almost) periodic in 119905 isin R uniformly for 119909 isin 119883Chang et al [57] studied the existence and uniqueness ofweighted pseudo almost automorphic solution to equation

(2) with Stepanov-like weighted pseudo almost automorphiccoefficient He et al [58] studied also the existence anduniqueness of weighted Stepanov-like pseudo almost auto-morphic mild solution to (2) Cao et al [59] studied theexistence and uniqueness of antiperiodic mild solution to(2) In [60] Cuevas et al showed sufficient conditions toensure the existence and uniqueness of mild solution for (2)in the following classes of vector-valued function spaces peri-odic functions asymptotically periodic functions pseudoperiodic functions almost periodic functions asymptoticallyalmost periodic functions pseudo almost periodic func-tions almost automorphic functions asymptotically almostautomorphic functions pseudo almost automorphic func-tions compact almost automorphic functions asymptoticallycompact almost automorphic functions pseudo compactalmost automorphic functions 119878-asymptotically 120596-periodicfunctions decay functions and mean decay functions

Recently Xia et al [61] established some sufficient criteriafor the existence and uniqueness of (120583 ])-pseudo almostautomorphic solution to the semilinear fractional differentialequation

D120572119905 119909 (119905) = 119860119909 (119905) + D120572minus1119905 119865 (119905 119861119909 (119905)) 119905 isin R (3)

where 1 lt 120572 lt 2 119860 is a sectorial operator of type 120596 lt 0 on acomplex Banach space119883 and 119861 is a bounded linear operatorThe fractional derivative is understood in the Riemann-Liouville senseTheir discussion is divided into two cases ie119865 R times 119883 997888rarr 119883 (119905 119909) 997888rarr 119865(119905 119909) is (120583 ])-pseudo almostautomorphic and 119865 R times 119883 997888rarr 119883 and (119905 119909) 997888rarr 119865(119905 119909)is Stepanov-like (120583 ])-pseudo almost automorphic Kavithaet al [62] studied weighted pseudo almost automorphicsolutions of the fractional integrodifferential equation

D120572119905 119909 (119905) = 119860119909 (119905) + D120572minus1119905 119865 (119905 119909 (119905) 119870119909 (119905)) 119905 isin R (4)

where 1 lt 120572 lt 2 and119870119909 (119905) = int119905

minusinfin119896 (119905 minus 119904) ℎ (119904 119909 (119904)) d119904 (5)

119860 is a linear densely defined sectorial operator on a complexBanach space 119883 119865 R times 119883 times 119883 997888rarr 119883 and (119905 119909 119910) 997888rarr119865(119905 119909 119910) is a weighted pseudo almost automorphic functionin 119905 isin R for each 119909 119910 isin 119883 satisfying suitable conditionsThe fractional derivative is understood in the Riemann-Liouville sense Mophou [63] investigated the existence anduniqueness of weighted pseudo almost automorphic mildsolution to the fractional differential equation

D120572119905 119909 (119905) = 119860119909 (119905) +D120572minus1119905 119865 (119905 119909 (119905) 119861119909 (119905)) 119905 isin R 1 lt 120572 lt 2 (6)

where 119860 119863(119860) sub 119883 997888rarr 119883 is a linear densely oper-ator of sectorial type on a complex Banach space 119883 119861 119883 997888rarr 119883 is a bounded linear operator and 119865 R times 119883 times119883 997888rarr 119883 and (119905 119909 119910) 997888rarr 119865(119905 119909 119910) is a weighted pseudoalmost automorphic function in 119905 isin R for each 119909 119910 isin 119883satisfying suitable conditions The fractional derivative D120572119905 isto be understood in Riemann-Liouville sense Chang et al

International Journal of Differential Equations 3

[64] investigated some existence results of 120583-pseudo almostautomorphic mild solutions to (6) assuming that 119865 R times119883times119883 997888rarr 119883 and (119905 119909 119910) 997888rarr 119865(119905 119909 119910) is a 120583-pseudo almostautomorphic function in 119905 isin R for each 119909 119910 isin 119883 satisfyingsuitable conditions For more on the almost periodicity andalmost automorphy for fractional differential equations andrelated issues we refer the reader to [65ndash67] and others

Equation (6) is motivated by physical problems Indeeddue to their applications in fields of sciencewhere characteris-tics of anomalous diffusion are presented type (6) equationsare attracting increasing interest (cf [68ndash70] and referencestherein) For example anomalous diffusion in fractals [69] orin macroeconomics [71] has been recently well studied in thesetting of fractional Cauchy problems like (6) For this reason(6) has gotten a considerable attention in recent years (cf [51ndash64 68ndash71] and the references therein)

To the best of our knowledge much less is knownabout the existence of asymptotically almost automorphicmild solutions to (6) when the nonlinearity 119865(119905 119909 119910) as awhole loses the Lipschitz continuity with respect to 119909 and119910 Motivated by the abovementioned works the purposeof this paper is to establish some new existence results ofasymptotically almost automorphic mild solutions to (6)In our results the nonlinearity 119865 R times 119883 times 119883 997888rarr119883 (119905 119909 119910) 997888rarr 119865(119905 119909 119910) does not have to satisfy a(locally) Lipschitz condition (see Remark 22) However inmany papers (for instance [11 51ndash64]) on almost periodictype and almost automorphic type solutions to fractionaldifferential equations to be able to apply the well-knownBanach contraction principle a (locally) Lipschitz conditionfor the nonlinearity of corresponding fractional differentialequations is needed As can be seen our results generalizethose as well as related research and have more broadapplications In particular as application and to illustrateour main results we will examine some sufficient conditionsfor the existence of asymptotically almost automorphic mildsolutions to the fractional relaxation-oscillation equationgiven by

120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909) + 120597120572minus1119905 [120583119886 (119905)

sdot sin( 12 + cos 119905 + cosradic2119905) [sin 119906 (119905 119909) + 119906 (119905 119909)]

+ ]119890minus|119905| [119906 (119905 119909) + sin 119906 (119905 119909)]] 119905 isin R 119909 isin [0 120587]

(7)

with boundary conditions 119906(119905 0) = 119906(119905 120587) = 0 119905 isin R where119886(119905) isin 119861119862(RR+) is a function and 119901 120583 and ] are positiveconstants

The rest of this paper is organized as follows In Section 2some concepts the related notations and some useful lem-mas are introduced and established In Section 3 we provethe existence of asymptotically almost automorphic mildsolutions to such problems The results obtained are utilizedto study the existence of asymptotically almost automorphicmild solutions to a fractional relaxation-oscillation equationgiven in Section 4

2 Preliminaries

This section is concerned with some notations definitionslemmas and preliminary facts which are used in whatfollows

From now on let (119883 sdot ) and (119884 sdot 119884) be two Banachspaces and 119861119862(R 119883) (resp 119861119862(R times 119884 times 119884119883)) is the spaceof all 119883-valued bounded continuous functions (resp jointlybounded continuous functions 119865 R times 119884 times 119884 997888rarr 119883)Furthermore 1198620(R 119883) (resp 1198620(R times 119884 times 119884119883)) is theclosed subspace of 119861119862(R 119883) (resp 119861119862(R times 119884 times 119884119883))consisting of functions vanishing at infinity (vanishing atinfinity uniformly in any compact subset of 119884 times 119884 in otherwords

lim|119905|997888rarr+infin

1003817100381710038171003817119892 (119905 119909 119910)1003817100381710038171003817 = 0 uniformly for (119909 119910) isin K (8)

whereK is an any compact subset of 119884times119884) Let also L(119883) bethe Banach space of all bounded linear operators from119883 intoitself endowed with the norm

119879L(119883) = sup 119879119909 119909 isin 119883 119909 = 1 (9)

For a bounded linear operator 119860 isin L(119883) let 120588(119860) and 119863(119860)stand for the resolvent and domain of 119860 respectively

First let us recall some basic definitions and results onalmost automorphic and asymptotically almost automorphicfunctions

Definition 1 ((Bochner) [1] (NrsquoGuerekata) [6]) A continuousfunction 119865 R 997888rarr 119883 is said to be almost automorphicif for every sequence of real numbers 1199041015840119899 there exists asubsequence 119904119899 such that

Θ (119905) = lim119899997888rarrinfin

119865 (119905 + 119904119899) (10)

is well defined for each 119905 isin R and

lim119899997888rarrinfin

Θ(119905 minus 119904119899) = 119865 (119905) for each 119905 isin R (11)

Denote by 119860119860(R 119883) the set of all such functions

Remark 2 (see [6]) By the point-wise convergence thefunctionΘ(119905) inDefinition 1 ismeasurable but not necessarilycontinuous Moreover if Θ(119905) is continuous then 119865(119905) isuniformly continuous (cf eg [17] Theorem 26) and ifthe convergence in Definition 1 is uniform on R one getsalmost periodicity (in the sense of Bochner and von Neu-mann) Almost automorphy is thus a more general conceptthan almost periodicity There exists an almost automorphicfunctionwhich is not almost periodicThe function119865 R 997888rarrR given by

119865 (119905) = sin( 12 + cos 119905 + cosradic2119905) (12)

is an example of such functions [72]

Lemma 3 (see [5]) 119860119860(R 119883) is a Banach space with thenorm 119865infin = sup119905isinR119865(119905)


Definition 4 (see [6]) A continuous function119865 Rtimes119884times119884 997888rarr119883 is said to be almost automorphic in 119905 isin R uniformly for all(119909 119910) isin 119870 where119870 is any bounded subset of119884times119884 if for everysequence of real numbers 1199041015840119899 there exists a subsequence 119904119899such that


119865 (119905 + 119904119899 119909 119910) = Θ (119905 119909 119910) exists

for each 119905 isin R and each (119909 119910) isin 119870 (13)

andlim119899997888rarrinfin

Θ(119905 minus 119904119899 119909 119910) = 119865 (119905 119909 119910) exists


The collection of those functions is denoted by 119860119860(R times 119884 times119884119883)Remark 5 The function 119865 R times 119883 times 119883 997888rarr 119883 given by

119865 (119905 119909 119910) = sin( 12 + cos 119905 + cosradic2119905) [sin (119909) + 119910] (15)

is almost automorphic in 119905 isin R uniformly for all (119909 119910) isin 119870where119870 is any bounded subset of 119883 times 119883 119883 = 1198712[0 120587]

Similar to Lemma 22 of [73] and Proposition 32 of[63] we have the following result on almost automorphicfunctions

Lemma 6 Let 119865 Rtimes119883times119883 997888rarr 119883 be almost automorphic in119905 isin R uniformly for all (119909 119910) isin 119870 where 119870 is any boundedsubset of 119883 times 119883 and assume that 119865(119905 119909 119910) is uniformlycontinuous on 119870 uniformly for 119905 isin R that is for any 120576 gt 0there exists 120575 gt 0 such that 1199091 1199092 1199101 1199102 isin 119870 and 1199091 minus 1199101 +1199092 minus 1199102 lt 120575 imply that1003817100381710038171003817119865 (119905 1199091 1199092) minus 119865 (119905 1199101 1199102)1003817100381710038171003817 lt 120576 forall119905 isin R (16)

Let 119909 119910 R 997888rarr 119883 be almost automorphic Then the functionΥ R 997888rarr 119883 defined by Υ(119905) = 119865(119905 119909(119905) 119910(119905)) is almostautomorphic

Proof Suppose that 119904119899 is a sequence of real numbers Thenby the definition of almost automorphic functions we canextract a subsequence 120591119899 of 119904119899 such that

(1198751) lim119899997888rarrinfin

119909 (119905 + 120591119899) = 119909 (119905) for each 119905 isin R(1198752) lim119899997888rarrinfin

119909 (119905 minus 120591119899) = 119909 (119905) for each 119905 isin R(1198753) lim119899997888rarrinfin



119865 (119905 + 120591119899 119909 119910) = 119865 (119905 119909 119910)for each 119905 isin R 119909 119910 isin 119883

(1198756) lim119899997888rarrinfin

119865 (119905 minus 120591119899 119909 119910) = 119865 (119905 119909 119910)for each 119905 isin R 119909 119910 isin 119883

(17)

Write

Υ (119905) fl 119865 (119905 119909 (119905) 119910 (119905)) 119905 isin R (18)

Then10038171003817100381710038171003817Υ (119905 + 120591119899) minus Υ (119905)10038171003817100381710038171003817

= 10038171003817100381710038171003817119865 (119905 + 120591119899 119909 (119905 + 120591119899) 119910 (119905 + 120591119899))minus 119865 (119905 119909 (119905) 119910 (119905))10038171003817100381710038171003817le 1003817100381710038171003817119865 (119905 + 120591119899 119909 (119905 + 120591119899) 119910 (119905 + 120591119899))minus 119865 (119905 + 120591119899 119909 (119905) 119910 (119905))1003817100381710038171003817 + 10038171003817100381710038171003817119865 (119905 + 120591119899 119909 (119905) 119910 (119905))minus 119865 (119905 119909 (119905) 119910 (119905))10038171003817100381710038171003817

(19)

Since 119909(119905) and 119910(119905) are almost automorphic then 119909(119905) 119910(119905)and 119909(119905) and 119910(119905) are bounded Therefore we can choose abounded subset119870 sub 119883 times 119883 such that

(119909 (119905) 119910 (119905)) isin 119870(119909 (119905) 119910 (119905)) isin 119870

forall119905 isin R(20)

By (1198751) (1198753) and the uniform continuity of 119865(119905 119909 119910) in(119909(119905) 119910(119905)) isin 119870 we have


1003817100381710038171003817119865 (119905 + 120591119899 119909 (119905 + 120591119899) 119910 (119905 + 120591119899))minus 119865 (119905 + 120591119899 119909 (119905) 119910 (119905))1003817100381710038171003817 = 0 (21)

Moreover by (1198755)lim119899997888rarrinfin

10038171003817100381710038171003817119865 (119905 + 120591119899 119909 (119905) 119910 (119905)) minus 119865 (119905 119909 (119905) 119910 (119905))10038171003817100381710038171003817 = 0 (22)

so remembering the above triangle inequality we deduce that


10038171003817100381710038171003817Υ (119905 + 120591119899) minus Υ (119905)10038171003817100381710038171003817 = 0 for each 119905 isin R (23)

Using the same argument we can prove that


10038171003817100381710038171003817Υ (119905 minus 120591119899) minus Υ (119905)10038171003817100381710038171003817 = 0 for each 119905 isin R (24)

This proves that Υ(119905) is almost automorphic by the definition

Remark 7 If 119865(119905 119909 119910) satisfies a Lipschitz condition withrespect to 119909 and 119910 uniformly in 119905 isin R ie for each pair1199091 1199092 1199101 1199102 isin 119883

1003817100381710038171003817119865 (119905 1199091 1199092) minus 119865 (119905 1199101 1199102)1003817100381710038171003817le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817)

(25)

uniformly in 119905 isin R where 119871 gt 0 is called the Lipschitzconstant for the function 119865(119905 119909 119910) then 119865(119905 119909 119910) is uni-formly continuous on 119870 uniformly for 119905 isin R where119870 is anybounded subset of 119883 times 119883


Remark 8 If 119865(119905 119909 119910) satisfies a local Lipschitz conditionwith respect to 119909 and 119910 uniformly in 119905 isin R ie for each pair1199091 1199092 1199101 1199102 isin 119883 119905 isin R

1003817100381710038171003817119865 (119905 1199091 1199092) minus 119865 (119905 1199101 1199102)1003817100381710038171003817le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817)

(26)

where 119871(119905) isin 119861119862(RR+) then 119865(119905 119909 119910) is uniformly con-tinuous on 119870 uniformly for 119905 isin R where 119870 is any boundedsubset of 119883 times 119883

Definition 9 (see [6]) A continuous function 119865 R 997888rarr 119883is said to be asymptotically almost automorphic if it can bedecomposed as 119865(119905) = 119866(119905) + Φ(119905) where

119866 (119905) isin 119860119860 (R 119883) Φ (119905) isin 1198620 (R 119883) (27)


Remark 10 The function 119865 R 997888rarr R defined by

119865 (119905) = 119866 (119905) + Φ (119905)= sin( 1

2 + cos 119905 + cosradic2119905) + 119890minus|119905| (28)

is an asymptotically almost automorphic function with

119866 (119905) = sin( 12 + cos 119905 + cosradic2119905) isin 119860119860 (RR)

Φ (119905) = 119890minus|119905| isin 1198620 (RR) (29)

Lemma 11 (see [6]) 119860119860119860(R 119883) is also a Banach space withthe supremum norm sdot infinDefinition 12 (see [6]) A continuous function 119865 R times 119884 times119884 997888rarr 119883 is said to be asymptotically almost automorphic ifit can be decomposed as 119865(119905 119909 119910) = 119866(119905 119909 119910) + Φ(119905 119909 119910)where

119866 (119905 119909 119910) isin 119860119860 (R times 119884 times 119884119883) Φ (119905 119909 119910) isin 1198620 (R times 119884 times 119884119883) (30)

Denote by119860119860119860(Rtimes119884times119884119883) the set of all such functionsRemark 13 The function 119865 R times 119883 times 119883 997888rarr 119883 given by

119865 (119905 119909 119910) = 119866 (119905 119909 119910) + Φ (119905 119909 119910)= sin( 1

2 + cos 119905 + cosradic2119905) [sin (119909) + 119910]+ 119890minus|119905| [119909 + sin (119910)]

(31)

is asymptotically almost automorphic in 119905 isin R uniformly forall (119909 119910) isin 119870 where119870 is any bounded subset of 119883 times 119883 119883 =1198712[0 120587] and

119866 (119905 119909 119910) = sin( 12 + cos 119905 + cosradic2119905) [sin (119909) + 119910]

isin 119860119860 (R times 119883 times 119883119883) Φ (119905 119909 119910) = 119890minus|119905| [119909 + sin (119910)] isin 1198620 (R times 119883 times 119883119883)

(32)

Next we give some basic definitions and properties ofthe fractional calculus theory which are used further in thispaper

Definition 14 (see [26]) The fractional integral of order 120572 gt 0with the lower limit 1199050 for a function 119891 is defined as

119868120572119891 (119905) = 1Γ (120572) int119905

1199050(119905 minus 119904)120572minus1 119891 (119904) d119904 119905 gt 1199050 120572 gt 0 (33)

provided that the right-hand side is point-wise defined on[1199050infin) where Γ is the Gamma function

Definition 15 (see [26]) Riemann-Liouville derivative oforder 120572 gt 0 with the lower limit 1199050 for a function 119891 [1199050infin) 997888rarr R can be written as

119863120572119905 119891 (119905) = 1Γ (119899 minus 120572)

d119899

d119905119899 int119905

1199050(119905 minus 119904)minus120572 119891 (119904) d119904

119905 gt 1199050 119899 minus 1 lt 120572 lt 119899(34)

The first and maybe the most important property ofRiemann-Liouville fractional derivative is that for 119905 gt 1199050and 120572 gt 0 one has 119863120572119905 (119868120572119891(119905)) = 119891(119905) which meansthat Riemann-Liouville fractional differentiation operator isa left inverse to the Riemann-Liouville fractional integrationoperator of the same order 120572

It is important to define sectorial operator for the defini-tion of mild solution of any fractional abstract equations Solet us nowgive the definitions of sectorial linear operators andtheir associated solution operators

Definition 16 ([74] sectorial operator) A closed and linearoperator 119860 is said to be sectorial of type 120596 and angle 120579 ifthere exist 0 lt 120579 lt 1205872 119872 gt 0 and 120596 isin R such that itsresolvent 120588(119860) exists outside the sector 120596 + 119878120579 fl 120596 + 120582 120582 isinC |arg(minus120582)| lt 120579 and

10038171003817100381710038171003817(120582 minus 119860)minus110038171003817100381710038171003817 le 119872|120582 minus 120596| 120582 notin 120596 + 119878120579 (35)

Sectorial operators are well studied in the literatureusually for the case 120596 = 0 For a recent reference includingseveral examples and properties we refer the reader to [74]Note that an operator 119860 is sectorial of type 120596 if and only if120596119868 minus 119860 is sectorial of type 0

Definition 17 (see [75]) Let119860 be a closed and linear operatorwith domain 119863(119860) defined on a Banach space 119883 We call 119860


the generator of a solution operator if there are 120596 isin R anda strongly continuous function 119878120572 R+ 997888rarr L(119883) such that120582120572 Re120582 gt 120596 sube 120588(119860) and

120582120572minus1 (120582120572 minus 119860)minus1 119909 = intinfin0

119890minus120582119905119878120572 (119905) 119909 d119905Re120582 gt 120596 119909 isin 119883

(36)

In this case 119878120572(119905) is called the solution operator generated by119860Note that if119860 is sectorial of type120596with 0 le 120579 le 120587(1minus1205722)

then 119860 is the generator of a solution operator given by

119878120572 (119905) fl 12120587119894 int120574 119890

minus120582119905120582120572minus1 (120582120572 minus 119860)minus1 d120582 (37)

where 120574 is a suitable path lying outside the sector 120596 + Σ120579 (cf[74])

Very recently Cuesta in [74](Theorem 1) has proved thatif 119860 is a sectorial operator of type 120596 lt 0 for some119872 gt 0 and0 le 120579 lt 120587(1 minus 1205722) then there exists 119862 gt 0 such that

1003817100381710038171003817119878120572 (119905)1003817100381710038171003817L(119883) le 1198621198721 + |120596| 119905120572 for 119905 ge 0 (38)

In the border case 120572 = 1 this is analogous to saying that 119860is the generator of a exponentially stable 1198620-semigroup Themain difference is that in the case 120572 gt 1 the solution family119878120572(119905) decays like 119905minus120572 Cuestarsquos result proves that 119878120572 (119905) is in factintegrable

In the following we present the following compactnesscriterion which is a special case of the general compactnessresult of Theorem 21 in [76]

Lemma 18 (see [76]) A set 119863 sub 1198620(R 119883) is relatively com-pact if

(1) 119863 is equicontinuous(2) lim|119905|997888rarrinfin119909(119905) = 0 uniformly for 119909 isin 119863(3) the set 119863(119905) fl 119909(119905) 119909 isin 119863 is relatively compact in119883 for every 119905 isin R

The following Krasnoselskiirsquos fixed point theorem plays akey role in the proofs of our main results which can be foundin many books

Lemma 19 (see [77]) Let 119880 be a bounded closed and convexsubset of119883 and 1198691 1198692 be maps of119880 into119883 such that 1198691119909+1198692119910 isin119880 for every pair 119909 119910 isin 119880 If 1198691 is a contraction and 1198692 iscompletely continuous then 1198691119909 + 1198692119909 = 119909 has a solution on1198803 Asymptotically Almost AutomorphicMild Solutions

In this section we study the existence of asymptoticallyalmost automorphic mild solutions for the semilinear frac-tional differential equations of the form


where 119860 119863(119860) sub 119883 997888rarr 119883 is a linear densely definedoperator of sectorial type of 120596 lt 0 on a complex Banachspace 119883 119861 119883 997888rarr 119883 is a bounded linear operator and119865 R times 119883 times 119883 997888rarr 119883 and (119905 119909 119910) 997888rarr 119865(119905 119909 119910) is a givenfunction to be specified later The fractional derivative D120572119905 isto be understood in Riemann-Liouville sense

We recall the following definition that will be essential forus

Definition 20 (see [63]) Assume that 119860 generates an inte-grable solution operator 119878120572(119905) A continuous function 119909 R 997888rarr 119883 satisfying the integral equation

119909 (119905) = int119905minusinfin

119878120572 (119905 minus 120590) 119865 (120590 119909 (120590) 119861119909 (120590))d120590 119905 isin R (40)

is called a mild solution on R to (39)

In the proofs of our results we need the followingauxiliary result

Lemma 21 Given 119884(119905) isin 119860119860(R 119883) and 119885(119905) isin 1198620(R 119883) letΦ1 (119905) fl int119905

minusinfin119878120572 (119905 minus 119904) 119884 (119904) 119889119904

Φ2 (119905) fl int119905minusinfin

119878120572 (119905 minus 119904) 119885 (119904) 119889119904119905 isin R

(41)

Then Φ1(119905) isin 119860119860(R 119883)Φ2(119905) isin 1198620(R 119883)Proof Firstly note that

intinfin0

11 + |120596| 119904120572 d119904 = |120596|minus1120572 120587

120572 sin (120587120572) for 1 lt 120572 lt 2 (42)

Then

1003817100381710038171003817Φ1 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817

= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591) 119884 (119905 minus 120591) d12059110038171003817100381710038171003817100381710038171003817

le 119862119872119884infin intinfin0

11 + |120596| 120591120572 d120591

= 119862119872 |120596|minus1120572 120587120572 sin (120587120572) 119884infin

(43)

which implies thatΦ1(119905) is well defined and continuous onRSince119884(119905) isin 119860119860(R 119883) then for any 120576 gt 0 and every sequenceof real numbers 1199041015840119899 there exist a subsequence 119904119899 a function(119905) and 119873 isin N such that

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 lt 120576for each 119899 gt 119873 and every 119904 isin R (44)


Define


119879 (119905 minus s) (119904) d119904 (45)

Then

10038171003817100381710038171003817Φ1 (119905 + 119904119899) minus Φ1 (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905+119904119899

minusinfin119878120572 (119905 + 119904119899 minus 119904) 119884 (119904) d119904

minus int119905minusinfin

119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (119904) 119884 (119905 + 119904119899 minus 119904) d119904

minus int+infin0

119878120572 (119904) 119884 (119905 minus 119904) d11990410038171003817100381710038171003817100381710038171003817le 119862119872intinfin

0

11 + |120596| 119904120572

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 d119904

le 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(46)

for each 119899 gt 119873 and every 119905 isin R This implies that

Φ1 (119905) = lim119899997888rarrinfin

Φ1 (119905 + 119904119899) (47)

is well defined for each 119905 isin RBy a similar argument one can obtain


Φ1 (119905 minus 119904119899) = Φ1 (119905) for each 119905 isin R (48)

ThusΦ1(119905) isin 119860119860(R 119883)Since 119885(119905) isin 1198620(R 119883) one can choose an 1198731 gt 0 such

that 119885(119905) lt 120576 for all 119905 gt 1198731 This enables us to concludethat for all 119905 gt 1198731

1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 le 100381710038171003817100381710038171003817100381710038171003817int1198731

minusinfin119878120572 (119905 minus 119904)119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817

+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198731119878120572 (119905 minus 119904) 119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817

le 119862119872119885infin int1198731minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576119862119872int1199051198731

11 + |120596| (119905 minus 119904)120572 d119904

le 119862119872119885infin|120596| int1198731minusinfin

1(119905 minus 119904)120572 d119904

+ 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

le 119862119872119885infin|120596|1

(120572 minus 1) (119905 minus 1198731)120572minus1

+ 119862119872|120596|minus1120572 120587120576120572 sin (120587120572)

(49)

which implies

lim119905997888rarr+infin

1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (50)

On the other hand from119885(119905) isin 1198620(R 119883) it follows that thereexists an 1198732 gt 0 such that 119885(119905) lt 120576 for all 119905 lt minus1198732This enables us to conclude that for all 119905 lt minus1198732

1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


le int119905minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 119885 (119904) d119904

le 119862119872120576int119905minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(51)

which implies

lim119905997888rarrminusinfin

1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (52)

Now we are in position to state and prove our first mainresult To prove ourmain result let us introduce the followingassumptions(1198671) 119865(119905 119909 119910) = 1198651(119905 119909 119910) + 1198652(119905 119909 119910) isin 119860119860119860(R times 119883 times119883119883) with

1198651 (119905 119909 119910) isin 119860119860 (R times 119883 times 119883119883) 1198652 (119905 119909 119910) isin 1198620 (R times 119883 times 119883119883) (53)

and there exists a constant 119871 gt 0 such that for all 119905 isin R and1199091 1199092 1199101 1199102 isin 11988310038171003817100381710038171198651 (119905 1199091 1199092) minus 1198651 (119905 1199101 1199102)1003817100381710038171003817

le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (54)

(1198672) There exist a function 120573(119905) isin 1198620(RR+) and anondecreasing function Φ R+ 997888rarr R+ such that for all119905 isin R and 119909 119910 isin 119883 with 119909 + 119910 le 119903

10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) Φ (119903)and lim inf119903997888rarr+infin

Φ (119903)119903 = 1205881

(55)

Remark 22 Assuming that 119865(119905 119909 119910) satisfies the assumption(1198671) it is noted that 119865(119905 119909 119910) does not have to meet the


Lipschitz continuity with respect to 119909 and 119910 Such class ofasymptotically almost automorphic functions 119865(119905 119909 119910) aremore complicated than those with Lipschitz continuity withrespect to 119909 and 119910 and little is known about them

Let 120573(119905) be the function involved in assumption (1198672)Define

120590 (119905) fl int119905minusinfin

120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)

Lemma 23 120590(119905) isin 1198620(RR+)Proof Since 120573(119905) isin 1198620(RR+) one can choose a 1198791 gt 0 suchthat 120573(119905) lt 120576 for all 119905 gt 1198791 This enables us to conclude thatfor all 119905 gt 1198791

120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|

1(120572 minus 1) (119905 minus 1198791)120572minus1 +

|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)

On the other hand from 120573(119905) isin 1198620(RR+) it follows thatthere exists a 1198792 gt 0 such that 120573(119905) lt 120576 for all 119905 lt minus1198792This enables us to conclude that for all 119905 lt minus1198792

120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

10038171003817100381710038171003817100381710038171003817le 120576int119905minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)

Theorem 24 Assume that 119860 is sectorial of type 120596 lt 0 Let119865 R times 119883 times 119883 997888rarr 119883 satisfy the hypotheses (1198671) and (1198672)Put 1205882 fl sup119905isinR120590(119905) Then (39) has at least one asymptoticallyalmost automorphic mild solution provided that

119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)

Proof The proof is divided into the following five steps

Step 1 Define a mapping Λ on 119860119860(R 119883) by

(ΛV) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)

and prove Λ has a unique fixed point V(119905) isin 119860119860(R 119883)Firstly since the function 119904 997888rarr 1198651(119904 V(119904) 119861V(119904)) is

bounded inR and

[ΛV] (119905) le int119905minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)

this implies that (ΛV)(119905) exists Moreover from 1198651(119905 119909 119910) isin119860119860(R times 119883 times 119883119883) satisfying (54) together with Lemma 6and Remark 7 it follows that

1198651 (sdot V (sdot) 119861V (sdot)) isin 119860119860 (R 119883)for every V (sdot) isin 119860119860 (R 119883) (64)

This together with Lemma 21 implies that Λ is well definedand maps 119860119860(R 119883) into itself

In the sequel we verify that Λ is continuousLet V119899(119905) V(119905) be in 119860119860(R 119883)with V119899(119905) 997888rarr V(119905) as 119899 997888rarrinfin then one has

1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905 minus 119904)

sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 le 119871int119905minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817sdot [1003817100381710038171003817V119899 (119904) minus V (119904)1003817100381710038171003817 + 1003817100381710038171003817119861V119899 (119904) minus 119861V (119904)1003817100381710038171003817]d119904le 119862119872119871int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)

minus V (119904)1003817100381710038171003817 d119904 le 119862119872119871 (1 + 119861L(119883)) 1003817100381710038171003817V119899 minus V1003817100381710038171003817infin


sdot int119905minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1003817100381710038171003817V119899 minus V1003817100381710038171003817infin

(65)

Therefore as 119899 997888rarr infin and ΛV119899 997888rarr ΛV hence Λ iscontinuous

Next we prove that Λ is a contraction on 119860119860(R 119883) andhas a unique fixed point V(119905) isin 119860119860(R 119883)

In fact let V1(119905) V2(119905) be in 119860119860(R 119883) and similar to theabove proof of the continuity of Λ one has

1003817100381710038171003817[ΛV1] (119905) minus [ΛV2] (119905)1003817100381710038171003817le 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 1003817100381710038171003817V1 minus V21003817100381710038171003817infin (66)

which implies1003817100381710038171003817[ΛV1] (119905) minus [ΛV2] (119905)1003817100381710038171003817infin

le 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1003817100381710038171003817V1 minus V2

1003817100381710038171003817infin (67)

Together with (61) this proves that Λ is a contraction on119860119860(R 119883) Thus Banachrsquos fixed point theorem implies that Λhas a unique fixed point V(119905) isin 119860119860(R 119883)Step 2 Set

Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)

For the above V(119905) define Γ fl Γ1 + Γ2 on 1198620(R 119883) as(Γ1120596) (119905) = int119905


sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)

and prove that Γ maps Ω1198960 into itself where 1198960 is a givenconstant

Firstly from (54) it follows that for all 119904 isin R and 120596(119904) isin119883 10038171003817100381710038171198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 [120596 (119904) + 119861120596 (119904)]le 119871 (1 + 119861L(119883)) 120596 (119904)

(70)

which implies that

1198651 (sdot V (sdot) + 120596 (sdot) 119861 (V (sdot) + 120596 (sdot))) minus 1198651 (sdot V (sdot) 119861V (sdot))isin 1198620 (R 119883) for every 120596 (sdot) isin 1198620 (R 119883) (71)

According to (55) one has

10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)

for all 119904 isin R and 120596(119904) isin 119883 with 120596(119904) le 119903 then1198652 (sdot V (sdot) + 120596 (sdot) 119861 (V (sdot) + 120596 (sdot))) isin 1198620 (R 119883)

as 120573 (sdot) isin 1198620 (RR+) (73)

Those together with Lemma 21 yield that Γ is well definedand maps 1198620(R 119883) into itself

On the other hand in view of (55) and (61) it is notdifficult to see that there exists a constant 1198960 gt 0 such that

119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)

This enables us to conclude that for any 119905 isin R and 1205961(119905)1205962(119905) isin Ω1198960 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 10038171003817100381710038171003817100381710038171003817int

119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904)+ 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904 + int119905

minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817sdot 10038171003817100381710038171198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817

+ V (119904) + 119861V (119904)) d119904 le 119862119872119871 (1 + 119861L(119883))sdot 100381710038171003817100381712059611003817100381710038171003817infin int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1

+ 119861L(119883)) (100381710038171003817100381712059621003817100381710038171003817infin + V (119904)infin))= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 120596infin+ 1198621198721205882Φ ((1 + 119861L(119883)) (100381710038171003817100381712059621003817100381710038171003817infin + V (119904)infin))le 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)

which implies that (Γ11205961)(119905) + (Γ21205962)(119905) isin Ω1198960 Thus Γ mapsΩ1198960 into itselfStep 3 Show that Γ1 is a contraction onΩ1198960

In fact for any 1205961(119905) 1205962(119905) isin Ω1198960 and 119905 isin R from (54) itfollows that

1003817100381710038171003817[1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]minus [1198651 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]1003817100381710038171003817 le 119871 [10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904) minus 1198611205962 (119904)1003817100381710038171003817] le 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)minus 1205962 (119904)1003817100381710038171003817

(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905 minus 119904) [(1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904))) minus 1198651 (119904 V (119904) 119861V (119904)))

minus (1198651 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) minus 1198651 (119904 V (119904) 119861V (119904)))]d11990410038171003817100381710038171003817100381710038171003817 le 119871int119905

minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)

minus 1205962 (119904)1003817100381710038171003817 d119904 le 119862119872119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin int119905minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817infinle 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)

Thus in view of (61) one obtains the conclusion

Step 4 Show that Γ2 is completely continuous onΩ1198960 Given 120576 gt 0 Let 120596119896+infin119896=1 sub Ω1198960 with 120596119896 997888rarr 1205960 in1198620(R 119883) as 119896 997888rarr +infin Since 120590(119905) isin 1198620(RR+) one may

choose a 1199051 gt 0 big enough such that for all 119905 ge 1199051Φ((1 + 119861L(119883)) (1198960 + Vinfin)) 120590 (119905) lt 120576

3119862119872 (79)

Also in view of (1198671) we have1198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)

for all 119904 isin (minusinfin 1199051] as 119896 997888rarr +infin and

10038171003817100381710038171198652 (sdot V (sdot) + 120596119896 (sdot) 119861 (V (sdot) + 120596119896 (sdot)))minus 1198652 (sdot V (sdot) + 1205960 (sdot) 119861 (V (sdot) + 1205960 (sdot)))1003817100381710038171003817le 2Φ ((1 + 119861L(119883)) (1198960 + Vinfin)) 120573 (sdot)isin 1198711 (minusinfin 1199051]

(81)

Hence by the Lebesgue dominated convergence theorem wededuce that there exists an119873 gt 0 such that

119862119872int1199051minusinfin

11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)


whenever 119896 ge 119873 Thus

10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905 minus 119904) 1198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))d119904 minus int119905minusinfin

119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 + 2119862119872Φ((1+ 119861L(119883)) (1198960 + Vinfin)) int

max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 + 2119862119872Φ((1+ 119861L(119883)) (1198960 + Vinfin)) 120590 (119905) le 120576

3 + 21205763 = 120576

(83)

whenever 119896 ge 119873 Accordingly Γ2 is continuous onΩ1198960 In the sequel we consider the compactness of Γ2Set 119861119903(119883) for the closed ball with center at 0 and radius 119903

in 119883 119881 = Γ2(Ω1198960) and 119911(119905) = Γ2(119906(119905)) for 119906(119905) isin Ω1198960 Firstfor all 120596(119905) isin Ω1198960 and 119905 isin R

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872120590 (119905)Φ ((1 + 119861L(119883)) (1198960 + Vinfin))

(84)

and in view of 120590(119905) isin 1198620(RR+) which follows fromLemma 23 one concludes that


(Γ2120596) (119905) = 0 uniformly for 120596 (119905) isin Ω1198960 (85)

As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)

sdot 1198652 (119905 minus 120591 V (119905 minus 120591)+ 120596 (119905 minus 120591) 119861 (V (119905 minus 120591) + 120596 (119905 minus 120591))) d12059110038171003817100381710038171003817100381710038171003817

(86)

Hence given 1205760 gt 0 one can choose a 120585 gt 0 such that

10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)

+ 120596 (119905 minus 120591) 119861 (V (119905 minus 120591) + 120596 (119905 minus 120591))) d12059110038171003817100381710038171003817100381710038171003817lt 1205760

(87)

Thus we get

119911 (119905) isin 120585119888 (119878120572 (120591) 1198652 (120582 V (120582) + 120596 (120582) 119861 (V (120582) + 120596 (120582))) 0 le 120591 le 120585 119905 minus 120585 le 120582 le 120585 120596infin le 119903) + 1198611205760 (119883) (88)

where 119888(119870) denotes the convex hull of 119870 Using that 119878120572(sdot) isstrongly continuous we infer that

119870 = 119878120572 (120591) 1198652 (120582 V (120582) + 120596 (120582) 119861 (V (120582) + 120596 (120582))) 0le 120591 le 120585 119905 minus 120585 le 120582 le 120585 120596infin le 119903 (89)

is a relatively compact set and 119881 sub 120585119888(119870) + 1198611205760(119883) whichimplies that 119881 is a relatively compact subset of 119883

Next we verify the equicontinuity of the set (Γ2120596)(119905) 120596(119905) isin Ω1198960Let 119896 gt 0 be small enough and 1199051 1199052 isin R and 120596(119905) isin Ω1198960

Then by (55) we have

10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904

+ int1199051minus119896minusinfin

1003817100381710038171003817[119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)]sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904+ int11990511199051minus119896

1003817100381710038171003817[119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)]sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872Φ((1 + 119861L(119883)) (1198960 + Vinfin))sdot int11990521199051

120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960

+ Vinfin)) sup119904isin[minusinfin1199051minus119896]

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)1003817100381710038171003817sdot int1199051minus119896minusinfin

120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960


+ Vinfin)) int1199051

1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0

as 1199052 minus 1199051 997888rarr 0 119896 997888rarr 0(90)

which implies the equicontinuity of the set (Γ2120596)(119905) 120596(119905) isinΩ1198960Now an application of Lemma 18 justifies the compact-

ness of Γ2Step 5 Show that (39) has at least one asymptotically almostautomorphic mild solution

Firstly the complete continuity of Γ2 together with theresults of Steps 2 and 3 as well as Lemma 19 yields that Γhas at least one fixed point 120596(119905) isin Ω1198960 furthermore 120596(119905) isin1198620(R 119883)

Then consider the following coupled system of integralequations

V (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin

119878120572 (119905 minus 119904)sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904 + int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)

From the result of Step 1 together with the above fixed point120596(119905) isin 1198620(R 119883) it follows that(V (119905) 120596 (119905)) isin 119860119860 (R 119883) times 1198620 (R 119883) (92)

is a solution to system (91) Thus

119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)

and it is a solution to the integral equation

119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)

that is 119909(119905) is an asymptotically almost automorphic mildsolution to (39)

Taking 119860 = minus120588120572119868 with 120588 gt 0 in (39) the above theoremgives the following corollary

Corollary 25 Let 119865 R times 119883 times 119883 997888rarr 119883 satisfy (1198671) and(1198672) Put 1205882 fl sup119905isinR120590(119905) Then (39) admits at least oneasymptotically almost automorphic mild solution whenever

119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)

Remark 26 It is interesting to note that the function 120572 997888rarr120572sin(120587120572)120588120587 is increasing from 0 to 2120588120587 in the interval 1 lt120572 lt 2 Therefore with respect to condition (61) the class ofadmissible terms 1198651(119905 119909(119905) 119861119909(119905)) is the best in the case 120572 = 2and the worst in the case 120572 = 1

Theorem 24 can be extended to the case of 1198651(119905 119909 119910)being locally Lipschitz continuous with respect to 119909 and 119910where we have the following result(11986710158401) 119865(119905 119909 119910) = 1198651(119905 119909 119910) + 1198652(119905 119909 119910) isin 119860119860119860(R times 119883 times119883119883) with


and for all 1199091 1199092 1199101 1199102 isin 119883 119905 isin R10038171003817100381710038171198651 (119905 1199091 1199092) minus 1198651 (119905 1199101 1199102)1003817100381710038171003817

le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)

where 119871(119905) is a function on R

Theorem 27 Assume that 119860 is sectorial of type 120596 lt 0 Let119865 R times 119883 times 119883 997888rarr 119883 satisfy the hypotheses (11986710158401) and (1198672)with 119871(119905) isin 119861119862(RR+) Put 1205882 fl sup119905isinR120590(119905) Let 119871 =sup119905isinR int119905+1119905 119871(119904)d119904 Then (39) has at least one asymptoticallyalmost automorphic mild solution provided that

119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)


Step 1 Define a mapping Λ on 119860119860(R 119883) by (62) and provethat Λ has a unique fixed point V(119905) isin 119860119860(R 119883)

Firstly similar to the proof in Step 1 of Theorem 24 wecan prove that (ΛV)(119905) exists Moreover from 1198651(119905 119909 119910) isin119860119860(R times 119883 times 119883119883) satisfying (97) together with Lemma 6and Remark 8 it follows that




1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 le int119905minusinfin

119871 (119904) 1003817100381710038171003817119878120572 (119905minus 119904)1003817100381710038171003817 [1003817100381710038171003817V119899 (119904) minus V (119904)1003817100381710038171003817 + 1003817100381710038171003817119861V119899 (119904) minus 119861V (119904)1003817100381710038171003817] d119904



119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)

minus V (119904)1003817100381710038171003817 d119904 le 119862119872(1 + 119861L(119883))sdot (+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) 1003817100381710038171003817V119899 minus V1003817100381710038171003817infin

le 119862119872(1 + 119861L(119883))sdot (+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) 1003817100381710038171003817V119899 minus V1003817100381710038171003817infin

le 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1003817100381710038171003817V119899 minus V1003817100381710038171003817infin (100)



In fact for V1(119905) V2(119905) in 119860119860(R 119883) similar to the aboveproof of the continuity of Λ one has

1003817100381710038171003817(ΛV1) (119905) minus (ΛV2) (119905)1003817100381710038171003817le 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1003817100381710038171003817V1 minus V2

1003817100381710038171003817infin (101)

which implies that1003817100381710038171003817(ΛV1) (119905) minus (ΛV2) (119905)1003817100381710038171003817infin


Hence by (98) together with the contraction principleΛ hasa unique fixed point V(119905) isin 119860119860(R 119883)Step 2 Set

Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)

For the above V(119905) define Γ fl Γ1+Γ2 on1198620(R 119883) as (69) andprove that ΓmapsΩ1198960 into itself where 1198960 is a given constant

Firstly from (97) it follows that for all 119904 isin R 120596(119904) isin 11988310038171003817100381710038171198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))

minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)

which together with 119871(119904) isin 119861119862(RR+) implies that

1198651 (sdot V (sdot) + 120596 (sdot) 119861 (V (sdot) + 120596 (sdot)))minus 1198651 (sdot V (sdot) 119861V (sdot)) isin 1198620 (R 119883)

for every 120596 (sdot) isin 1198620 (R 119883) (105)

According to (55) one has10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))

sdot 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862119872int119905minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1


+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin int119905minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904

+ 119862119872120590 (119905) Φ((1 + 119861L(119883)) (100381710038171003817100381712059621003817100381710038171003817infin+ sup119904isinR

V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))

sdot (100381710038171003817100381712059621003817100381710038171003817infin + sup119904isinR

V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)



1003817100381710038171003817[1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]minus [1198651 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]1003817100381710038171003817 le 119871 (119904) [10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904) minus 1198611205962 (119904)1003817100381710038171003817] le 119871 (119904) (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)minus 1205962 (119904)1003817100381710038171003817

(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


minus (1198651 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) minus 1198651 (119904 V (119904) 119861V (119904)))] d11990410038171003817100381710038171003817100381710038171003817 le int119905minusinfin

119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)

minus 1205962 (119904)1003817100381710038171003817d119904 le 119862119872int119905minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin le 119862119872(+infinsum

119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin

(111)

which implies that

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817infinle 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin

(112)


Step 4 Show that Γ2 is completely continuous on Ω1198960 The proof is similar to the proof in Step 4 of Theorem 24

Step 5 Show that (39) has at least one asymptotically almostautomorphic mild solution


The proof is similar to the proof in Step 5 of Theorem 24

Taking 119860 = minus120588120572119868 with 120588 gt 0 in (39) Theorem 27 givesthe following corollary

Corollary 28 Let 119865 R times 119883 times 119883 997888rarr 119883 satisfy (11986710158401) and(1198672) with 119871(119905) isin 119861119862(RR+) Put 1205882 fl sup119905isinR120590(119905) Let 119871 =sup119905isinR int119905+1119905 119871(119904)d119904Then (39) admits at least one asymptotical-ly almost automorphic mild solution whenever

119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)

Now we consider a more general case of equations intro-ducing a new class of functions 119871(119905) We have the followingresult

(11986710158402) There exists a function 120573(119905) isin 1198620(RR+) such thatfor all 119905 isin R and 119909 119910 isin 119883

10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)

Theorem 29 Assume that 119860 is sectorial of type 120596 lt 0Let 119865 R times 119883 times 119883 997888rarr 119883 satisfy the hypotheses (11986710158401)and (11986710158402) with 119871(119905) isin 119861119862(RR+) Moreover the integralint119905minusinfinmax119871(119904) 120573(119904)d119904 exists for all 119905 isin R Then (39) has atleast one asymptotically almost automorphic mild solution



Firstly similar to the proof in Step 1 ofTheorem 27we canprove that Λ is well defined and maps 119860119875(R 119883) into itselfmoreover Λ is continuous


In fact for V1(119905) V2(119905) is in 119860119860(R 119883) and defines a newnorm

|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)

where 120583(119905) fl 119890minus119896int119905minusinfin max119871(119904)120573(119904)d119904 and 119896 is a fixed positiveconstant Let 119862120572 fl sup119905isinR119878120572(119905) then we have

120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905

minusinfin120583 (119905) 119871 (120590) [1003817100381710038171003817V1 (120590) minus V2 (120590)1003817100381710038171003817 + 1003817100381710038171003817119861V1 (120590)

minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)

sdot 120583 (120590)minus1 (1 + 119861L(119883)) 1003817100381710038171003817V1 (120590) minus V2 (120590)1003817100381710038171003817 d120590le 119862120572 (1 + 119861L(119883)) 10038161003816100381610038161003817100381710038171003817V1 minus V2

10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590

= 119862120572 (1 + 119861L(119883))119896 (1 minus 119890minus119896int119905minusinfin 119871(120591)d120591) 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that

10038161003816100381610038161003817100381710038171003817Λ119909 (119905) minus Λ119910 (119905)10038161003816100381610038161003817100381710038171003817 le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817119909 minus 11991010038161003816100381610038161003817100381710038171003817 (117)

Hence Λ has a unique fixed point 119909 isin 119860119860(R 119883) when 119896 isgreater than 119862120572(1 + 119861L(119883))Step 2 Set Θ119903 fl 120596(119905) isin 1198620(R 119883) |120596(119905)| le 119903 For theabove V(119905) define Γ fl Γ1 + Γ2 on 1198620(R 119883) as (69) and provethat Γmaps Θ1198960 into itself where 1198960 is a given constant


minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)



According to (114) one has10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)

sdot (120596 (119904) + 119861120596 (119904) + V (119904) + 119861V (119904)) le 120573 (119904)sdot ((1 + 119861L(119883)) 120596 (119904) + (1 + 119861L(119883)) V (119904))le 120573 (119904) ((1 + 119861L(119883)) [120596 (119904) + V (119904)])

(120)



as 120573 (sdot) isin 1198620 (RR+) (121)


On the other hand it is not difficult to see that there existsa constant 1198960 gt 0 such that

2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)

when 119896 is large enough This enables us to conclude that forany 119905 isin R and 1205961(119905) 1205962(119905) isin Θ1198960

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905

minusinfin120583 (119905) 120583 (119904) 119871 (119904) 120583 (119904)minus1

sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)

sdot 120573 (119904) 120583 (119904)minus1 (1 + 119861L(119883)) (100381710038171003817100381712059621003817100381710038171003817 + V (119904)) d119904le 119862120572 (1 + 119861L(119883)) 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905

minusinfin120583 (119905) 120583 (119904)minus1 119871 (119904) d119904

+ 119862120572 (1 + 119861L(119883)) (10038161003816100381610038161003817100381710038171003817120596210038161003816100381610038161003817100381710038171003817 + |V (119904)| ) int119905minusinfin

120583 (119905)

sdot 120583 (119904)minus1 120573 (119904) d119904 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817sdot int119905minusinfin

119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1

+ 119861L(119883)) (10038161003816100381610038161003817100381710038171003817120596210038161003816100381610038161003817100381710038171003817 + |V (119904)| )sdot int119905minusinfin

119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817sdot int119905minusinfin

119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))

sdot (10038161003816100381610038161003817100381710038171003817120596210038161003816100381610038161003817100381710038171003817 + |V (119904)| ) int119905minusinfin

119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904

+ 119862120572 (1 + 119861L(119883)) (10038161003816100381610038161003817100381710038171003817120596210038161003816100381610038161003817100381710038171003817 + |V (119904)| )sdot int119905minusinfin

dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904

= 119862120572 (1 + 119861L(119883))119896 (1 minus 119890minus119896int119905minusinfin 119871(120591)d120591) 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817+ 119862120572 (1 + 119861L(119883))119896 (1 minus 119890minus119896int119905minusinfin 120573(120591)d120591) (10038161003816100381610038161003817100381710038171003817120596210038161003816100381610038161003817100381710038171003817+ |V (119904)| ) le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 + 119862120572119896 (1+ 119861L(119883)) (10038161003816100381610038161003817100381710038171003817120596210038161003816100381610038161003817100381710038171003817 + |V (119904)| ) le 1198960

(123)

which implies that (Γ11205961)(119905) + (Γ21205962)(119905) isin Θ1198960 Thus Γ mapsΘ1198960 into itselfStep 3 Show that Γ1 is a contraction on Θ1198960

In fact for any 1205961(119905) 1205962(119905) isin Θ1198960 and 119905 isin R from (97) itfollows that

1003817100381710038171003817[1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]minus [1198651 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]1003817100381710038171003817 le 119871 (119904) [10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904) minus 1198611205962 (119904)1003817100381710038171003817] le 119871 (119904) (1 + 119861L(119883))sdot 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817

(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


minus (1198651 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) minus 1198651 (119904 V (119904) 119861V (119904)))] d11990410038171003817100381710038171003817100381710038171003817 le 119862120572 int

119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590) 120583 (120590)minus1 (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590) minus 1205962 (120590)1003817100381710038171003817 d120590 le 119862120572 (1 + 119861L(119883)) 100381610038161003816100381610038171003817100381710038171205961

minus 120596210038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 int119905minusinfin

119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 int119905minusinfin

119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd120590 (119890119896int120590119905 119871(120591)d120591) d120590

= 119862120572 (1 + 119861L(119883))119896 (1 minus 119890minus119896int119905minusinfin 119871(120591)d120591) 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (125)

which implies1003816100381610038161003816100381610038171003817100381710038171003817 (Γ11205961) (119905) minus (Γ11205962) (119905)1003816100381610038161003816100381610038171003817100381710038171003817

le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)

Thus when 119896 is greater than 119862120572(1 + 119861L(119883)) one obtains theconclusion

Step 4 Show that Γ2 is completely continuous on Θ1198960 Given 120576 gt 0 Let 120596119899+infin119899=1 sub Θ1198960 with 120596119899 997888rarr 1205960 in Θ1198960 as119899 997888rarr +infin Since 120590(119905) isin 1198620(RR+) one may choose a 1199051 gt 0

big enough such that for all 119905 ge 1199051(1 + 119861L(119883)) (1198960 + |V| ) 120590 (119905) lt 120576

3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)


120583 (sdot) 10038171003817100381710038171198652 (sdot V (sdot) + 120596119899 (sdot) 119861 (V (sdot) + 120596119899 (sdot)))minus 1198652 (sdot V (sdot) + 1205960 (sdot) 119861V (sdot) + 1205960 (sdot)) )1003817100381710038171003817 le 120583 (sdot)sdot 120573 (sdot) (1003817100381710038171003817120596119899 (sdot)1003817100381710038171003817 + V (sdot) + 1003817100381710038171003817119861120596119899 (sdot)1003817100381710038171003817 + 119861V (sdot)+ 10038171003817100381710038171205960 (sdot)1003817100381710038171003817 + V (sdot) + 10038171003817100381710038171198611205960 (sdot)1003817100381710038171003817 + 119861V (sdot)) le 120573 (sdot)sdot (1003816100381610038161003816100381710038171003817100381712059611989910038161003816100381610038161003817100381710038171003817 + |V| + 1003816100381610038161003816100381710038171003817100381711986112059611989910038161003816100381610038161003817100381710038171003817 + |119861V| + 10038161003816100381610038161003817100381710038171003817120596010038161003816100381610038161003817100381710038171003817+ |V| + 10038161003816100381610038161003817100381710038171003817119861120596010038161003816100381610038161003817100381710038171003817 + |119861V| ) le 120573 (sdot)sdot (2 (1 + 119861L(119883)) (1198960 + |V| )) isin 1198711 (minusinfin 1199051]

(129)

Hence by the Lebesgue dominated convergence theorem wededuce that there exists an 119873 gt 0 such that


11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) d119904minus int119905minusinfin

S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817d119904+ 119862119872(2 (1 + 119861L(119883)) (1198960 + |V| ))sdot intmax1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817d119904+ 119862119872120590 (119905) (2 (1 + 119861L(119883)) (1198960 + |V| )) le 120576

3+ 2120576

3 = 120576

(131)

whenever 119896 ge 119873 Accordingly Γ2 is continuous on Θ1198960 In the sequel we consider the compactness of Γ2


Set 119861119903(119883) for the closed ball with center at 0 and radius 119903in 119883 119881 = Γ2(Θ1198960) and 119911(119905) = Γ2(119906(119905)) for 119906(119905) isin Θ1198960 Firstfor all 120596(119905) isin Θ1198960 and 119905 isin R

120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)

in view of 120590(119905) isin 1198620(RR+) which follows from Lemma 23one concludes that


(Γ2120596) (119905) = 0 uniformly for 120596 (119905) isin Θ1198960 (133)

as

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904 = int+infin

0119878120572 (120591)


(134)

Hence for given 1205760 gt 0 one can choose a 120585 gt 0 such that

1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591) + 120596 (119905 minus 120591) 119861 (V (119905 minus 120591) + 120596 (119905 minus 120591))) d12059110038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get

119911 (119905) isin 120585119888 (119878120572 (120591) 1198652 (120582 V (120582) + 120596 (120582) 119861 (V (120582) + 120596 (120582))) 0 le 120591 le 120585 119905 minus 120585 le 120582 le 120585 |120596| le 1198960) + 1198611205760 (Θ1198960) (136)

where 119888(119870) denotes the convex hull of 119870 Using the fact that119878120572(sdot) is strongly continuous we infer that119870 = 119878120572 (120591) 1198652 (120582 V (120582) + 120596 (120582) 119861 (V (120582) + 120596 (120582))) 0

le 120591 le 120585 119905 minus 120585 le 120582 le 120585 |120596| le 1198960 (137)

is a relatively compact set and 119881 sub 120585119888(119870) + 1198611205760(Θ1198960) whichimplies that 119881 is a relatively compact subset of Θ1198960

Next we verify the equicontinuity of the set (Γ2120596)(119905) 120596(119905) isin Θ1198960 given 1205761 gt 0 In view of (114) together with thecontinuity of 119878120572(119905)119905gt0 there exists an 120578 gt 0 such that for all120596(119905) isin Ω1198960 and 1199052 ge 1199051 with 1199052 minus 1199051 lt 120578

int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578

10038161003816100381610038161003817100381710038171003817 [119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)] 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614 (138)

Also one can choose a 119896 gt 0 such that

int1199051minus1205781199051minus119896

10038161003816100381610038161003817100381710038171003817 [119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)] 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614(1 + 119861L(119883)) (1198960 + |V| ) sup

119904isin[minusinfin1199051minus119896]

1003817100381710038171003817119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)1003817100381710038171003817 int1199051minus119896

minusinfin120573 (119904) d119904 lt 12057614

(139)


which implies that for all 120596(119905) isin Ω1198960 and 1199052 ge 1199051

int1199051minus119896minusinfin

10038161003816100381610038161003817100381710038171003817 [119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)] 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

le (1 + 119861L(119883)) (1198960 + |V| ) sup119904isin[minusinfin1199051minus119896]


minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052

minusinfin119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904 minus int1199051

minusinfin119878120572 (1199051 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896

10038161003816100381610038161003817100381710038171003817 [119878120572 (1199052 minus 119904) minus 119878120572 (1199051 minus 119904)] 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 1205761

(141)

which implies the equicontinuity of the set (Γ2120596)(119905) 120596(119905) isinΘ1198960Now an application of Lemma 18 justifies the compact-




Corollary 30 Let 119865 R times 119883 times 119883 997888rarr 119883 satisfy (11986710158401)and (11986710158402) with 119871(119905) isin 119861119862(RR+) Moreover the integralint119905minusinfinmax119871(119904) 120573(119904)d119904 exists for all 119905 isin R Then (39) has atleast one asymptotically almost automorphic mild solution

4 Applications

In this section we give an example to illustrate the aboveresults

Consider the following fractional relaxation-oscillationequation

120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)

times [sin 119906 (119905 119909) + 119906 (119905 119909)]+ ]119890minus|119905| [119906 (119905 119909) + sin 119906 (119905 119909)]]

119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)

where 119886(119905) isin 119861119862(RR+) is a function and 119901 120583 and ] arepositive constants

Take 119883 = 1198712([0 120587]) and define the operator 119860 by

119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)

It is well known that 119861119906 = 11990610158401015840 is self-adjoint with compactresolvent and is the infinitesimal generator of an analyticsemigroup on119883 Hence119901119868minus119861 is sectorial of type120596 = minus119901 lt 0Let

1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1

2 + cos 119905 + cosradic2119905) [sin 119909 (120585) + 119910 (120585)] 1198652 (119905 119909 (120585) 119910 (120585)) fl ]119890minus|119905| [119909 (120585) + sin 119910 (120585)]

(145)


Then it is easy to verify that 1198651 1198652 R times 119883 times 119883 997888rarr 119883 arecontinuous and 1198651(119905 119909 119910) isin 119860119860(R times 119883 times 119883119883) satisfying

10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162

sdot 1003816100381610038161003816[sin 1199091 (119904) + 1199101 (119904)] minus [sin 1199092 (119904) + 1199102 (119904)]1003816100381610038161003816 d119904le 12058321198862 (119905) 10038161003816100381610038161003816100381610038161003816sin( 1

2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172

le 120583119886 (119905) (10038171003817100381710038171199091 minus 119909210038171003817100381710038172 + 10038171003817100381710038171199101 minus 119910210038171003817100381710038172)forall119905 isin R 1199091 1199101 1199092 1199102 isin 119883

(147)

furthermore10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172

le 120583 119886infin (10038171003817100381710038171199091 minus 119909210038171003817100381710038172 + 10038171003817100381710038171199101 minus 119910210038171003817100381710038172)forall119905 isin R 1199091 1199101 1199092 1199102 isin 119883

(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)

forall119905 isin R 119909 119910 isin 119883 (150)

which implies 1198652(119905 119909 119910) isin 1198620(R times 119883 times 119883119883) Furthermore

119865 (119905 119909 119910) = 1198651 (119905 119909 119910) + 1198652 (119905 119909 119910)isin 119860119860119860 (R times 119883 times 119883119883) (151)

Thus (142) can be reformulated as the abstract problem (39)and the assumptions (1198671) and (1198672) hold with

119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)

the assumption (11986710158401) holds with 119871(119905) = 120583119886(119905) and theassumption (11986710158402) holds

In consequence the fractional relaxation-oscillationequation (142) has at least one asymptotically almost auto-morphic mild solutions if either

120583119862119872119886infin 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (153)

(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)

(Theorem 27) where 119886 = sup119905isinR int119905+1119905 119886(119904)d119904 or the integralint119905minusinfin

max 120583119886 (119904) ]119890minus|119905| d119904 (155)

exists for all 119905 isin R (Theorem 29)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the NNSF of China (no11561009) and (no 41665006) the Guangdong ProvinceNatural Science Foundation (no 2015A030313896) the Char-acteristic Innovation Project (Natural Science) of GuangdongProvince (no 2016KTSCX094) the Science and Technol-ogy Program Project of Guangzhou (no 201707010230)and the Guangxi Province Natural Science Foundation (no2016GXNSFAA380240)

References

[1] S Bochner ldquoContinuous mappings of almost automorphic andalmost periodic functionsrdquoProceedings of theNational Acadamyof Sciences of the United States of America vol 52 pp 907ndash9101964

[2] S Bochner ldquoUniform convergence of monotone sequences offunctionsrdquo Proceedings of the National Acadamy of Sciences ofthe United States of America vol 47 pp 582ndash585 1961

[3] S Bochner ldquoA new approach to almost periodicityrdquo Proceedingsof the National Acadamy of Sciences of the United States ofAmerica vol 48 pp 2039ndash2043 1962

[4] S Bochner and J Von Neumann ldquoOn compact solutions ofoperational-differential equationsrdquo I Annals of MathematicsSecond Series vol 36 no 1 pp 255ndash291 1935

[5] G M NrsquoGuerekata Almost Automorphic Functions and AlmostPeriodic Functions in Abstract Spaces Kluwer AcademicPle-num Publishers New York London Moscow 2001

[6] G M NrsquoGuerekata Topics in Almost Automorphy SpringerNew York NY USA 2005

[7] WAVeech ldquoAlmost automorphic functionsrdquoProceedings of theNational Acadamy of Sciences of the United States of Americavol 49 pp 462ndash464 1963

[8] J Campos and M Tarallo ldquoAlmost automorphic linear dynam-ics by Favard theoryrdquo Journal of Differential Equations vol 256no 4 pp 1350ndash1367 2014


[9] T Caraballo and D Cheban ldquoAlmost periodic and almost auto-morphic solutions of linear differentialdifference equationswithout Favardrsquos separation conditionrdquo I Journal of DifferentialEquations vol 246 no 1 pp 108ndash128 2009

[10] L Mahto and S Abbas ldquoPC-almost automorphic solutionof impulsive fractional differential equationsrdquo MediterraneanJournal of Mathematics vol 12 no 3 pp 771ndash790 2015

[11] D Araya and C Lizama ldquoAlmost automorphic mild solutionsto fractional differential equationsrdquo Nonlinear Analysis TheoryMethods amp Applications An International MultidisciplinaryJournal vol 69 no 11 pp 3692ndash3705 2008

[12] G M Mophou and G M NrsquoGuerekata ldquoOn some classes ofalmost automorphic functions and applications to fractionaldifferential equationsrdquo Computers ampMathematics with Applica-tions An International Journal vol 59 no 3 pp 1310ndash1317 2010

[13] L Abadias and C Lizama ldquoAlmost automorphic mild solutionsto fractional partial difference-differential equationsrdquo Applica-ble Analysis An International Journal vol 95 no 6 pp 1347ndash1369 2016

[14] M Fu and Z Liu ldquoSquare-mean almost automorphic solutionsfor some stochastic differential equationsrdquo Proceedings of theAmerican Mathematical Society vol 138 no 10 pp 3689ndash37012010

[15] J Cao Q Yang and Z Huang ldquoExistence and exponentialstability of almost automorphic mild solutions for stochasticfunctional differential equationsrdquo Stochastics An InternationalJournal of Probability and Stochastic Processes vol 83 no 3 pp259ndash275 2011

[16] Z Liu andK Sun ldquoAlmost automorphic solutions for stochasticdifferential equations driven by Levy noiserdquo Journal of Func-tional Analysis vol 266 no 3 pp 1115ndash1149 2014

[17] G M Nrsquoguerekata ldquoComments on almost automorphic andalmost periodic functions in Banach spacesrdquo Far East Journal ofMathematical Sciences (FJMS) vol 17 no 3 pp 337ndash344 2005

[18] G M NrsquoGuerekata ldquoSur les solutions presqu automorphesdrsquoequations differentielles abstraitesrdquo Annales des SciencesMathematiques du Quebec no 1 pp 69ndash79 1981

[19] D Bugajewski and G M NrsquoGuerekata ldquoOn the topologicalstructure of almost automorphic and asymptotically almostautomorphic solutions of differential and integral equationsin abstract spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 59 no 8 pp 1333ndash1345 2004

[20] T Diagana E M Hernandez and J dos Santos ldquoExistence ofasymptotically almost automorphic solutions to some abstractpartial neutral integro-differential equationsrdquo Nonlinear Analy-sis Theory Methods amp Applications An International Multidis-ciplinary Journal vol 71 no 1-2 pp 248ndash257 2009

[21] H-SDing T-J Xiao and J Liang ldquoAsymptotically almost auto-morphic solutions for some integrodifferential equations withnonlocal initial conditionsrdquo Journal of Mathematical Analysisand Applications vol 338 no 1 pp 141ndash151 2008

[22] J-Q Zhao Y-K Chang and G M Nrsquoguerekata ldquoExistence ofasymptotically almost automorphic solutions to nonlinear delayintegral equationsrdquo Dynamic Systems and Applications vol 21no 2-3 pp 339ndash349 2012

[23] Y-K Chang and C Tang ldquoAsymptotically almost automorphicsolutions to stochastic differential equations driven by a Levyprocessrdquo Stochastics An International Journal of Probability andStochastic Processes vol 88 no 7 pp 980ndash1011 2016

[24] Z-H Zhao Y-K Chang and J J Nieto ldquoSquare-mean asymp-totically almost automorphic process and its application to

stochastic integro-differential equationsrdquo Dynamic Systems andApplications vol 22 no 2-3 pp 269ndash284 2013

[25] G M NrsquoGuerekata Spectral Theory for Bounded Functions andApplications to Evolution Equations Nova Science PublishersNY USA 2017

[26] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 24 ofNorth-Holland Mathematics Studies Elsevier Science BV Am-sterdam 2006

[27] S G Samko A A Kilbas andO I MarichevFractional integraland derivatives Theory and applications Gordon and BreachScience Publishers Switzerland 1993

[28] K Diethelm The Analysis of Fractional Differential EquationsLecture Notes in Mathematics Springer Verlag Berlin Heidel-berg Germany 2010

[29] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[30] I Podlubny Fractional Differential Equations Academic PressCA USA 1999

[31] K SMiller and B RossAn Introduction to the Fractional Calcu-lus and Fractional Differential Equations A Wiley IntersciencePublication John Wiley amp Sons NY USA 1993

[32] Y Zhou Basicheory of Fractional Diferential Equations WorldScientiic Singapore 2014

[33] R P Agarwal M Belmekki and M Benchohra ldquoA surveyon semilinear differential equations and inclusions involvingRiemann-Liouville fractional derivativerdquoAdvances inDifferenceEquations vol 2009 Article ID 981728 47 pages 2009

[34] R P Agarwal V Lakshmikantham and J J Nieto ldquoOn theconcept of solution for fractional differential equations withuncertaintyrdquo Nonlinear Analysis Theory Methods amp Applica-tions An International Multidisciplinary Journal vol 72 no 6pp 2859ndash2862 2010

[35] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[36] R P Agarwal M Benchohra and S Hamani ldquoA survey onexistence results for boundary value problems of nonlinear frac-tional differential equations and inclusionsrdquo Acta ApplicandaeMathematicae vol 109 no 3 pp 973ndash1033 2010

[37] M M El-Borai ldquoSome probability densities and fundamentalsolutions of fractional evolution equationsrdquo Chaos Solitons ampFractals vol 14 no 3 pp 433ndash440 2002

[38] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 60 pp 3337ndash3343 2008

[39] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis Theory Meth-ods amp Applications An International Multidisciplinary Journalvol 69 no 8 pp 2677ndash2682 2008

[40] V Lakshmikantham and A S Vatsala ldquoTheory of fractionaldifferential inequalities and applicationsrdquo Communications inApplied Analysis vol 11 no 3-4 pp 395ndash402 2007

[41] V Lakshmikantham and J V Devi ldquoTheory of fractionaldifferential equations in a Banach spacerdquo European Journal ofPure and Applied Mathematics vol 1 no 1 pp 38ndash45 2008

[42] G Mophou O Nakoulima and G M NrsquoGuerekata ldquoExistenceresults for some fractional differential equations with nonlocalconditionsrdquoNonlinear StudiesThe International Journal vol 17no 1 pp 15ndash21 2010


[43] G M Mophou and G M NrsquoGuerekata ldquoExistence of the mildsolution for some fractional differential equationswith nonlocalconditionsrdquo Semigroup Forum vol 79 no 2 pp 315ndash322 2009

[44] GMMophou andGMNrsquoGuerekata ldquoOn integral solutions ofsome nonlocal fractional differential equations with nondensedomainrdquo Nonlinear Analysis Theory Methods amp ApplicationsAn International Multidisciplinary Journal vol 71 no 10 pp4668ndash4675 2009

[45] G M Mophou ldquoExistence and uniqueness of mild solutions toimpulsive fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications An International Multidisci-plinary Journal vol 72 no 3-4 pp 1604ndash1615 2010

[46] G M NrsquoGuerekata ldquoA Cauchy problem for some fractionalabstract differential equation with non local conditionsrdquo Non-linear Analysis Theory Methods amp Applications An Interna-tional Multidisciplinary Journal vol 70 no 5 pp 1873ndash18762009

[47] Y Zhou and L Peng ldquoOn the time-fractional Navier-Stokesequationsrdquo Computers amp Mathematics with Applications AnInternational Journal vol 73 no 6 pp 874ndash891 2017

[48] Y Zhou and L Peng ldquoWeak solutions of the time-fractionalNavier-Stokes equations and optimal controlrdquo Computers ampMathematics withApplications An International Journal vol 73no 6 pp 1016ndash1027 2017

[49] Y Zhou and L Zhang ldquoExistence and multiplicity resultsof homoclinic solutions for fractional Hamiltonian systemsrdquoComputers amp Mathematics with Applications An InternationalJournal vol 73 no 6 pp 1325ndash1345 2017

[50] Y Zhou V Vijayakumar and R Murugesu ldquoControllability forfractional evolution inclusionswithout compactnessrdquoEvolutionEquations and Control Theory vol 4 no 4 pp 507ndash524 2015

[51] C Cuevas and C Lizama ldquoAlmost automorphic solutions toa class of semilinear fractional differential equationsrdquo AppliedMathematics Letters vol 21 no 12 pp 1315ndash1319 2008

[52] R P Agarwal B de Andrade and C Cuevas ldquoOn typeof periodicity and ergodicity to a class of fractional orderdifferential equationsrdquoAdvances inDifference Equations ArticleID 179750 2010

[53] R P Agarwal C Cuevas and H Soto ldquoPseudo-almost periodicsolutions of a class of semilinear fractional differential equa-tionsrdquo Applied Mathematics and Computation vol 37 no 1-2pp 625ndash634 2011

[54] R P Agarwal B de Andrade and C Cuevas ldquoWeightedpseudo-almost periodic solutions of a class of semilinear frac-tional differential equationsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 5 pp 3532ndash3554 2010

[55] H-S Ding J Liang and T-J Xiao ldquoAlmost automorphicsolutions to abstract fractional differential equationsrdquoAdvancesin Difference Equations Article ID 508374 2010

[56] C Cuevas A Sepulveda and H Soto ldquoAlmost periodic andpseudo-almost periodic solutions to fractional differential andintegro-differential equationsrdquo Applied Mathematics and Com-putation vol 218 no 5 pp 1735ndash1745 2011

[57] Y-K Chang R Zhang and G M NrsquoGuerekata ldquoWeightedpseudo almost automorphic mild solutions to semilinear frac-tional differential equationsrdquo Computers amp Mathematics withApplications An International Journal vol 64 no 10 pp 3160ndash3170 2012

[58] B He J Cao and B Yang ldquoWeighted Stepanov-like pseudo-almost automorphic mild solutions for semilinear fractionaldifferential equationsrdquo Advances in Difference Equations vol2015 74 pages 2015

[59] J Cao Q Yang and Z Huang ldquoExistence of anti-periodic mildsolutions for a class of semilinear fractional differential equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 277ndash283 2012

[60] C Lizama and F Poblete ldquoRegularity of mild solutions for aclass of fractional order differential equationsrdquo Applied Math-ematics and Computation vol 224 pp 803ndash816 2013

[61] Z Xia M Fan and R P Agarwal ldquoPseudo almost automorphyof semilinear fractional differential equations in Banach spacesrdquoFractional Calculus and Applied Analysis vol 19 no 3 pp 741ndash764 2016

[62] S Abbas V Kavitha and RMurugesu ldquoStepanov-likeweightedpseudo almost automorphic solutions to fractional orderabstract integro-differential equationsrdquo Journal of FractionalCalculus and Applications vol 4 pp 1ndash19 2013

[63] G M Mophou ldquoWeighted pseudo almost automorphicmild solutions to semilinear fractional differential equationsrdquoApplied Mathematics and Computation vol 217 no 19 pp7579ndash7587 2011

[64] Y-K Chang and X-X Luo ldquoPseudo almost automorphicbehavior of solutions to a semi-linear fractional differentialequationrdquoMathematical Communications vol 20 no 1 pp 53ndash68 2015

[65] V Kavitha S Abbas and R Murugesu ldquo(12058311205832)-pseudo almostautomorphic solutions of fractional order neutral integro-differential equationsrdquo Nonlinear Stud vol 24 pp 669ndash6852017

[66] V Kavitha S Abbas and R Murugesu ldquoAsymptotically almostautomorphic solutions of fractional order neutral integro-differential equationsrdquo Bulletin of the Malaysian MathematicalSciences Society vol 39 no 3 pp 1075ndash1088 2016

[67] S Abbas ldquoWeighted pseudo almost automorphic solutions offractional functional differential equationsrdquo Cubo (Temuco)vol 16 pp 21ndash35 2014

[68] E Bazhlekova Fractional Evolution Equations in Banach SpacesEindhoven University of Technology 2001

[69] S D Eidelman and A N Kochubei ldquoCauchy problem forfractional diffusion equationsrdquo Journal ofDifferential Equationsvol 199 no 2 pp 211ndash255 2004

[70] R Gorenflo and F Mainardi ldquoFractional calculus Integraland differential equations of fractional orderrdquo in Fractals andFractional Calculus in Continuum Mechanics A Carpinteriand F Mainardi Eds pp 223ndash276 Springer-Verlag NY USAVienna Austria 1997

[71] V V Anh and R Mcvinish ldquoFractional differential equationsdriven by Levy noiserdquo Journal of Applied Mathematics andStochastic Analysis vol 16 no 2 pp 97ndash119 2003

[72] H-S Ding J Liang and T-J Xiao ldquoSome properties ofStepanov-like almost automorphic functions and applicationsto abstract evolution equationsrdquo Applicable Analysis An Inter-national Journal vol 88 no 7 pp 1079ndash1091 2009

[73] J Liang J Zhang and T-J Xiao ldquoComposition of pseudoalmost automorphic and asymptotically almost automorphicfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 340 no 2 pp 1493ndash1499 2008

[74] M Haase ldquoThe functional calculus for sectorial operatorsrdquo inOperatorTheory Advances and Applications vol 169 BirkhuserVerlag Basel Switzerland 2006

[75] E Cuesta ldquoAsymptotic behaviour of the solutions of fractionalintegro-differential equations and some time discretizationsrdquoDiscrete and Continuous Dynamical Systems - Series A pp 277ndash285 2007


[76] W M Ruess and W H Summers ldquoCompactness in spacesof vector valued continuous functions and asymptotic almostperiodicityrdquoMathematischeNachrichten vol 135 pp 7ndash33 1988

[77] D R Smart Fixed Point Theorems Cambridge University PressLondon UK 1980

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With motivation coming from a wide range of engineer-ing and physical applications fractional differential equationshave recently attracted great attention of mathematiciansand scientists This kind of equations is a generalization ofordinary differential equations to arbitrary noninteger ordersFractional differential equations find numerous applicationsin the field of viscoelasticity feedback amplifiers electri-cal circuits electro analytical chemistry fractional multi-poles neuron modelling encompassing different branches ofphysics chemistry and biological sciences [26ndash32] Manyphysical processes appear to exhibit fractional order behaviorthat may vary with time or space In recent years therehas been a significant development in ordinary and partialdifferential equations involving fractional derivatives weonly enumerate here the monographs of Kilbas et al [26 27]Diethelm [28] Hilfer [29] Podlubny [30] Miller [31] andZhou [32] and the papers of Agarwal et al [33 34] Benchohraet al [35 36] El-Borai [37] Lakshmikantham et al [38ndash41]Mophou et al [42ndash45]NrsquoGuerekata [46] andZhou et al [47ndash50] and the reference therein

The study of almost periodic and almost automorphictype solutions to fractional differential equations was initi-ated by Araya and Lizama [11] In their work the authorsinvestigated the existence and uniqueness of an almostautomorphic mild solution of the semilinear fractional dif-ferential equation

D120572119905 119909 (119905) = 119860119909 (119905) + 119865 (119905 119909 (119905)) 119905 isin R 1 lt 120572 lt 2 (1)

when 119860 is a generator of an 120572-resolvent family and D120572119905 is theRiemann-Liouville fractional derivative In [51] Cuevas andLizama considered the fractional differential equation

D120572119905 119909 (119905) = 119860119909 (119905) +D120572minus1119905 119865 (119905 119909 (119905)) 119905 isin R 1 lt 120572 lt 2 (2)

where 119860 is a linear operator of sectorial negative type ona complex Banach space 119883 and the fractional derivative isunderstood in the Riemann-Liouville sense Under suitableconditions on 119865(119905 119909) the authors proved the existence anduniqueness of an almost automorphic mild solution to (2)Cuevas et al [52 53] studied respectively the pseudo almostperiodic and pseudo almost periodic class infinity mildsolutions to (2) assuming that 119865 R times 119883 997888rarr 119883 and(119905 119909) 997888rarr 119865(119905 119909) is a pseudo almost periodic and pseudoalmost periodic of class infinity function satisfying suitableconditions in 119909 isin 119883 Agarwal et al [54] studied the existenceand uniqueness of a weighted pseudo almost periodic mildsolution to equation (2) Ding et al [55] investigated theexistence and uniqueness of almost automorphic solution to(2) assuming that 119865 R times 119883 997888rarr 119883 and (119905 119909) 997888rarr 119865(119905 119909) isStepanov-like almost automorphic in 119905 isin R satisfying somekind of Lipschitz conditions Cuevas et al [56] studied theexistence of almost periodic (resp pseudo almost periodic)mild solutions to equation (2) assuming that 119865 Rtimes119883 997888rarr 119883and (119905 119909) 997888rarr 119865(119905 119909) is Stepanov almost (resp Stepanov-like pseudo almost) periodic in 119905 isin R uniformly for 119909 isin 119883Chang et al [57] studied the existence and uniqueness ofweighted pseudo almost automorphic solution to equation

(2) with Stepanov-like weighted pseudo almost automorphiccoefficient He et al [58] studied also the existence anduniqueness of weighted Stepanov-like pseudo almost auto-morphic mild solution to (2) Cao et al [59] studied theexistence and uniqueness of antiperiodic mild solution to(2) In [60] Cuevas et al showed sufficient conditions toensure the existence and uniqueness of mild solution for (2)in the following classes of vector-valued function spaces peri-odic functions asymptotically periodic functions pseudoperiodic functions almost periodic functions asymptoticallyalmost periodic functions pseudo almost periodic func-tions almost automorphic functions asymptotically almostautomorphic functions pseudo almost automorphic func-tions compact almost automorphic functions asymptoticallycompact almost automorphic functions pseudo compactalmost automorphic functions 119878-asymptotically 120596-periodicfunctions decay functions and mean decay functions

Recently Xia et al [61] established some sufficient criteriafor the existence and uniqueness of (120583 ])-pseudo almostautomorphic solution to the semilinear fractional differentialequation

D120572119905 119909 (119905) = 119860119909 (119905) + D120572minus1119905 119865 (119905 119861119909 (119905)) 119905 isin R (3)

where 1 lt 120572 lt 2 119860 is a sectorial operator of type 120596 lt 0 on acomplex Banach space119883 and 119861 is a bounded linear operatorThe fractional derivative is understood in the Riemann-Liouville senseTheir discussion is divided into two cases ie119865 R times 119883 997888rarr 119883 (119905 119909) 997888rarr 119865(119905 119909) is (120583 ])-pseudo almostautomorphic and 119865 R times 119883 997888rarr 119883 and (119905 119909) 997888rarr 119865(119905 119909)is Stepanov-like (120583 ])-pseudo almost automorphic Kavithaet al [62] studied weighted pseudo almost automorphicsolutions of the fractional integrodifferential equation

D120572119905 119909 (119905) = 119860119909 (119905) + D120572minus1119905 119865 (119905 119909 (119905) 119870119909 (119905)) 119905 isin R (4)

where 1 lt 120572 lt 2 and119870119909 (119905) = int119905

minusinfin119896 (119905 minus 119904) ℎ (119904 119909 (119904)) d119904 (5)

119860 is a linear densely defined sectorial operator on a complexBanach space 119883 119865 R times 119883 times 119883 997888rarr 119883 and (119905 119909 119910) 997888rarr119865(119905 119909 119910) is a weighted pseudo almost automorphic functionin 119905 isin R for each 119909 119910 isin 119883 satisfying suitable conditionsThe fractional derivative is understood in the Riemann-Liouville sense Mophou [63] investigated the existence anduniqueness of weighted pseudo almost automorphic mildsolution to the fractional differential equation


where 119860 119863(119860) sub 119883 997888rarr 119883 is a linear densely oper-ator of sectorial type on a complex Banach space 119883 119861 119883 997888rarr 119883 is a bounded linear operator and 119865 R times 119883 times119883 997888rarr 119883 and (119905 119909 119910) 997888rarr 119865(119905 119909 119910) is a weighted pseudoalmost automorphic function in 119905 isin R for each 119909 119910 isin 119883satisfying suitable conditions The fractional derivative D120572119905 isto be understood in Riemann-Liouville sense Chang et al





120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909) + 120597120572minus1119905 [120583119886 (119905)



(7)



2 Preliminaries






119879L(119883) = sup 119879119909 119909 isin 119883 119909 = 1 (9)




Θ (119905) = lim119899997888rarrinfin

119865 (119905 + 119904119899) (10)






119865 (119905) = sin( 12 + cos 119905 + cosradic2119905) (12)






119865 (119905 + 119904119899 119909 119910) = Θ (119905 119909 119910) exists



Θ(119905 minus 119904119899 119909 119910) = 119865 (119905 119909 119910) exists









(1198751) lim119899997888rarrinfin






(1198756) lim119899997888rarrinfin


(17)

Write

Υ (119905) fl 119865 (119905 119909 (119905) 119910 (119905)) 119905 isin R (18)

Then10038171003817100381710038171003817Υ (119905 + 120591119899) minus Υ (119905)10038171003817100381710038171003817


(19)


(119909 (119905) 119910 (119905)) isin 119870(119909 (119905) 119910 (119905)) isin 119870




1003817100381710038171003817119865 (119905 + 120591119899 119909 (119905 + 120591119899) 119910 (119905 + 120591119899))minus 119865 (119905 + 120591119899 119909 (119905) 119910 (119905))1003817100381710038171003817 = 0 (21)


10038171003817100381710038171003817119865 (119905 + 120591119899 119909 (119905) 119910 (119905)) minus 119865 (119905 119909 (119905) 119910 (119905))10038171003817100381710038171003817 = 0 (22)










(25)





(26)



119866 (119905) isin 119860119860 (R 119883) Φ (119905) isin 1198620 (R 119883) (27)



119865 (119905) = 119866 (119905) + Φ (119905)= sin( 1




Φ (119905) = 119890minus|119905| isin 1198620 (RR) (29)




119865 (119905 119909 119910) = 119866 (119905 119909 119910) + Φ (119905 119909 119910)= sin( 1


(31)




(32)



119868120572119891 (119905) = 1Γ (120572) int119905




119863120572119905 119891 (119905) = 1Γ (119899 minus 120572)

d119899

d119905119899 int119905

1199050(119905 minus 119904)minus120572 119891 (119904) d119904

119905 gt 1199050 119899 minus 1 lt 120572 lt 119899(34)










119890minus120582119905119878120572 (119905) 119909 d119905Re120582 gt 120596 119909 isin 119883

(36)



119878120572 (119905) fl 12120587119894 int120574 119890




1003817100381710038171003817119878120572 (119905)1003817100381710038171003817L(119883) le 1198621198721 + |120596| 119905120572 for 119905 ge 0 (38)












119909 (119905) = int119905minusinfin

119878120572 (119905 minus 120590) 119865 (120590 119909 (120590) 119861119909 (120590))d120590 119905 isin R (40)






119878120572 (119905 minus 119904) 119885 (119904) 119889119904119905 isin R

(41)


intinfin0

11 + |120596| 119904120572 d119904 = |120596|minus1120572 120587

120572 sin (120587120572) for 1 lt 120572 lt 2 (42)

Then

1003817100381710038171003817Φ1 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591) 119884 (119905 minus 120591) d12059110038171003817100381710038171003817100381710038171003817


11 + |120596| 120591120572 d120591

= 119862119872 |120596|minus1120572 120587120572 sin (120587120572) 119884infin

(43)




Define


119879 (119905 minus s) (119904) d119904 (45)

Then

10038171003817100381710038171003817Φ1 (119905 + 119904119899) minus Φ1 (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905+119904119899



119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (119904) 119884 (119905 + 119904119899 minus 119904) d119904

minus int+infin0


0

11 + |120596| 119904120572

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 d119904

le 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(46)


Φ1 (119905) = lim119899997888rarrinfin

Φ1 (119905 + 119904119899) (47)






1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 le 100381710038171003817100381710038171003817100381710038171003817int1198731


+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198731119878120572 (119905 minus 119904) 119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 d119904

+ 120576119862119872int1199051198731

11 + |120596| (119905 minus 119904)120572 d119904


1(119905 minus 119904)120572 d119904

+ 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

le 119862119872119885infin|120596|1


+ 119862119872|120596|minus1120572 120587120576120572 sin (120587120572)

(49)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (50)


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 119885 (119904) d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(51)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (52)




le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (54)



Φ (119903)119903 = 1205881

(55)






120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)


120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|


|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)


120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)


119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)




119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)


bounded inR and


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)





1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904



minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)




11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018












120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909) + 120597120572minus1119905 [120583119886 (119905)



(7)



2 Preliminaries






119879L(119883) = sup 119879119909 119909 isin 119883 119909 = 1 (9)




Θ (119905) = lim119899997888rarrinfin

119865 (119905 + 119904119899) (10)






119865 (119905) = sin( 12 + cos 119905 + cosradic2119905) (12)






119865 (119905 + 119904119899 119909 119910) = Θ (119905 119909 119910) exists



Θ(119905 minus 119904119899 119909 119910) = 119865 (119905 119909 119910) exists









(1198751) lim119899997888rarrinfin






(1198756) lim119899997888rarrinfin


(17)

Write

Υ (119905) fl 119865 (119905 119909 (119905) 119910 (119905)) 119905 isin R (18)

Then10038171003817100381710038171003817Υ (119905 + 120591119899) minus Υ (119905)10038171003817100381710038171003817


(19)


(119909 (119905) 119910 (119905)) isin 119870(119909 (119905) 119910 (119905)) isin 119870




1003817100381710038171003817119865 (119905 + 120591119899 119909 (119905 + 120591119899) 119910 (119905 + 120591119899))minus 119865 (119905 + 120591119899 119909 (119905) 119910 (119905))1003817100381710038171003817 = 0 (21)


10038171003817100381710038171003817119865 (119905 + 120591119899 119909 (119905) 119910 (119905)) minus 119865 (119905 119909 (119905) 119910 (119905))10038171003817100381710038171003817 = 0 (22)










(25)





(26)



119866 (119905) isin 119860119860 (R 119883) Φ (119905) isin 1198620 (R 119883) (27)



119865 (119905) = 119866 (119905) + Φ (119905)= sin( 1




Φ (119905) = 119890minus|119905| isin 1198620 (RR) (29)




119865 (119905 119909 119910) = 119866 (119905 119909 119910) + Φ (119905 119909 119910)= sin( 1


(31)




(32)



119868120572119891 (119905) = 1Γ (120572) int119905




119863120572119905 119891 (119905) = 1Γ (119899 minus 120572)

d119899

d119905119899 int119905

1199050(119905 minus 119904)minus120572 119891 (119904) d119904

119905 gt 1199050 119899 minus 1 lt 120572 lt 119899(34)










119890minus120582119905119878120572 (119905) 119909 d119905Re120582 gt 120596 119909 isin 119883

(36)



119878120572 (119905) fl 12120587119894 int120574 119890




1003817100381710038171003817119878120572 (119905)1003817100381710038171003817L(119883) le 1198621198721 + |120596| 119905120572 for 119905 ge 0 (38)












119909 (119905) = int119905minusinfin

119878120572 (119905 minus 120590) 119865 (120590 119909 (120590) 119861119909 (120590))d120590 119905 isin R (40)






119878120572 (119905 minus 119904) 119885 (119904) 119889119904119905 isin R

(41)


intinfin0

11 + |120596| 119904120572 d119904 = |120596|minus1120572 120587

120572 sin (120587120572) for 1 lt 120572 lt 2 (42)

Then

1003817100381710038171003817Φ1 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591) 119884 (119905 minus 120591) d12059110038171003817100381710038171003817100381710038171003817


11 + |120596| 120591120572 d120591

= 119862119872 |120596|minus1120572 120587120572 sin (120587120572) 119884infin

(43)




Define


119879 (119905 minus s) (119904) d119904 (45)

Then

10038171003817100381710038171003817Φ1 (119905 + 119904119899) minus Φ1 (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905+119904119899



119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (119904) 119884 (119905 + 119904119899 minus 119904) d119904

minus int+infin0


0

11 + |120596| 119904120572

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 d119904

le 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(46)


Φ1 (119905) = lim119899997888rarrinfin

Φ1 (119905 + 119904119899) (47)






1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 le 100381710038171003817100381710038171003817100381710038171003817int1198731


+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198731119878120572 (119905 minus 119904) 119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 d119904

+ 120576119862119872int1199051198731

11 + |120596| (119905 minus 119904)120572 d119904


1(119905 minus 119904)120572 d119904

+ 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

le 119862119872119885infin|120596|1


+ 119862119872|120596|minus1120572 120587120576120572 sin (120587120572)

(49)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (50)


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 119885 (119904) d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(51)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (52)




le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (54)



Φ (119903)119903 = 1205881

(55)






120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)


120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|


|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)


120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)


119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)




119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)


bounded inR and


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)





1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904



minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)




11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018











119865 (119905 + 119904119899 119909 119910) = Θ (119905 119909 119910) exists



Θ(119905 minus 119904119899 119909 119910) = 119865 (119905 119909 119910) exists









(1198751) lim119899997888rarrinfin






(1198756) lim119899997888rarrinfin


(17)

Write

Υ (119905) fl 119865 (119905 119909 (119905) 119910 (119905)) 119905 isin R (18)

Then10038171003817100381710038171003817Υ (119905 + 120591119899) minus Υ (119905)10038171003817100381710038171003817


(19)


(119909 (119905) 119910 (119905)) isin 119870(119909 (119905) 119910 (119905)) isin 119870




1003817100381710038171003817119865 (119905 + 120591119899 119909 (119905 + 120591119899) 119910 (119905 + 120591119899))minus 119865 (119905 + 120591119899 119909 (119905) 119910 (119905))1003817100381710038171003817 = 0 (21)


10038171003817100381710038171003817119865 (119905 + 120591119899 119909 (119905) 119910 (119905)) minus 119865 (119905 119909 (119905) 119910 (119905))10038171003817100381710038171003817 = 0 (22)










(25)





(26)



119866 (119905) isin 119860119860 (R 119883) Φ (119905) isin 1198620 (R 119883) (27)



119865 (119905) = 119866 (119905) + Φ (119905)= sin( 1




Φ (119905) = 119890minus|119905| isin 1198620 (RR) (29)




119865 (119905 119909 119910) = 119866 (119905 119909 119910) + Φ (119905 119909 119910)= sin( 1


(31)




(32)



119868120572119891 (119905) = 1Γ (120572) int119905




119863120572119905 119891 (119905) = 1Γ (119899 minus 120572)

d119899

d119905119899 int119905

1199050(119905 minus 119904)minus120572 119891 (119904) d119904

119905 gt 1199050 119899 minus 1 lt 120572 lt 119899(34)










119890minus120582119905119878120572 (119905) 119909 d119905Re120582 gt 120596 119909 isin 119883

(36)



119878120572 (119905) fl 12120587119894 int120574 119890




1003817100381710038171003817119878120572 (119905)1003817100381710038171003817L(119883) le 1198621198721 + |120596| 119905120572 for 119905 ge 0 (38)












119909 (119905) = int119905minusinfin

119878120572 (119905 minus 120590) 119865 (120590 119909 (120590) 119861119909 (120590))d120590 119905 isin R (40)






119878120572 (119905 minus 119904) 119885 (119904) 119889119904119905 isin R

(41)


intinfin0

11 + |120596| 119904120572 d119904 = |120596|minus1120572 120587

120572 sin (120587120572) for 1 lt 120572 lt 2 (42)

Then

1003817100381710038171003817Φ1 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591) 119884 (119905 minus 120591) d12059110038171003817100381710038171003817100381710038171003817


11 + |120596| 120591120572 d120591

= 119862119872 |120596|minus1120572 120587120572 sin (120587120572) 119884infin

(43)




Define


119879 (119905 minus s) (119904) d119904 (45)

Then

10038171003817100381710038171003817Φ1 (119905 + 119904119899) minus Φ1 (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905+119904119899



119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (119904) 119884 (119905 + 119904119899 minus 119904) d119904

minus int+infin0


0

11 + |120596| 119904120572

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 d119904

le 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(46)


Φ1 (119905) = lim119899997888rarrinfin

Φ1 (119905 + 119904119899) (47)






1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 le 100381710038171003817100381710038171003817100381710038171003817int1198731


+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198731119878120572 (119905 minus 119904) 119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 d119904

+ 120576119862119872int1199051198731

11 + |120596| (119905 minus 119904)120572 d119904


1(119905 minus 119904)120572 d119904

+ 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

le 119862119872119885infin|120596|1


+ 119862119872|120596|minus1120572 120587120576120572 sin (120587120572)

(49)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (50)


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 119885 (119904) d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(51)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (52)




le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (54)



Φ (119903)119903 = 1205881

(55)






120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)


120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|


|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)


120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)


119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)




119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)


bounded inR and


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)





1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904



minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)




11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018











(26)



119866 (119905) isin 119860119860 (R 119883) Φ (119905) isin 1198620 (R 119883) (27)



119865 (119905) = 119866 (119905) + Φ (119905)= sin( 1




Φ (119905) = 119890minus|119905| isin 1198620 (RR) (29)




119865 (119905 119909 119910) = 119866 (119905 119909 119910) + Φ (119905 119909 119910)= sin( 1


(31)




(32)



119868120572119891 (119905) = 1Γ (120572) int119905




119863120572119905 119891 (119905) = 1Γ (119899 minus 120572)

d119899

d119905119899 int119905

1199050(119905 minus 119904)minus120572 119891 (119904) d119904

119905 gt 1199050 119899 minus 1 lt 120572 lt 119899(34)










119890minus120582119905119878120572 (119905) 119909 d119905Re120582 gt 120596 119909 isin 119883

(36)



119878120572 (119905) fl 12120587119894 int120574 119890




1003817100381710038171003817119878120572 (119905)1003817100381710038171003817L(119883) le 1198621198721 + |120596| 119905120572 for 119905 ge 0 (38)












119909 (119905) = int119905minusinfin

119878120572 (119905 minus 120590) 119865 (120590 119909 (120590) 119861119909 (120590))d120590 119905 isin R (40)






119878120572 (119905 minus 119904) 119885 (119904) 119889119904119905 isin R

(41)


intinfin0

11 + |120596| 119904120572 d119904 = |120596|minus1120572 120587

120572 sin (120587120572) for 1 lt 120572 lt 2 (42)

Then

1003817100381710038171003817Φ1 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591) 119884 (119905 minus 120591) d12059110038171003817100381710038171003817100381710038171003817


11 + |120596| 120591120572 d120591

= 119862119872 |120596|minus1120572 120587120572 sin (120587120572) 119884infin

(43)




Define


119879 (119905 minus s) (119904) d119904 (45)

Then

10038171003817100381710038171003817Φ1 (119905 + 119904119899) minus Φ1 (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905+119904119899



119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (119904) 119884 (119905 + 119904119899 minus 119904) d119904

minus int+infin0


0

11 + |120596| 119904120572

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 d119904

le 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(46)


Φ1 (119905) = lim119899997888rarrinfin

Φ1 (119905 + 119904119899) (47)






1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 le 100381710038171003817100381710038171003817100381710038171003817int1198731


+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198731119878120572 (119905 minus 119904) 119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 d119904

+ 120576119862119872int1199051198731

11 + |120596| (119905 minus 119904)120572 d119904


1(119905 minus 119904)120572 d119904

+ 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

le 119862119872119885infin|120596|1


+ 119862119872|120596|minus1120572 120587120576120572 sin (120587120572)

(49)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (50)


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 119885 (119904) d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(51)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (52)




le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (54)



Φ (119903)119903 = 1205881

(55)






120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)


120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|


|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)


120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)


119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)




119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)


bounded inR and


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)





1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904



minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)




11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018











119890minus120582119905119878120572 (119905) 119909 d119905Re120582 gt 120596 119909 isin 119883

(36)



119878120572 (119905) fl 12120587119894 int120574 119890




1003817100381710038171003817119878120572 (119905)1003817100381710038171003817L(119883) le 1198621198721 + |120596| 119905120572 for 119905 ge 0 (38)












119909 (119905) = int119905minusinfin

119878120572 (119905 minus 120590) 119865 (120590 119909 (120590) 119861119909 (120590))d120590 119905 isin R (40)






119878120572 (119905 minus 119904) 119885 (119904) 119889119904119905 isin R

(41)


intinfin0

11 + |120596| 119904120572 d119904 = |120596|minus1120572 120587

120572 sin (120587120572) for 1 lt 120572 lt 2 (42)

Then

1003817100381710038171003817Φ1 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591) 119884 (119905 minus 120591) d12059110038171003817100381710038171003817100381710038171003817


11 + |120596| 120591120572 d120591

= 119862119872 |120596|minus1120572 120587120572 sin (120587120572) 119884infin

(43)




Define


119879 (119905 minus s) (119904) d119904 (45)

Then

10038171003817100381710038171003817Φ1 (119905 + 119904119899) minus Φ1 (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905+119904119899



119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (119904) 119884 (119905 + 119904119899 minus 119904) d119904

minus int+infin0


0

11 + |120596| 119904120572

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 d119904

le 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(46)


Φ1 (119905) = lim119899997888rarrinfin

Φ1 (119905 + 119904119899) (47)






1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 le 100381710038171003817100381710038171003817100381710038171003817int1198731


+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198731119878120572 (119905 minus 119904) 119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 d119904

+ 120576119862119872int1199051198731

11 + |120596| (119905 minus 119904)120572 d119904


1(119905 minus 119904)120572 d119904

+ 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

le 119862119872119885infin|120596|1


+ 119862119872|120596|minus1120572 120587120576120572 sin (120587120572)

(49)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (50)


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 119885 (119904) d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(51)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (52)




le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (54)



Φ (119903)119903 = 1205881

(55)






120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)


120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|


|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)


120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)


119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)




119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)


bounded inR and


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)





1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904



minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)




11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018









Define


119879 (119905 minus s) (119904) d119904 (45)

Then

10038171003817100381710038171003817Φ1 (119905 + 119904119899) minus Φ1 (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905+119904119899



119878120572 (119905 minus 119904) 119884 (119904) d11990410038171003817100381710038171003817100381710038171003817= 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (119904) 119884 (119905 + 119904119899 minus 119904) d119904

minus int+infin0


0

11 + |120596| 119904120572

10038171003817100381710038171003817119884 (119904 + 119904119899) minus (119904)10038171003817100381710038171003817 d119904

le 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(46)


Φ1 (119905) = lim119899997888rarrinfin

Φ1 (119905 + 119904119899) (47)






1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 le 100381710038171003817100381710038171003817100381710038171003817int1198731


+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198731119878120572 (119905 minus 119904) 119885 (119904) d119904100381710038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 d119904

+ 120576119862119872int1199051198731

11 + |120596| (119905 minus 119904)120572 d119904


1(119905 minus 119904)120572 d119904

+ 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

le 119862119872119885infin|120596|1


+ 119862119872|120596|minus1120572 120587120576120572 sin (120587120572)

(49)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (50)


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 119885 (119904) d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872 |120596|minus1120572 120587120576120572 sin (120587120572)

(51)

which implies


1003817100381710038171003817Φ2 (119905)1003817100381710038171003817 = 0 (52)




le 119871 (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (54)



Φ (119903)119903 = 1205881

(55)






120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)


120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|


|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)


120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)


119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)




119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)


bounded inR and


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)





1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904



minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)




11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018












120573 (s)1 + |120596| (119905 minus 119904)120572 d119904 119905 isin R (56)


120590 (119905) le 100381710038171003817100381710038171003817100381710038171003817int1198791

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817int119905

1198791

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904

100381710038171003817100381710038171003817100381710038171003817le 10038171003817100381710038171205731003817100381710038171003817infin int1198791

minusinfin

11 + |120596| (119905 minus 119904)120572 d119904

+ 120576int1199051198791

11 + |120596| (119905 minus 119904)120572 d119904

le10038171003817100381710038171205731003817100381710038171003817infin|120596| int1198791

minusinfin

1(119905 minus 119904)120572 d119904 +

|120596|minus1120572 120587120576120572 sin (120587120572)

le10038171003817100381710038171205731003817100381710038171003817infin|120596|


|120596|minus1120572 120587120576120572 sin (120587120572)

(57)

which implies


120590 (119905) = 0 (58)


120590 (119905) = 10038171003817100381710038171003817100381710038171003817int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 d119904 = |120596|minus1120572 120587120576

120572 sin (120587120572) (59)

which implies


120590 (119905) = 0 (60)


119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572)

+ 119862119872(1 + 119861L(119883)) 12058811205882 lt 1(61)




119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904119905 isin R

(62)


bounded inR and


1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 d119904


11 + |120596| (119905 minus 119904)120572 d119904

= 119862119872119871 |120596|minus1120572 120587 100381710038171003817100381711986511003817100381710038171003817infin120572 sin (120587120572)

(63)





1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))] d119904



minusinfin

11 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)




11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










11 + |120596| (119905 minus 119904)120572 d119904


(65)








1003817100381710038171003817infin (67)


Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (68)



sdot [1198651 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

(Γ2120596) (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 1198652 (119904 V (119904)+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904

(69)



(70)

which implies that



10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 le 120573 (119904)sdot Φ(120596 (119904) + 119861120596 (119904) + sup

119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) = 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(72)


as 120573 (sdot) isin 1198620 (RR+) (73)



119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587120572 sin (120587120572) 1198960

+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup119904isinR

V (119904)))le 1198960

(74)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))] d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817



minusinfin


minusinfin

11 + |120596| (119905 minus 119904)120572 [10038171003817100381710038171205961 (119904)1003817100381710038171003817


+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018









+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904+ 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ(10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817


minusinfin

11 + |120596| (119905 minus 119904)120572 d119904 + 119862119872120590 (119905) Φ ((1


120572 sin (120587120572) 1198960 + 1198621198721205882Φ((1+ 119861L(119883)) (1198960 + V (119904)infin)) le 1198960

(75)




(76)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



minusinfin

1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


11 + |120596| (119905 minus 119904)120572 d119904 = 119862119872119871 (1 + 119861L(119883)) |120596|minus1120572 120587

120572 sin (120587120572) 10038171003817100381710038171205961minus 12059621003817100381710038171003817infin

(77)

which implies that


120572 sin (120587120572) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin (78)




3119862119872 (79)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (80)



(81)



11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)

+ 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904))) minus 1198652 (119904 V (119904)+ 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904 le 120576

3

(82)



10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


max1199051199051

1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572 10038171003817100381710038171198652 (119904 V (119904)


3 + 21205763 = 120576

(83)



10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



(84)




As

10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


+ 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int+infin

0119878120572 (120591)


(86)


10038171003817100381710038171003817100381710038171003817int+infin

120585119878120572 (120591) 1198652 (119905 minus 120591 V (119905 minus 120591)


(87)

Thus we get







10038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)10038171003817100381710038171003817 le int11990521199051

1003817100381710038171003817119878120572 (1199052 minus 119904)sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904




120573 (119904)1 + |120596| (1199052 minus 119904)120572 d119904 + Φ ((1 + 119861L(119883)) (1198960



120573 (119904) d119904 + 119862119872Φ((1 + 119861L(119883)) (1198960



1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










1199051minus119896( 120573 (119904)1 + |120596| (1199052 minus 119904)120572

+ 120573 (119904)1 + |120596| (1199051 minus 119904)120572) d119904 997888rarr 0







119878120572 (119905 minus 119904) 1198651 (119904 V (119904) 119861V (119904)) d119904 119905 isin R

120596 (119905) = int119905minusinfin



sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d119904119905 isin R

(91)



119909 (119905) fl V (119905) + 120596 (119905) isin 119860119860119860 (R 119883) (93)


119909 (119905) = int119905minusinfin

119878120572 (119905 minus 119904) 119865 (119904 119909 (119904) 119861119909 (119904)) d119904 119905 isin R (94)




119862119871 (1 + 119861L(119883)) 120588120587120572 sin (120587120572) + 119862 (1 + 119861L(119883)) 12058811205882 lt 1 (95)





le 119871 (119905) (10038171003817100381710038171199091 minus 11991011003817100381710038171003817 + 10038171003817100381710038171199092 minus 11991021003817100381710038171003817) (97)



119862119872119871 119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572)+ 11986211987212058811205882 (1 + 119861L(119883)) lt 1

(98)







1003817100381710038171003817[ΛV119899] (119905) minus [ΛV] (119905)1003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V119899 (119904) 119861V119899 (119904))minus 1198651 (119904 V (119904) 119861V (119904))]d119904





119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 1003817100381710038171003817V119899 (119904)


int119905minus119898119905minus(119898+1)



11 + |120596|119898120572 int

119905minus119898







1003817100381710038171003817infin (101)




Ω119903 fl 120596 (119905) isin 1198620 (R 119883) 120596 (119905) le 119903 (103)



minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904)

(104)





119904isinRV (119904) + 119861V (119904))

le 120573 (119904) Φ((1 + 119861L(119883)) 120596 (119904)

+ (1 + 119861L(119883)) sup119904isinR

V (119904)) le 120573 (119904)

sdot Φ((1 + 119861L(119883)) [120596 (119904) + sup119904isinR

V (119904)])

(106)


as 120573 (sdot) isin 1198620 (RR+) (107)



119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960

(108)


119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904))) d11990410038171003817100381710038171003817100381710038171003817


119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 [10038171003817100381710038171205961 (119904)1003817100381710038171003817+ 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817] d119904 + 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572Φ

sdot (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205962 (119904)1003817100381710038171003817 + V (119904) + 119861V (119904))d119904le 119862119872int119905

minusinfin

119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883))


120573 (119904)1 + |120596| (119905 minus 119904)120572Φ ((1

+ 119861L(119883)) (10038171003817100381710038171205962 (119904)1003817100381710038171003817 + V (119904)) d119904 le 119862119872(1



119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904


V (119904)))

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)

119871 (119904)1 + |120596| (119905 minus 119904)120572 d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883))


V (119904)))

le 119862119872(+infinsum119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1

+ 119861L(119883)) 100381710038171003817100381712059611003817100381710038171003817infin + 1198621198721205882Φ((1 + 119861L(119883)) (1198960

+ sup119904isinR

V (119904))) le 119862119872(+infinsum119898=0

11 + |120596|119898120572) 119871 (1

+ 119861L(119883)) 1198960 + 1198621198721205882Φ((1 + 119861L(119883))(1198960

+ sup119904isinR

V (119904)))

= 119862119872119871 |120596|minus1120572 120587 (1 + 119861L(119883))120572 sin (120587120572) 1198960+ 1198621198721205882Φ((1 + 119861L(119883)) (1198960 + sup

119904isinRV (119904)))

le 1198960(109)




(110)

Thus

10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 10038171003817100381710038171003817100381710038171003817int119905



119871 (119904) 1003817100381710038171003817119878120572 (119905 minus 119904)1003817100381710038171003817 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)


119871 (119904)1 + |120596| (119905 minus 119904)120572 (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904) minus 1205962 (119904)1003817100381710038171003817 d119904

le 119862119872(+infinsum119898=0

int119905minus119898119905minus(119898+1)


119898=0

11 + |120596|119898120572 int

119905minus119898

119905minus(119898+1)119871 (119904) d119904) (1


11 + |120596|119898120572) 119871 (1 + 119861L(119883)) 10038171003817100381710038171205961 minus 12059621003817100381710038171003817infin


(111)

which implies that


(112)








119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018












119862 119871 120588120587 (1 + 119861L(119883))120572 sin (120587120572) + 11986212058811205882 (1 + 119861L(119883)) lt 1 (113)



10038171003817100381710038171198652 (119905 119909 119910)1003817100381710038171003817 le 120573 (119905) (119909 + 10038171003817100381710038171199101003817100381710038171003817) (114)







|119909| fl sup119905isinR

120583 (119905) 119909 (119905) (115)


120583 (119905) 1003817100381710038171003817ΛV1 (119905) minus ΛV2 (119905)1003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (120590 V1 (120590) 119861V1 (120590))minus 1198651 (120590 V2 (120590) 119861V2 (120590))] d120590

10038171003817100381710038171003817100381710038171003817le 119862120572 int

119905


minus 119861V2 (120590)1003817100381710038171003817] d120590 = 119862120572 int119905

minusinfin120583 (119905) 120583 (120590) 119871 (120590)


10038161003816100381610038161003817100381710038171003817 int119905minusinfin

120583 (119905) 120583 (120590)minus1

sdot 119871 (120590) d120590 = 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590max119871(120591)120573(120591)d120591119871 (120590) d120590

le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


119896119890minus119896int119905120590 119871(120591)d120591119871 (120590) d120590

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817


dd120590 (119890119896int120590119905 119871(120591)d120591) d120590


le 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817V1 minus V210038161003816100381610038161003817100381710038171003817

(116)

which implies that




minus 1198651 (119904 V (119904) 119861V (119904))1003817100381710038171003817 le 119871 (119904) [120596 (119904) + 119861120596 (119904)]le 119871 (119904) (1 + 119861L(119883)) 120596 (119904) + 119861120596 (119904)

(118)





(120)



as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










as 120573 (sdot) isin 1198620 (RR+) (121)



2119862120572 (1 + 119861L(119883))119896 1198960 + 119862120572 (1 + 119861L(119883))119896 |V (119904)|le 1198960

(122)


120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) + (Γ21205962) (119905)10038171003817100381710038171003817 le 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot [1198651 (119904 V (119904) + 1205961 (119904) 119861 (V (119904) + 1205961 (119904)))minus 1198651 (119904 V (119904) 119861V (119904))]d119904

10038171003817100381710038171003817100381710038171003817 + 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905

minusinfin119878120572 (119905

minus 119904) 1198652 (119904 V (119904) + 1205962 (119904) 119861 (V (119904) + 1205962 (119904)))d11990410038171003817100381710038171003817100381710038171003817

le 119862120572 int119905

minusinfin120583 (119905) 119871 (119904) (10038171003817100381710038171205961 (119904)1003817100381710038171003817 + 10038171003817100381710038171198611205961 (119904)1003817100381710038171003817) d119904

+ 119862120572 int119905

minusinfin120583 (119905) 120573 (119904) (100381710038171003817100381712059621003817100381710038171003817 + V (119904) + 100381710038171003817100381711986112059621003817100381710038171003817

+ 119861V (119904)) d119904 = 119862120572 int119905


sdot (1 + 119861L(119883)) 10038171003817100381710038171205961 (119904)1003817100381710038171003817 d119904 + 119862120572 int119905

minusinfin120583 (119905) 120583 (119904)




120583 (119905)


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591119871 (119904) d119904 + 119862120572 (1


119896119890minus119896int119904119905 max119871(120591)120573(120591)d120591120573 (119904) d119904


119896119890minus119896int119904119905 119871(120591)d120591119871 (119904) d119904 + 119862120572 (1 + 119861L(119883))


119896119890minus119896int119904119905 120573(120591)d120591120573 (119904) d119904

= 119862120572 (1 + 119861L(119883))119896 10038161003816100381610038161003817100381710038171003817120596110038161003816100381610038161003817100381710038171003817 int119905minusinfin

dd119904 (119890119896 int119904119905 119871(120591)d120591) d119904


dd119904 (119890119896 int119904119905 120573(120591)d120591) d119904


(123)




(124)

Thus

120583 (119905) 10038171003817100381710038171003817(Γ11205961) (119905) minus (Γ11205962) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



119905

minusinfin120583 (119905) 119871 (120590) (1 + 119861L(119883)) 10038171003817100381710038171205961 (120590)


minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018









minus 1205962 (120590)1003817100381710038171003817 d120590 = 119862120572 int119905




119896119890minus119896int119905120590 max119871(120591)120573(120591)d120591119871 (120590) d120590



dd120590 (119890119896int120590119905 119871(120591)d120591) d120590



le 119862120572 (1 + 119861L(119883))119896 100381610038161003816100381610038171003817100381710038171205961 minus 120596210038161003816100381610038161003817100381710038171003817 (126)




3119862119872 (127)


997888rarr 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) (128)



(129)



11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596119896 (119904) 119861 (V (119904) + 120596119896 (119904)))

minus 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904)))1003817100381710038171003817 d119904le 120576

3(130)


120583 (119905) 10038171003817100381710038171003817(Γ2120596119896) (119905) minus (Γ21205960) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905



S120572 (119905 minus 119904)

sdot 1198652 (119904 V (119904) + 1205960 (119904) 119861 (V (119904) + 1205960 (119904))) d11990410038171003817100381710038171003817100381710038171003817


11 + |120596| (119905 minus 119904)120572120583 (119905)


1199051

120573 (119904)1 + |120596| (119905 minus 119904)120572 d119904


11 + |120596| (119905 minus 119904)120572120583 (119905)


3+ 2120576

3 = 120576

(131)




120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










120583 (119905) 10038171003817100381710038171003817(Γ2120596) (119905)10038171003817100381710038171003817 = 120583 (119905) 10038171003817100381710038171003817100381710038171003817int119905


sdot 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904))) d11990410038171003817100381710038171003817100381710038171003817le 119862119872int119905

minusinfin

11 + |120596| (119905 minus 119904)120572 120583 (119905)

sdot 10038171003817100381710038171198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))1003817100381710038171003817 d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (V (119904) + 120596 (119904) )

+ 119861V (119904) + 119861120596 (119904) )) d119904le 119862119872int119905

minusinfin

120573 (119904)1 + |120596| (119905 minus 119904)120572 120583 (119905) (1 + 119861L(119883))

sdot (V (119904) + 120596 (119904) ) ) d119904 le 119862119872120590 (119905) (1+ 119861L(119883)) (1198960 + |V (119904)| )

(132)




as

(Γ2120596) (119905) = int119905minusinfin


0119878120572 (120591)


(134)


1003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817100381710038171003817int+infin


10038171003817100381710038171003817100381710038171003817 lt 1205760 (135)

Thus we get






int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904 lt 12057614

int11990511199051minus120578



int1199051minus1205781199051minus119896




minusinfin120573 (119904) d119904 lt 12057614

(139)







minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018














minusinfin120573 (119904) d119904 lt 12057614

(140)

Then one has

1003816100381610038161003816100381610038171003817100381710038171003817(Γ2120596) (1199052) minus (Γ2120596) (1199051)1003816100381610038161003816100381610038171003817100381710038171003817= 10038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817100381710038171003817int1199052



10038171003817100381710038171003817100381710038171003817le int11990521199051

10038161003816100381610038161003817100381710038171003817119878120572 (1199052 minus 119904) 1198652 (119904 V (119904) + 120596 (119904) 119861 (V (119904) + 120596 (119904)))10038161003816100381610038161003817100381710038171003817 d119904

+ int11990511199051minus120578




+ int1199051minus1205781199051minus119896


(141)






4 Applications



120597120572119905 119906 (119905 119909) = 1205972119909119906 (119905 119909) minus 119901119906 (119905 119909)+ 120597120572minus1119905 [120583119886 (119905) sin( 1

2 + cos 119905 + cosradic2119905)


119905 isin R 119909 isin [0 120587] 119906 (119905 0) = 119906 (119905 120587) = 0 119905 isin R

(142)



119860120593 fl 12059310158401015840 minus 119901120593 120593 isin 119863 (119860) (143)

where

119863 (119860) fl 120593 isin 119883 12059310158401015840 isin 119883 120593 (0) = 120593 (120587) sub 119883 (144)


1198651 (119905 119909 (120585) 119910 (120585)) fl 120583119886 (119905)sdot sin( 1


(145)



10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018










10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)100381710038171003817100381722le int1205870

1205832 10038161003816100381610038161003816100381610038161003816119886 (119905) sin( 12 + cos 119905 + cosradic2119905)

100381610038161003816100381610038161003816100381610038162


2 + cos 119905 + cosradic2119905)100381610038161003816100381610038161003816100381610038162

sdot (10038171003817100381710038171199091 minus 1199092100381710038171003817100381722 + 10038171003817100381710038171199101 minus 1199102100381710038171003817100381722)

(146)

that is10038171003817100381710038171198651 (119905 1199091 1199101) minus 1198651 (119905 1199092 1199102)10038171003817100381710038172


(147)



(148)

And

10038171003817100381710038171198652 (119905 119909 119910)100381710038171003817100381722 le int1205870]2119890minus2|119905| 1003816100381610038161003816119909 (119904) + sin 119910 (119904)1003816100381610038161003816 d119904

le ]2119890minus2|119905| (11990922 + 1003817100381710038171003817119910100381710038171003817100381722) (149)

that is10038171003817100381710038171198652 (119905 119909 119910)10038171003817100381710038172 le ]119890minus|119905| (1199092 + 100381710038171003817100381711991010038171003817100381710038172)





119871 = 120583 119886infin Φ (119903) = 119903120573 (119905) = ]119890minus|119905|

1205881 = 11205882 le ]

(152)




(Theorem 24) or

120583119862119872119886 120587 10038161003816100381610038161199011003816100381610038161003816minus1120572120572 sin (120587120572) + 119862119872] lt 12 (154)


max 120583119886 (119904) ]119890minus|119905| d119904 (155)




Acknowledgments


References

























































































Journal of












Journal of







Volume 2018






Volume 2018
























































































Journal of












Journal of







Volume 2018






Volume 2018




















































Journal of












Journal of







Volume 2018






Volume 2018


















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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