Research Article Existence of Mild Solutions for Impulsive...

Research ArticleExistence of Mild Solutions for Impulsive Fractional StochasticDifferential Inclusions with State-Dependent Delay

Toufik Guendouzi and Ouahiba Benzatout

Laboratory of Stochastic Models Statistic and Applications Tahar Moulay University PO Box 138 En-Nasr 20000 Saida Algeria

Correspondence should be addressed to Toufik Guendouzi tfguendouzigmailcom

Received 10 December 2013 Accepted 2 January 2014 Published 17 February 2014

Academic Editors Z Guo and J Zhu

Copyright copy 2014 T Guendouzi and O Benzatout This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We study the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with state-dependentdelay Sufficient conditions for the existence of solutions are derived by using the nonlinear alternative of Leray-Schauder type formultivalued maps due to OrsquoRegan An example is given to illustrate the theory

1 Introduction

During the past two decades fractional differential equationshave gained considerable importance due to their applicationin various sciences such as physics mechanics and engi-neering [1ndash3] There has been a great deal of interest in thesolutions of fractional differential equations in analytical andnumerical senses One can see themonographs of Kilbas et al[2] Miller and Ross [4] Podlubny [5] and Lakshmikanthamet al [6] and the survey of Agarwal et al [7 8]

To study the theory of abstract differential equations withfractional derivatives in infinite dimensional spaces the firststep is how to introduce new concepts of mild solutionsA pioneering work has been reported by El-Borai [9 10]Very recently Hernandez et al [11] showed that some recentpapers of fractional differential equations in Banach spaceswere incorrect and used another approach to treat abstractequations with fractional derivatives based on the well-developed theory of resolvent operators for integral equa-tions MoreoverWang and Zhou [12] Zhou and Jiao [13] alsointroduced a suitable definition of mild solutions based onLaplace transform and probability density functions

On the other hand the theory of impulsive differentialequations or inclusions has become an active area of inves-tigation due to its applications in fields such as mechanicselectrical engineering medicine biology and ecology Onecan refer to [14 15] and the references therein Recentlythe problems of existence of solutions and controllability of

impulsive differential equations and differential inclusionshave been extensively studied [16 17] Benedetti in [18]proved an existence result for impulsive functional differen-tial inclusions in Banach spaces Obukhovskii and Yao [19]considered local and global existence results for semilinearfunctional differential inclusions with infinite delay andimpulse characteristics in a Banach space Some existenceresults were obtained for certain classes of functional dif-ferential equations and inclusions in Banach spaces underassumption that the linear part generates an compact semi-group (see eg [20ndash22]) The existence results of impulsivedifferential equations and inclusions have been generalizedto stochastic differential equations with impulsive conditions[23 24] and for stochastic impulsive differential inclusions[25ndash27]

We would like to mention that the impulsive effectsalso widely exist in fractional stochastic differential systems[28ndash30] and it is important and necessary to discuss thequalitative properties for stochastic fractional equations withimpulsive perturbations with state-dependent delay How-ever to the authorsrsquo knowledge no result has been reportedon the existence problem of impulsive fractional stochasticdifferential inclusionswith state-dependent delay and the aimof this paper is to fill this gap

Motivated by this consideration in this paper we will dis-cuss the existence of mild solutions for a class of impul-sive fractional stochastic differential inclusions with state-dependent delay in Hilbert spaces Specifically sufficient

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2014 Article ID 981714 13 pageshttpdxdoiorg1011552014981714

2 Chinese Journal of Mathematics

conditions for the existence are given by means of thenonlinear alternative of Leray-Schauder type for multivaluedmaps due to OrsquoRegan

2 Preliminaries and Basic Properties

In this section we provide definitions lemmas and notationsnecessary to establish our main results Throughout thispaper we use the following notations Let (ΩFP) be acomplete probability space equipped with a normal filtrationF

119905 119905 isin 119869 = [0 119887] satisfying the usual conditions (ie right

continuous and F0containing all P-null sets) We consider

two real separable Hilbert spaces H K with inner product(sdot sdot)H (sdot sdot)K and norm sdot H sdot K Let 119908 = (119908

119905)119905ge0

bea 119876-Wiener process defined on (ΩF

119887P) with the linear

bounded covariance operator 119876 such that Tr(119876) lt infinAssume that there exists a complete orthonormal system119890

119896119896ge1

in K a bounded sequence of nonnegative real num-bers 120582

119896 such that 119876119890

119896= 120582

119896119890119896 119896 = 1 2 and a sequence

120573119896119896ge1

of independent Brownian motions such that

(119908 (119905) 119890)K =

infin

sum119896=1

radic120582119896(119890

119896 119890)

K120573119896(119905) 119890 isin K 119905 isin [0 119887]

(1)

and F119905= F119908

119905 where F119908

119905is the sigma algebra generated

by 119908(119904) 0 le 119904 le 119905 Let 119871(KH) denote the space of allbounded linear operators from K to H equipped with theusual operator norm sdot

119871(KH) For 120595 isin 119871(KH) we define

100381710038171003817100381712059510038171003817100381710038172

119876= Tr (120595119876120595lowast

) =

infin

sum119896=1

1003817100381710038171003817100381710038171003817radic120582

119896120595119890

119896

1003817100381710038171003817100381710038171003817

2

(2)

If 1205952119876lt infin then 120595 is called a 119876-Hilbert-Schmidt operator

Let 119871119876(KH) denote the space of all 119876-Hilbert-Schmidt

operators 120595 The completion 119871119876(KH) of 119871(KH) with

respect to the topology is induced by the norm sdot 119876where

1205952

119876= (120595 120595) is a Hilbert space with the above norm topol-

ogy Let 1198712(ΩH) be a Banach space of all strongly

measurable square integrable H-valued random variablesequipped with the norm 119909(sdot)

1198712

= (E119909(sdot 120596)2) where E(sdot)

denote the expectation with respect to the measure PLet C(119869 119871

2(ΩH)) be the Banach space of all continu-

ous maps from 119869 to 1198712(ΩH) satisfying the condition

sup0le119905le119887

E119909(119905)2lt infin Let 1198710

2(ΩH) denote the family of all

F0-measurableH-valued random variables 119909(0)The purpose of this paper is to investigate the existence

of mild solutions for a class of impulsive fractional stochasticdifferential inclusions with state-dependent delay of the form

119863120572

119905119909 (119905) minus 119860119909 (119905) isin 119891 (119905 119909

120588(119905119909119905)) + Σ (119905 119909

120588(119905119909119905))119889119908 (119905)

119889119905

119905 isin 119869 = [0 119887] 119905 = 119905119896 119896 = 1 119898

Δ119909 (119905119896) = 119868

119896(119909

119905119896

) 119896 = 1 119898

1199090= 120601 isin B

(3)

where 119863120572

119905is the Caputo fractional derivative of order 120572 0 lt

120572 lt 1 119909(sdot) takes the value in the separable Hilbert space Hand 119860 D(119860) sub H rarr H is the generator of an 120572-resolventoperator family 119878

120572(119905) 119905 ge 0The history 119909

119905 (minusinfin 0] rarr H

119909119905(120579) = 119909(119905 + 120579) 120579 le 0 belongs to an abstract phase spaceB

defined axiomatically 119891 Σ 120588 and 119868119896 119896 = 1 119898 are given

functions to be specified later Here 0 = 1199050lt 119905

1lt sdot sdot sdot lt 119905

119898lt

119905119898+1

= 119887Δ119909(119905119896) = 119909(119905

+

119896)minus119909(119905

minus

119896) 119909(119905+

119896) = lim

ℎrarr0119909(119905

119896+ℎ) and

119909(119905minus119896) = lim

ℎrarr0119909(119905

119896minus ℎ) represent the right and left limits of

119909(119905) at 119905 = 119905119896 respectively The initial data 120601 = 120601(119905) 119905 isin

(minusinfin 0] is an F0-measurable B-valued random variable

independent of 119908 with finite second momentsRecall the following known definitions For more details

see [2]

Definition 1 The fractional integral of order 120572 with the lowerlimit 0 for a function 119891 is defined as

119868120572

119905119891 (119905) =

1

Γ (120572)int119905

0

119891 (119904)

(119905 minus 119904)1minus120572

119889119904 119905 gt 0 120572 gt 0 (4)

provided the right-hand side is pointwise defined on [0infin)where Γ is the gamma function

Definition 2 Riemann-Liouville derivative of order 120572 withlower limit zero for a function119891 [0infin) rarr R can bewrittenas

119871119863

120572119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int119905

0

119891 (119904)

(119905 minus 119904)120572+1minus119899

119889119904

119905 gt 0 119899 minus 1 lt 120572 lt 119899

(5)

Definition 3 The Caputo derivative of order 120572 for a function119891 [0infin) rarr R can be written as

119888119863

120572119891 (119905) =

119871119863

120572(119891 (119905) minus

119899minus1

sum119896=0

119905119896

119896119891119896(0))

119905 gt 0 119899 minus 1 lt 120572 lt 119899

(6)

If 119891(119905) isin 119862119899[0infin) then

119888119863

120572119891 (119905) =

1

Γ (119899 minus 120572)int119905

0

(119905 minus 119904)119899minus120572minus1

119891119899(119904) 119889119904 = 119868

119899minus120572119891119899(119904)

119905 gt 0 119899 minus 1 lt 120572 lt 119899

(7)

Obviously theCaputo derivative of a constant is equal to zeroThe Laplace transform of the caputo derivative of order 120572 gt 0is given as

119871 119888119863

120572119891 (119905) 119904 = 119904

120572119891 (119904) minus

119899minus1

sum119896=0

119904120572minus119896minus1

119891(119896)(0)

119899 minus 1 le 120572 lt 119899

(8)

Chinese Journal of Mathematics 3

Definition 4 (see [31]) A closed and linear operator 119860 is saidto be sectorial if there are constants 120596 isin R 120579 isin [1205872 120587]119872 gt 0 such that the following two conditions are satisfied

(i) 120588(119860) sub Ψ120579120596

= 120582 isin C 120582 = 120596 | arg (120582 minus 120596)| lt 120579

(ii) 119877(120582 119860) = (120582 minus 119860)minus1 le 119872|120582 minus 120596| 120582 isin Ψ120579120596

Definition 5 (see [30]) Let 119860 be a closed and linear operatorwith the domain119863(119860) defined in a Banach space119883 Let 120588(119860)be the resolvent set of 119860 We say that 119860 is the generator ofan 120572-resolvent family if there exist 120596 ge 0 and a stronglycontinuous function 119878

120572 R

+rarr 119871(119883) where 119871(119883) is a

Banach space of all bounded linear operators from 119883 to 119883and the corresponding norm is denoted by sdot such that120582

120572 Re 120582 gt 120596 sub 120588(119860) and

(120582120572119868 minus 119860)

minus1

119909 = intinfin

0

119890120582119905119878120572(119905) 119909 119889119905 Re 120582 gt 120596 119909 isin 119883

(9)

where 119878120572(119905) is called the 120572-resolvent family generated by 119860

Definition 6 Let 119878120572be an 120572-resolvent operator family on

Banach space 119883 with generator 119860 Then the following asser-tions hold

(i) 119878120572(119905)D(119860) sub D(119860) and 119860119878

120572(119905)119909 = 119878

120572(119905)119860119909 for all

119909 isin D(119860) and 119905 ge 0(ii) for all 119909 isin 119883 119868120572

119905119878120572(119905)119909 isin D(119860) and 119878

120572(119905)119909 = 119909 +

119860119868120572119905119878120572(119905)119909 119905 ge 0

(iii) 119909 isin D(119860) and 119860119909 = 119909 if and only if 119878120572(119905)119909 = 119909 +

119868120572119905119860119878

120572(119905)119909 119905 ge 0

(iv) 119860 is closed densely defined

Definition 7 (see [30]) Let 119860 be a closed and linear operatorwith the domain119863(119860)defined in a Banach space119883 and120572 gt 0We say that 119860 is the generator of a solution operator if thereexist 120596 ge 0 and a strongly continuous function 119878

120572 R

+rarr

119871(119883) such that 120582120572 Re 120582 gt 120596 sub 120588(119860) and

120582120572minus1

(120582120572119868 minus 119860)

minus1

119909 = intinfin

0

119890120582119905119878120572(119905) 119909 119889119905

Re 120582 gt 120596 119909 isin 119883(10)

where 119878120572(119905) is called the solution operator generated by 119860

The concept of the solution operator is closely related tothe concept of a resolvent family For more details on 120572-resolvent family and solution operators we refer the readerto [2]

Definition 8 We say that a function 119909 [119886 119887] rarr H is anormalized piecewise continuous function on [119886 119887] if 119909 ispiecewise continuous and left continuous on (119886 119887]

We denote by PC([119886 119887]H) the space formed by nor-malized piecewise continuous F

119905-adapted measurable pro-

cesses from [119886 119887] into H In particular we introduce thespace PC formed by F

119905-adapted measurable H-valued

stochastic processes 119909(119905) 119905 isin [0 119887] such that 119909 is continu-ous at 119905 = 119905

119896 119909(119905

119896) = 119909(119905minus

119896) and 119909(119905+

119896) exists for 119896 = 1 119898

In this paper we assume that PC is endowed with thenorm 119909PC = (sup

0le119905le119887E119909(119905)

2)12 Then (PC sdot PC) is

a Banach space [32]We denote by 119909

119896isin C([119905

119896 119905

119896+1] 119871

2(ΩH)) 119896 = 0 119898

the function given by

119909119896(119905) =

119909 (119905) for 119905 isin (119905119896 119905

119896+1]

119909 (119905+119896) for 119905 = 119905

119896

(11)

Moreover for 119861 sube PC we denote by 119861119896 119896 = 0 119898 the

set 119861119896= 119909

119896 119909 isin 119861 It is proved in [32] that 119861 sube PC is

relatively compact inPC if and only if the set119861119896is relatively

compact in C([119905119896 119905

119896+1] 119871

2(ΩH)) for every 119896 = 0 119898

The notation 119861119903(119909H) stands for the closed ball with center

at 119909 and radius 119903 gt 0 inHThroughout this paper we assume that the phase space

(B sdot B) is a seminormed linear space of F0-measurable

functions mapping (minusinfin 0] intoH and satisfying the follow-ing fundamental axioms [33]

(i) If 119909 (minusinfin 120591 + 119887] rarr H 119887 gt 0 120591 isin R is continuouson [120591 120591 + 119887) and 119909

120591isin B then for every 119905 isin [120591 120591 + 119887)

the following conditions hold

(a) 119909119905is inB

(b) 119909(119905) le 119867119909119905B

(c) 119909119905B le 119870(119905 minus 120591) sup119909(119904) 120591 le 119904 le 119905 +

119873(119905minus120591)119909120591B where119867 gt 0 is a constant119870119873

[0infin) rarr [1infin) 119870 is continuous119873 is locallybounded and119867119870119873 are independent of 119909(sdot)

(ii) For the function 119909(sdot) in (i) the function 119905 rarr 119909119905is

continuous from [120591 120591 + 119887] intoB(iii) The spaceB is complete

The next result is a consequence of the phase space axiomsThe reader can refer to [34] for the proof

Lemma 9 Let 119909 (minusinfin 119887] rarr H be an F119905-adapted mea-

surable process such that the F0-adapted process 119909

0= 120601(119905) isin

11987102(ΩB) and 119909|

119869isin PC(119869H) Then

10038171003817100381710038171199091199041003817100381710038171003817B le 119873

119887E10038171003817100381710038171206011003817100381710038171003817119861 + 119870119887

sup0le119904le119887

E 119909 (119904) (12)

where 119870119887= sup119870(119905) 0 le 119905 le 119887 and 119873

119887= 119873(119905) 0 le 119905 le

119887

In what follows we use the notationsP(H) for the familyof all nonempty subsets ofH and denote

P119888119897(H) = 119884 isin P (H) 119884 is closed

P119887119889(H) = 119884 isin P (H) 119884 is bounded

P119888V (H) = 119884 isin P (H) 119884 is convex


P119888119901(H) = 119884 isin P (H) 119884 is compact

P119888119889(H) = 119884 isin P (H) 119884 is compact-acyclic

(13)

Now we briefly introduce some facts onmultivalued analysisFor details one can see [35]

A multivalued map 119866 H rarr P(H) is convex (closed)valued if119866(119909) is convex (closed) for all 119909 isin H119866 is boundedon bounded sets if 119866(119861) = ⋃

119909isin119861119866(119909) is bounded in H for

any bounded set 119861 of H that is sup119909isin119861

sup119910 isin 119866(119909) lt

infinFor 119909 isin H and 119884119885 isin P

119887119889119888119897(H) we denote by 119889(119909 119884) =

inf119909 minus 119910 119910 isin 119884 and 120581(119884 119885) = sup119886isin119884

119889(119886 119885) and theHausdorff metric 119867

119889 P

119887119889119888119897(H) times P

119887119889119888119897(H) rarr R

+by

119867119889(119860 119861) = max120581(119860 119861) 120581(119861 119860)A multivalued map 119866 is called upper semicontinuous

(usc for short) on H if for each 1199090isin H the set 119866(119909

0) is

a nonempty closed subset ofH and if for each open set 119861 ofH containing119866(119909

0) there exists an open neighborhood of

1199090such that 119866() sube 119861119866 is said to be completely continuous if 119866(119861) is relatively

compact for every bounded subset 119861 sube HIf the multivalued map 119866 is completely continuous with

nonempty compact values then119866 is usc if and only if119866 hasa closed graph that is 119909

119899rarr 119909

lowast 119910

119899rarr 119910

lowast 119910

119899isin 119866(119909

119899) imply

119910lowastisin 119866(119909

lowast)

A multivalued map 119866 119869 rarr P119887119889119888119897119888V(H) is said to be

measurable if for each 119909 isin H the function 119905 rarr 119889(119909 119866(119905)) ismeasurable function on 119869

Definition 10 (see [35]) Let 119866 H rarr P119887119889119888119897

(H) be a multi-valued map Then 119866 is called a multivalued contraction ifthere exists a constant 120599 isin (0 1) such that for each 119909 119910 isin H

119867119889(119866 (119909) minus 119866 (119910)) le 120599

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (14)

The constant 120599 is called a contraction constant of 119866

Next wemention the statement of a nonlinear alternativeof Leray-Schauder type formultivaluedmaps due toOrsquoRegan

Lemma 11 (see [36]) LetH be aHilbert space with119881 an openconvex subset ofH and 119910 isin H Suppose that

(a) Φ 119881 rarr P119888119889(H) has closed graph

(b) Φ 119881 rarr P119888119889(H) is a condensing map with Φ(119881) a

subset of a bounded set inH hold Then either

(i) Φ has a fixed point in 119881 or(ii) there exist 119910 isin 120597119881 and 120582 isin (0 1) with 119910 isin

120582Φ(119910) + (1 minus 120582)1199100

3 The Mild Solution and Existence

Before stating and proving the main result we present thedefinition of the mild solution to the system (3)ndash(3) based onthe paper [30 31]

Let SΣ120595

= 120590 isin 1198712(119871(KH)) 120590(119905) isin Σ(119905 120595) for ae 119905 isin119869 be the set of selections of Σ for each 120595 isin B and 119909(119905+

119896) =

119909(119905minus

119896) + 119868

119896(119909

119905119896

) 119896 = 1 2 119898

Definition 12 An F119905-adapted stochastic process 119909 (minusinfin

119887] rarr H is called a mild solution of the system (3) if 1199090= 120601

119909120588(119905119909119905)isin B for every 119905 isin 119869Δ119909(119905

119896) = 119868

119896(119909

119905119896

) 119896 = 1 119898 therestriction of 119909(sdot) to the interval (119905

119896 119905

119896+1] (119896 = 0 1 119898) is

continuous and there exists 120590 isin SΣ119909120588

such that 119909(119905) satisfiesthe following integral equation

119909 (119905) =

119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119909 (119905minus

1) + 119868

1(119909

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119909 (119905minus

119898) + 119868

119898(119909

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(15)

where 119879120572(119905) = (12120587119894) int

119861119903

119890120582119905120582120572minus1(120582120572 minus 119860)minus1119889120582 119878

120572(119905) =

(12120587119894) int119861119903

119890120582119905(120582120572 minus 119860)minus1119889120582 and 119861

119903denotes the Bromwich

path 119878120572(119905) is called the 120572-resolvent family and 119879

120572(119905) is the

solution operator generated by 119860

The following result on the operator 119878120572(119905) appeared and

is proved in [31]

Theorem 13 If 120572 isin (0 1) and 119860 isin A120572(1205790 120596

0) is a sectorial

operator then for any 119909 isin H and 119905 gt 0 one has

1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 119905 gt 0 120596 gt 120596

0 (16)

where 119862 is a constant depending only on 120579 and 120596

In order to establish the results we first assume that thefunction 120588 is continuous from 119869 times B into (minusinfin 119887] and weimpose the following additional hypotheses

(H1) If 120572 isin (0 1) and 119860 isin A120572(1205790 120596

0) is a sectorial

operator then for 119909 isin H and 119905 gt 0

1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119872119890

120596119905

1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 120596 gt 120596

0

(17)


If 119879= sup

0le119905le119887119879

120572(119905) and

119878= sup

0le119905le119887119862119890120596119905(1 + 1199051minus120572)

we have

1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119879

1003817100381710038171003817119878120572 (119905)

1003817100381710038171003817 le 119905120572minus1

119878

(for more details see [31]) (18)

(H2)The function 119905 rarr 120601119905is continuous from Z(120588minus) =

120588(119904 120595) le 0 (119904 120595) isin 119869 times B to B and there exists acontinuous and bounded function W120601 Z(120588minus) rarr (0infin)

such that 120601119905B le W120601(119905)E120601B for each 119905 isin Z(120588minus)

(H3)The multivalued map 119865 119869 timesB rarr P119887119889119888119897119888V(119871(K

H)) is Carathedory that is

(i) 119905 997891rarr 119865(119905 120595) is measurable for each 120595 isin B

(ii) 120595 997891rarr 119865(119905 120595) is upper semicontinuous (usc) foralmost all 119905 isin 119869

and for each fixed 120595 isin B the set SΣ120595

of selections of Σ isnonempty

(H4)There exists a positive integrable function ] isin 1198711([0119887]R+) such that

lim sup1205952

Brarrinfin

1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

] (119905) 100381710038171003817100381712059510038171003817100381710038172

B

= Λ (19)

uniformly in 119905 isin 119869 for a nonnegative constant Λ where

1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

= sup E1205902 120590 isin Σ (119905 120595) (20)

(H5) The function 119891 119869 times B rarr H is continuous andthere exists119872

119891gt 0 such that

E1003817100381710038171003817119891 (119905 1205951

) minus 119891 (119905 1205952)10038171003817100381710038172

le 119872119891

10038171003817100381710038171205951minus 120595

2

10038171003817100381710038172

B

119905 isin 119869 1205951 120595

2isin B

E1003817100381710038171003817119891 (119905 120595)

10038171003817100381710038172

le 119872119891(1 +

100381710038171003817100381712059510038171003817100381710038172

B) 119905 isin 119869 120595 isin B

(21)

(H6) The functions 119868119896

B rarr H are completelycontinuous and there exist constants 120576

119896such that

lim sup1205952

Brarrinfin

1003817100381710038171003817119868119896 (120595)10038171003817100381710038172

100381710038171003817100381712059510038171003817100381710038172

B

= 120576119896 (22)

for every 120595 isin B 119896 = 1 2 119898

Remark 14 The condition (H2) is frequently verified bycontinuous and bounded functions For more details see forinstance [34] (Proposition 711)

The following lemma is required for the main result Thereader can refer to [37 38] for the lemma and to [32] for moredetails about the proof

Lemma 15 Let 119909 (minusinfin 119887] rarr H such that 1199090= 120601 and

119909|119869isin PC(119869H) If (H2) holds then

10038171003817100381710038171199091199041003817100381710038171003817B le (119873

119887+W

120601

0)10038171003817100381710038171206011003817100381710038171003817B

+ 119870119887sup 119909 (120579) 120579 isin [0max 0 119904]

119904 isin Z (120588minus) cup 119869

(23)

whereW120601

0= sup

119905isinZ(120588minus)W120601(119905)

Lemma 16 (see [39]) Let 119869 be a compact interval and H aHilbert space Let Σ be a multivalued map satisfying (H3) andΓ a linear continuous operator from 1198712(119869H) toC(119869H)Thenthe operator Γ ∘ S

120590 C(119869H) rarr P

119888119901119888V(C(119869H)) is a closedgraph inC(119869H) timesC(119869H)

Theorem 17 Assume that (H1)ndash(H6) hold and 1199090isin 1198710

2(Ω

H) with 120588(119905 120595) le 119905 for every (119905 120595) isin 119869timesB Then the problem(3) has a mild solution on 119869 provided that

max1le119896le119898

62

119879(1 + 2119870

2

119887120576119896) lt 1 (24)

Proof Consider the space BPC = 119909 (minusinfin 119887) rarr H

1199090= 0 119909|

119869isin PC endowed with the uniform convergence

topology and define the multivalued map Φ BPC rarr

P(BPC) by Φ119909 = 119911 isin BPC such that

119911 (119905) =

0 119905 isin (minusinfin 0]

119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119911 (119905

minus

1) + 119868

1(119911

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(25)

where 120590 isin SΣ119911120588

= 120590 isin 1198712(119871(KH)) 120590(119905) isin Σ(119905

119911120588(119905119911119905)) for ae 119905 isin 119869 and 119911 (minusinfin 0] rarr H such that 119911

0= 120601

and 119911 = 119909 on 119869We shall show that Φ has a fixed point which is then a

mild solution for the problem (3) To this end we show that


Φ satisfies all the conditions of Lemma 11 For the sake ofconvenience we divide the proof into several steps

Step 1 We show that there exists an open set 119881 sube BPC with119909 isin 120582Φ119909 for 120582 isin (0 1) and 119909 notin 120597119881 Let 120582 isin (0 1) and 119909 isin

120582Φ119909 then there exists an 120590 isin SΣ119911120588

such that

119909 (119905) =

120582119879120572(119905) 120601 (0)

+120582int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

120582119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)]

+120582int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

120582119879

120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+120582int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(26)

From assumption (H4) it follows that there exist two non-negative real numbers 119888

1and 119888

2such that for any 120595 isin B and

119905 isin 1198691003817100381710038171003817Σ (119905 120595)

10038171003817100381710038172

le 1198881] (119905) + 119888

2] (119905) 1003817100381710038171003817120595

10038171003817100381710038172

B (27)

From assumption (H6) we conclude that there exist positiveconstants 119886

119896(119896 = 1 119898) 119888

3such that for 1205952B gt 119888

3

E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

le (120576119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B

max1le119896le119898

62

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896)) lt 1

(28)

Let

1198621= 120595

100381710038171003817100381712059510038171003817100381710038172

Ble 119888

3 119862

2= 120595

100381710038171003817100381712059510038171003817100381710038172

Bgt 119888

3

1198623= max E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

120595 isin 1198621

(29)

We have

E1003817100381710038171003817119868119896(120595)

10038171003817100381710038172

le 1198623+ (120576

119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B (30)

By assumption (H5) (27) and (30) we have for 119905 isin [0 1199051]

E119909 (119905)2

le 3E1003817100381710038171003817119879120572 (119905) 120601 (0)

10038171003817100381710038172

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 32

119879E1003817100381710038171003817120601 (0)

10038171003817100381710038172

+ 32

119878int119905

0

(119905 minus 119904)120572minus1

119889119904

times int119905

0

(119905 minus 119904)120572minus1

119872119891(1 +

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) int

119905

0

(119905 minus 119904)2(120572minus1)

[1198881] (119904) + 119888

2] (119904)

times10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B] 119889119904

le 32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878119872

119891

119887120572

120572

times int119905

0

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(31)

Similarly for any 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

E119909(119905)2

le 3E10038171003817100381710038171003817119879120572(119905 minus 119905

119896) [119911 (119905

minus

119896) + 119868

119896(119911

119905119896

)]10038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119879[E1003817100381710038171003817119911(119905

minus

119896)10038171003817100381710038172

+ 1198623+ (120576

119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

119887120572

120572int119905

119905119896

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(32)

Then for all 119905 isin [0 119887] we have

E119909 (119905)2

le 119872lowast+ 6

2

119879[E1003817100381710038171003817119911 (119905

minus

119896)10038171003817100381710038172

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

1198872120572

2120572int119905

0

(119905 minus 119904)120572minus110038171003817100381710038171003817

119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(33)


where

119872lowast= max32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878Tr (119876) 119888

1

times int119905

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904 62

1198791198623

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(34)

By Lemmas 9 and 15 it follows that 120588 (119904 119911119904) le 119904 119904 isin [0 119905]

119905 isin [0 119887] and10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887sup0le119904le119887

E119909 (119904)2

(35)

For each 119905 isin [0 119887] we have

E119909 (119905)2

le 119872lowast+ [6

2

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896))]

times sup0le119905le119887

E119909 (119905)2+ 6

2

119878119872

119891119870

2

119887

1198872120572

2120572

times int119905

0

(119905 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+ 62

119878119870

2

119887Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904)


E119909 (120591)2119889119904

(36)

where119872

lowast= 119872

lowast+ 6

2

119879[119862

3+ (120576

119896+ 119886

119896) 119862

4]

+ 32

119878119872

1198911198624

1198873120572

3120572+ 3

2

1198781198624Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904

1198624= 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

(37)

Since 119897 = max1le119896le119898

62

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 we have

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897+62

119878119872

1198911198702

1198871198872120572

(1 minus 119897) 2120572

times int119887

0

(119887 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+62

1198781198702

119887Tr (119876) 119888

1

1 minus 119897

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) sup

0le120591le119904

E119909 (120591)2119889119904

(38)

Applying Gronwallrsquos inequality we get

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)

(39)

Therefore

1199092

PC

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)ltinfin

(40)

Then there exists 119903lowast such that 1199092PC = 119903lowast Set 119881 = 119909 isin

BPC 1199092

PC lt 119903lowast Thus from the choice of 119881 there isno 119909 isin 120597119881 such that 119909 isin 120582120601119909 for 120582 isin (0 1)

Step 2 Φ has a closed graphLet 119909(119899) rarr 119909lowast 119911

119899isin Φ119909(119899) 119909(119899) isin 119881 = 119861

119903lowast(0BPC)

and 119911119899rarr 119911

lowast It is easy to see that (119911(119899))

119904rarr 119911lowast

119904uniformly

for 119904 isin (minusinfin 119887] as 119899 rarr infin We need to show that 119911lowastisin Φ119909lowast

Now 119911119899isin Φ119909(119899) means that there exists 120590

119899isin S

Σ119911(119899)

120588

suchthat for each 119905 isin [0 119905

1]

119911119899(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(41)

Wemust show that there exists 120590lowastisin S

Σ119909lowast

120588

such that for each119905 isin [0 119905

1]

119911lowast(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(42)

SetΘ119899(119905) = 119911

119899(119905) minus 119879

120572120601(0) minus int

119905

0119878120572(119905 minus 119904)119891(119904 (119911(119899))

120588(119904119911(119899)

119904))119889119904 minus

int119905

0119878120572(119905 minus 119904)120590

119899(119904)119889119908(119904) andΘ

lowast(119905) = 119911

lowast(119905) minus119879

120572120601(0) minus int

119905

0119878120572(119905 minus

119904)119891(119904 (119911lowast)120588(119904119911lowast

119904))119889119904 minus int

119905

0119878120572(119905 minus 119904)120590

lowast(119904)119889119908(119904)

We have for every 119905 isin [0 1199051]

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172

PC997888rarr 0 as 119899 997888rarr infin (43)

Consider the linear continuous operator Γ 1198712([0 1199051]H) rarr

C([0 1199051]H) defined by

Γ (120590) (119905) = int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (44)


From Lemma 16 and the definition of Γ it follows that Γ ∘SΣ

is a closed graph operator and for every 119905 isin [0 1199051] Θ

119899(119905) isin

Γ(SΣ119911(119899))

Since (119911(119899))119904rarr 119911lowast

119904and Γ ∘S

Σis a closed graph operator

then there exists 120590lowastisin S

Σ119911lowast

120588

such that for every 119905 isin [0 1199051]

119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)

We must show that there exists 120590lowastisin S

Σ119911lowast

120588

such that forevery 119905 isin (119905

119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911

lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)

Now for every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we consider

the linear continuous operator Γ 1198712((119905119896 119905

119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)

From Lemma 16 it follows that Γ ∘ SΣis a closed graph

operator and for every 119905 isin (119905119896 119905

119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588

such that for every 119905 isin (119905119896 119905

119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)

Hence Φ has a closed graph

Step 3 We show that the operator Φ is condensing Let Φ1

119881 rarr P(BPC) and Φ2 119881 rarr P(BPC) be defined by

Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)

We first show thatΦ1is a contractionwhileΦ

2is a completely

continuous operator

Claim 1Φ1is a contraction onBPC Let119906 V isin BPC From

(H6) Lemmas 9 and 15 we have for every 119905 isin (119905119896 119905

119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887

times sup E119906(120591) minus V(120591)2 0 le 120591 le 119905 ]

le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2] since 119906 = 119906 V = V on 119869

= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)

Thus for all 119905 isin [0 119887] we have

1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 HenceΦ

1

is a contraction onBPC

Claim 2Φ2is convex for each 119909 isin 119881 Indeed if 1199111

2 1199112

2belong

to Φ2119909 then there exist 120590

1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)

Let 0 le 120582 le 1 For each 119905 isin [0 1199051] we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)

Similarly for 0 le 120582 le 1 and any 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588

is convex (because Σ has convex values) we have(1205821199111

2+ (1 minus 120582)1199112

2) isin Φ

2119909

Claim 3 Φ2(119881) is completely continuous First We need to

show that Φ2(119881) is equicontinuous Let 119909 isin 119881 Then from

Lemmas 9 and 15 it follows that

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ

2119909 there exists

120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572

times int1205911minus120598

0

1003817100381710038171003817119878120572 (1205912 minus 119904) minus 119878120572 (1205911 minus 119904)10038171003817100381710038172

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0

1003817100381710038171003817119878120572minus1 (1205912 minus 119904) minus 119878120572 (1205911 minus 119904)10038171003817100381710038172

times (1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)

Similarly for any 1205911 120591

2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896

1003817100381710038171003817119878120572(1205912 minus 119904) minus 119878120572 (1205911 minus 119904)10038171003817100381710038172

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)

Therefore from the above inequalities for 120598 sufficiently smallthe right-hand side of E119911

2(120591

2) minus 119911

2(120591

1)

2 tends to zero as1205912minus 120591

1rarr 0 since 119868

119896 119896 = 1 119898 are completely

continuous in H and the compactness of 119878120572(119905) for 119905 gt 0 (119878

120572

is generated by the dense operator 119860) implies the continuityin the uniform operator topology Thus the set Φ

2119909 119909 isin 119881

is equicontinuousSecondwe show that120601

2(119881) is relatively compact for every

119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905

1be fixed and let 120598 be a real number

satisfying 0 lt 120598 lt 119905 For 119909 isin 119881 we define

1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588

Using the compactness of 119879120572(119905) and 119878

120572(119905)

for 119905 gt 0 we deduce that the set 119880120598(119905) = 119911

2120598(119905) 119909 isin 119881 is

relatively compact inH for every 120598 0 lt 120598 lt 119905 Moreover forevery 119909 isin 119881 we have

E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1

be fixed and let 120598 be a real number satisfying 0 lt 120598 lt 119905For 119909 isin 119881 we define

1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588

From the compactness of 119879120572(119905) and 119878

120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)

The right-hand side of the above inequality tends to zero as120598 rarr 0 This implies that there are relatively compact setsarbitrarily close to the set 119880(119905) = 119911

1(119905) 119909 isin 119881 Hence

119880(119905) is relatively compact inH ByArzela-Ascoli theoremweconclude that the operatorΦ

2(119881) is completely continuous

As a consequence of the above Claims 1ndash3 we concludethatΦ is a condensingmap All of the conditions of Lemma 11are satisfied we deduce that Φ has a fixed point 119909 in BPCwhich is a mild solution of the problem (3)

4 An Example

To apply our abstract results we consider the followingimpulsive fractional stochastic partial differential inclusionswith state-dependent delay of the form

119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)

where 120573(119905) is a standard cylindrical Wiener process in Hdefined on a stochastic space (ΩF F

119905P) 119863120572

119905is the

Caputo fractional derivative of order 0 lt 120572 lt 1 120601 iscontinuous and 0 lt 119905

1lt 119905


119898lt 119887 are prefixed

numbersLetH = 119871

2([0 120587])with the norm sdot Define119860 D(119860) sub

H rarr H by 119860119910 = 11991010158401015840 with the domain

D (119860) = 119910 isin H 119910 1199101015840 are absolutely continuous

11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)

Then 119860119910 = suminfin

119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =

radic(2120587) sin(119899119909) 119899 = 1 2 is the orthogonal set of eigenvec-tors of119860 It is well known that119860 is the infinitesimal generatorof an analytic semigroup (119879(119905))

119905ge0inH is given by

119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)

It follows from the above expressions that (119879(119905))119905ge0

is auniformly bounded compact semigroup so that 119877(120582 119860) =

(120582 minus 119860)minus1 is a compact operator for all 120582 in the resolvent set

of 119860119903 ge 0 119901 ge 1 and let 119911 (minusinfin 119903] rarr R be a nonnegative

measurable function which satisfies the conditions (H-5) and(H-6) in the terminology of Hino et al [34] Briefly thismeans that 119911 is locally integrable and there is a non-negativelocally bounded function ℎ on (minusinfin 0] such that 119911(120585 + 120591) leℎ(120585)119911(120591) for all 120591 le 0 and 120579 isin (minusinfin minus119903) 119873

120585 where

119873120585sube (minusinfin minus119903) is a set whose Lebesguersquos measure is zero Let

PC119903times1198712(119911H) be the set consisting of all classes of functions

120601 (minusinfin 0] rarr H such that 120601|[minus1199030]

isin PC([minus119903 0]H) 120601(sdot)is Lebesgue measurable on (minusinfin minus119903) and 119911120601119901 is Lebesgueintegrable on (minusinfin minus119903) The seminorm is given by

10038171003817100381710038171206011003817100381710038171003817B = sup

minus119903le120591le0

1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)

B = PC119903times1198712(119911H) satisfies the fundamental axioms given

in Section 2 When 119903 = 0 and 119901 = 2 we can take 119867 = 1

119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that

(i) the functions 120588119894 [0infin) rarr [0infin) 119894 = 1 2 are con-

tinuous(ii) the functions 120583

119894 R3 rarr R 119894 = 1 2 are continuous

with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12

lt infin(iii) the functions ]

119896 R rarr R 119896 = 1 2 119898 are con-

tinuous with 119871119896= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12

lt infin forevery 119896 = 1 2 119898

andBwill be the phase spacePC0times1198712(119911H) Set 120601(120579)(119909) =

120601(120579 119909) isin B Define 119891 [0 119887] timesB rarr H Σ [0 119887] timesB rarr

P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)

Thus 119891 Σ are bounded operators onB with 119891 le 1198971 Σ le

1198972and 119868

119896 le 119871

119896 119896 = 1 2 119898 Therefore the problem (4)

can be written in the abstract form of (3) All conditions ofTheorem 17 are now fulfilled so we deduce that the system(4) has a mild solution on [0 119887]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Hilfer Applications of Fractional Calculus in Physics WorldScientific River Edge NJ USA 2000

[2] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Science Amster-dam The Netherlands 2006

[3] V E Tarasov Fractional Dynamics Application of FractionalCalculus to Dynamics of Particales Fields and Media SpringerHeidelberg Germany 2010

[4] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[5] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[6] V Lakshmikantham S Leela and J Vasundhara DeviTheory ofFractionalDynamic Systems CambridgeAcademic CambridgeUK 2009

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

[8] R P Agarwal M Belmekki and M Benchohra ldquoA survey onsemilinear differential equations and inclusions involving Rie-mann-Liouville fractional derivativerdquo Advances in DifferenceEquations vol 2009 Article ID 918728 47 pages 2009

[9] M M El-Borai ldquoThe fundamental solutions for fractional evo-lution equations of parabolic typerdquo Journal of Applied Mathe-matics and Stochastic Analysis vol 3 Article ID 197211 2004

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

[11] E Hernandez D OrsquoRegan and K Balachandran ldquoOn recentdevelopments in the theory of abstract differential equationswith fractional derivativesrdquoNonlinearAnalysisTheoryMethodsand Applications vol 73 no 10 pp 3462ndash3471 2010

[12] J R Wang and Y Zhou ldquoA class of fractional evolution equa-tions and optimal controlsrdquo Nonlinear Analysis Real WorldApplications vol 12 pp 262ndash272 2011

[13] Y Zhou and F Jiao ldquoExistence of mild solutions for fractionalneutral evolution equationsrdquo Computers and Mathematics withApplications vol 59 no 3 pp 1063ndash1077 2010

[14] Y-K Chang A Anguraj and M Mallika Arjunan ldquoExistenceresults for impulsive neutral functional differential equationswith infinite delayrdquo Nonlinear Analysis Hybrid Systems vol 2no 1 pp 209ndash218 2008

[15] S K Ntouyas ldquoExistence results for impulsive partial neutralfunctional differential inclusionsrdquo Electronic Journal of Differ-ential Equations vol 30 pp 1ndash11 2005

[16] N Abada M Benchohra and H Hammouche ldquoExistence andcontrollability results for nondensely defined impulsive semi-linear functional differential inclusionsrdquo Journal of DifferentialEquations vol 246 no 10 pp 3834ndash3863 2009

[17] Y-K Chang and W-T Li ldquoExistence results for second orderimpulsive functional differential inclusionsrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 2 pp 477ndash4902005

[18] I Benedetti ldquoAn existence result for impulsive functional differ-ential inclusions in Banach spacesrdquo Discussiones MathematicaeDifferential Inclusions Control andOptimization vol 24 pp 13ndash30 2004

[19] V Obukhovskii and J-C Yao ldquoOn impulsive functional differ-ential inclusions with Hille-Yosida operators in Banach spacesrdquoNonlinear Analysis Theory Methods and Applications vol 73no 6 pp 1715ndash1728 2010

[20] N Abada R P Agarwal M Benchohra and H HammoucheldquoExistence results for nondensely defined impulsive semilinearfunctional differential equations with state-dependent delayrdquoAsian-European Journal of Mathematics vol 1 no 4 pp 449ndash468 2008

[21] N Abada M Benchohra and H Hammouche ldquoExistence andcontrollability results for impulsive partial functional differen-tial inclusionsrdquoNonlinear Analysis Theory Methods amp Applica-tions vol 69 pp 2892ndash2909 2008

[22] N Abada M Benchohra and H Hammouche ldquoNonlinearimpulsive partial functional differential inclusions with state-dependent delay and multivalued jumpsrdquo Nonlinear AnalysisHybrid Systems vol 4 no 4 pp 791ndash803 2010

[23] A Anguraj and A Vinodkumar ldquoExistence uniqueness andstability results of impulsive stochastic semilinear neutral func-tional differential equations with infinite delaysrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 67pp 1ndash13 2009

[24] L Hu and Y Ren ldquoExistence results for impulsive neutral sto-chastic functional integro-differential equations with infinitedelaysrdquo Acta Applicandae Mathematicae vol 111 no 3 pp 303ndash317 2010


[25] P Balasubramaniam and D Vinayagam ldquoExistence of solutionsof nonlinear neutral stochastic differential inclusions in aHilbert spacerdquo Stochastic Analysis and Applications vol 23 no1 pp 137ndash151 2005

[26] A Lin and L Hu ldquoExistence results for impulsive neutral sto-chastic functional integro-differential inclusions with nonlocalinitial conditionsrdquo Computers and Mathematics with Applica-tions vol 59 no 1 pp 64ndash73 2010

[27] Y Ren L Hu and R Sakthivel ldquoControllability of impulsiveneutral stochastic functional differential inclusions with infinitedelayrdquo Journal of Computational and Applied Mathematics vol235 no 8 pp 2603ndash2614 2011

[28] T Guendouzi ldquoExistence and controllability of fractional-orderimpulsive stochastic system with infinite delayrdquo DiscussionesMathematicae Differential Inclusions Control and Optimizationvol 33 no 1 2013

[29] T Guendouzi and K Mehdi ldquoExistence of mild solutions forimpulsive fractional stochastic equations with infinite delayrdquoMalaya Journal of Matematik vol 4 no 1 pp 30ndash43 2013

[30] R Sakthivel P Revathi and Y Ren ldquoExistence of solutions fornonlinear fractional stochastic differential equationsrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 81 pp 70ndash862013

[31] X-B Shu Y Lai and Y Chen ldquoThe existence of mild solutionsfor impulsive fractional partial differential equationsrdquo Nonlin-ear Analysis Theory Methods and Applications vol 74 no 5pp 2003ndash2011 2011

[32] Z Yan and H Zhang ldquoExistence of solutions to impulsive frac-tional partial neutral stochastic integro-differential inclusionswith state-dependent delayrdquo Electronic Journal of DifferentialEquations vol 81 pp 1ndash21 2013

[33] J K Hale and J Kato ldquoPhase spaces for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj vol 21 pp 11ndash41 1978

[34] Y Hino S Murakami and T Naito Functional DifferentialEquations with Infinite Delay vol 1473 of Lecture Notes inMathematicsSpringer Berlin Germany 1991

[35] K Deimling Multivalued Differential Equations De GruyterNew York NY USA 1992

[36] D OrsquoRegan ldquoNonlinear alternatives for multivalued maps withapplications to operator inclusions in abstract spacesrdquo Proceed-ings of the American Mathematical Society vol 127 no 12 pp3557ndash3564 1999

[37] R P Agarwal B De Andrade and G Siracusa ldquoOn fractionalintegro-differential equations with state-dependent delayrdquoComputers andMathematics with Applications vol 62 no 3 pp1143ndash1149 2011

[38] M Benchohra S Litimein and G M NrsquoGuerekata ldquoOn frac-tional integro-differential inclusionswith state-dependent delayin Banach spacesrdquo Applicable Analysis vol 92 no 2 pp 335ndash350 2013

[39] A Lasota and Z Opial ldquoAn application of the Kakutani-Ky-Fantheorem in the theory of ordinary differential equationsrdquo Bul-letin de lrsquoAcademie Polonaise des Sciences vol 13 pp 781ndash7861965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


conditions for the existence are given by means of thenonlinear alternative of Leray-Schauder type for multivaluedmaps due to OrsquoRegan

2 Preliminaries and Basic Properties

In this section we provide definitions lemmas and notationsnecessary to establish our main results Throughout thispaper we use the following notations Let (ΩFP) be acomplete probability space equipped with a normal filtrationF

119905 119905 isin 119869 = [0 119887] satisfying the usual conditions (ie right

continuous and F0containing all P-null sets) We consider

two real separable Hilbert spaces H K with inner product(sdot sdot)H (sdot sdot)K and norm sdot H sdot K Let 119908 = (119908

119905)119905ge0

bea 119876-Wiener process defined on (ΩF

119887P) with the linear

bounded covariance operator 119876 such that Tr(119876) lt infinAssume that there exists a complete orthonormal system119890

119896119896ge1

in K a bounded sequence of nonnegative real num-bers 120582

119896 such that 119876119890

119896= 120582

119896119890119896 119896 = 1 2 and a sequence

120573119896119896ge1

of independent Brownian motions such that

(119908 (119905) 119890)K =

infin

sum119896=1

radic120582119896(119890

119896 119890)

K120573119896(119905) 119890 isin K 119905 isin [0 119887]

(1)

and F119905= F119908

119905 where F119908

119905is the sigma algebra generated

by 119908(119904) 0 le 119904 le 119905 Let 119871(KH) denote the space of allbounded linear operators from K to H equipped with theusual operator norm sdot

119871(KH) For 120595 isin 119871(KH) we define

100381710038171003817100381712059510038171003817100381710038172

119876= Tr (120595119876120595lowast

) =

infin

sum119896=1

1003817100381710038171003817100381710038171003817radic120582

119896120595119890

119896

1003817100381710038171003817100381710038171003817

2

(2)

If 1205952119876lt infin then 120595 is called a 119876-Hilbert-Schmidt operator

Let 119871119876(KH) denote the space of all 119876-Hilbert-Schmidt

operators 120595 The completion 119871119876(KH) of 119871(KH) with

respect to the topology is induced by the norm sdot 119876where

1205952

119876= (120595 120595) is a Hilbert space with the above norm topol-

ogy Let 1198712(ΩH) be a Banach space of all strongly

measurable square integrable H-valued random variablesequipped with the norm 119909(sdot)

1198712

= (E119909(sdot 120596)2) where E(sdot)

denote the expectation with respect to the measure PLet C(119869 119871

2(ΩH)) be the Banach space of all continu-

ous maps from 119869 to 1198712(ΩH) satisfying the condition

sup0le119905le119887

E119909(119905)2lt infin Let 1198710

2(ΩH) denote the family of all

F0-measurableH-valued random variables 119909(0)The purpose of this paper is to investigate the existence

of mild solutions for a class of impulsive fractional stochasticdifferential inclusions with state-dependent delay of the form

119863120572

119905119909 (119905) minus 119860119909 (119905) isin 119891 (119905 119909

120588(119905119909119905)) + Σ (119905 119909

120588(119905119909119905))119889119908 (119905)

119889119905

119905 isin 119869 = [0 119887] 119905 = 119905119896 119896 = 1 119898

Δ119909 (119905119896) = 119868

119896(119909

119905119896

) 119896 = 1 119898

1199090= 120601 isin B

(3)

where 119863120572

119905is the Caputo fractional derivative of order 120572 0 lt

120572 lt 1 119909(sdot) takes the value in the separable Hilbert space Hand 119860 D(119860) sub H rarr H is the generator of an 120572-resolventoperator family 119878

120572(119905) 119905 ge 0The history 119909

119905 (minusinfin 0] rarr H

119909119905(120579) = 119909(119905 + 120579) 120579 le 0 belongs to an abstract phase spaceB

defined axiomatically 119891 Σ 120588 and 119868119896 119896 = 1 119898 are given

functions to be specified later Here 0 = 1199050lt 119905


119898lt

119905119898+1

= 119887Δ119909(119905119896) = 119909(119905

+

119896)minus119909(119905

minus

119896) 119909(119905+

119896) = lim

ℎrarr0119909(119905

119896+ℎ) and

119909(119905minus119896) = lim

ℎrarr0119909(119905

119896minus ℎ) represent the right and left limits of

119909(119905) at 119905 = 119905119896 respectively The initial data 120601 = 120601(119905) 119905 isin

(minusinfin 0] is an F0-measurable B-valued random variable

independent of 119908 with finite second momentsRecall the following known definitions For more details

see [2]

Definition 1 The fractional integral of order 120572 with the lowerlimit 0 for a function 119891 is defined as

119868120572

119905119891 (119905) =

1

Γ (120572)int119905

0

119891 (119904)

(119905 minus 119904)1minus120572

119889119904 119905 gt 0 120572 gt 0 (4)

provided the right-hand side is pointwise defined on [0infin)where Γ is the gamma function

Definition 2 Riemann-Liouville derivative of order 120572 withlower limit zero for a function119891 [0infin) rarr R can bewrittenas

119871119863

120572119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int119905

0

119891 (119904)

(119905 minus 119904)120572+1minus119899

119889119904

119905 gt 0 119899 minus 1 lt 120572 lt 119899

(5)

Definition 3 The Caputo derivative of order 120572 for a function119891 [0infin) rarr R can be written as

119888119863

120572119891 (119905) =

119871119863

120572(119891 (119905) minus

119899minus1

sum119896=0

119905119896

119896119891119896(0))

119905 gt 0 119899 minus 1 lt 120572 lt 119899

(6)

If 119891(119905) isin 119862119899[0infin) then

119888119863

120572119891 (119905) =

1

Γ (119899 minus 120572)int119905

0

(119905 minus 119904)119899minus120572minus1

119891119899(119904) 119889119904 = 119868

119899minus120572119891119899(119904)

119905 gt 0 119899 minus 1 lt 120572 lt 119899

(7)

Obviously theCaputo derivative of a constant is equal to zeroThe Laplace transform of the caputo derivative of order 120572 gt 0is given as

119871 119888119863

120572119891 (119905) 119904 = 119904

120572119891 (119904) minus

119899minus1

sum119896=0

119904120572minus119896minus1

119891(119896)(0)

119899 minus 1 le 120572 lt 119899

(8)



(i) 120588(119860) sub Ψ120579120596




120572 R

+rarr 119871(119883) where 119871(119883) is a


120572 Re 120582 gt 120596 sub 120588(119860) and

(120582120572119868 minus 119860)

minus1

119909 = intinfin

0

119890120582119905119878120572(119905) 119909 119889119905 Re 120582 gt 120596 119909 isin 119883

(9)




(i) 119878120572(119905)D(119860) sub D(119860) and 119860119878

120572(119905)119909 = 119878

120572(119905)119860119909 for all


119905119878120572(119905)119909 isin D(119860) and 119878

120572(119905)119909 = 119909 +

119860119868120572119905119878120572(119905)119909 119905 ge 0


119868120572119905119860119878

120572(119905)119909 119905 ge 0



120572 R

+rarr


120582120572minus1

(120582120572119868 minus 119860)

minus1

119909 = intinfin

0

119890120582119905119878120572(119905) 119909 119889119905

Re 120582 gt 120596 119909 isin 119883(10)









119896 119909(119905

119896) = 119909(119905minus

119896) and 119909(119905+

119896) exists for 119896 = 1 119898


0le119905le119887E119909(119905)



119896isin C([119905

119896 119905

119896+1] 119871

2(ΩH)) 119896 = 0 119898


119909119896(119905) =

119909 (119905) for 119905 isin (119905119896 119905

119896+1]

119909 (119905+119896) for 119905 = 119905

119896

(11)


set 119861119896= 119909



compact in C([119905119896 119905

119896+1] 119871

2(ΩH)) for every 119896 = 0 119898








(a) 119909119905is inB

(b) 119909(119905) le 119867119909119905B









0= 120601(119905) isin

11987102(ΩB) and 119909|


10038171003817100381710038171199091199041003817100381710038171003817B le 119873

119887E10038171003817100381710038171206011003817100381710038171003817119861 + 119870119887

sup0le119904le119887

E 119909 (119904) (12)


119887= 119873(119905) 0 le 119905 le

119887








(13)





sup119910 isin 119866(119909) lt


119887119889119888119897(H) we denote by 119889(119909 119884) =



119889 P

119887119889119888119897(H) times P

119887119889119888119897(H) rarr R

+by



0) is






119899rarr 119909

lowast 119910

119899rarr 119910

lowast 119910

119899isin 119866(119909

119899) imply

119910lowastisin 119866(119909

lowast)





119867119889(119866 (119909) minus 119866 (119910)) le 120599

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (14)








120582Φ(119910) + (1 minus 120582)1199100



Let SΣ120595


119896) =

119909(119905minus

119896) + 119868

119896(119909

119905119896

) 119896 = 1 2 119898




119896) = 119868

119896(119909

119905119896


119896 119905

119896+1] (119896 = 0 1 119898) is



119909 (119905) =

119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119909 (119905minus

1) + 119868

1(119909

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119909 (119905minus

119898) + 119868

119898(119909

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(15)

where 119879120572(119905) = (12120587119894) int

119861119903


120572(119905) =

(12120587119894) int119861119903

119890120582119905(120582120572 minus 119860)minus1119889120582 and 119861



120572(119905) is the



is proved in [31]


0) is a sectorial


1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 119905 gt 0 120596 gt 120596

0 (16)




0) is a sectorial


1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119872119890

120596119905

1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 120596 gt 120596

0

(17)


If 119879= sup

0le119905le119887119879

120572(119905) and

119878= sup

0le119905le119887119862119890120596119905(1 + 1199051minus120572)

we have

1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119879

1003817100381710038171003817119878120572 (119905)

1003817100381710038171003817 le 119905120572minus1

119878












lim sup1205952

Brarrinfin

1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

] (119905) 100381710038171003817100381712059510038171003817100381710038172

B

= Λ (19)


1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

= sup E1205902 120590 isin Σ (119905 120595) (20)



E1003817100381710038171003817119891 (119905 1205951

) minus 119891 (119905 1205952)10038171003817100381710038172

le 119872119891

10038171003817100381710038171205951minus 120595

2

10038171003817100381710038172

B

119905 isin 119869 1205951 120595

2isin B

E1003817100381710038171003817119891 (119905 120595)

10038171003817100381710038172

le 119872119891(1 +

100381710038171003817100381712059510038171003817100381710038172

B) 119905 isin 119869 120595 isin B

(21)



119896such that

lim sup1205952

Brarrinfin

1003817100381710038171003817119868119896 (120595)10038171003817100381710038172

100381710038171003817100381712059510038171003817100381710038172

B

= 120576119896 (22)

for every 120595 isin B 119896 = 1 2 119898





10038171003817100381710038171199091199041003817100381710038171003817B le (119873

119887+W

120601

0)10038171003817100381710038171206011003817100381710038171003817B

+ 119870119887sup 119909 (120579) 120579 isin [0max 0 119904]

119904 isin Z (120588minus) cup 119869

(23)

whereW120601

0= sup

119905isinZ(120588minus)W120601(119905)


120590 C(119869H) rarr P



2(Ω


max1le119896le119898

62

119879(1 + 2119870

2

119887120576119896) lt 1 (24)


1199090= 0 119909|




119911 (119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119911 (119905

minus

1) + 119868

1(119911

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(25)

where 120590 isin SΣ119911120588

= 120590 isin 1198712(119871(KH)) 120590(119905) isin Σ(119905


0= 120601







such that

119909 (119905) =

120582119879120572(119905) 120601 (0)

+120582int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

120582119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)]

+120582int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

120582119879

120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+120582int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(26)


1and 119888


119905 isin 1198691003817100381710038171003817Σ (119905 120595)

10038171003817100381710038172

le 1198881] (119905) + 119888

2] (119905) 1003817100381710038171003817120595

10038171003817100381710038172

B (27)


119896(119896 = 1 119898) 119888


3

E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

le (120576119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B

max1le119896le119898

62

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896)) lt 1

(28)

Let

1198621= 120595

100381710038171003817100381712059510038171003817100381710038172

Ble 119888

3 119862

2= 120595

100381710038171003817100381712059510038171003817100381710038172

Bgt 119888

3

1198623= max E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

120595 isin 1198621

(29)

We have

E1003817100381710038171003817119868119896(120595)

10038171003817100381710038172

le 1198623+ (120576

119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B (30)


E119909 (119905)2

le 3E1003817100381710038171003817119879120572 (119905) 120601 (0)

10038171003817100381710038172

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 32

119879E1003817100381710038171003817120601 (0)

10038171003817100381710038172

+ 32

119878int119905

0

(119905 minus 119904)120572minus1

119889119904

times int119905

0

(119905 minus 119904)120572minus1

119872119891(1 +

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) int

119905

0

(119905 minus 119904)2(120572minus1)

[1198881] (119904) + 119888

2] (119904)

times10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B] 119889119904

le 32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878119872

119891

119887120572

120572

times int119905

0

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(31)


119896+1] 119896 = 1 119898 we have

E119909(119905)2

le 3E10038171003817100381710038171003817119879120572(119905 minus 119905

119896) [119911 (119905

minus

119896) + 119868

119896(119911

119905119896

)]10038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119879[E1003817100381710038171003817119911(119905

minus

119896)10038171003817100381710038172

+ 1198623+ (120576

119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

119887120572

120572int119905

119905119896

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(32)


E119909 (119905)2

le 119872lowast+ 6

2

119879[E1003817100381710038171003817119911 (119905

minus

119896)10038171003817100381710038172

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

1198872120572

2120572int119905

0

(119905 minus 119904)120572minus110038171003817100381710038171003817

119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(33)


where

119872lowast= max32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878Tr (119876) 119888

1

times int119905

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904 62

1198791198623

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(34)


119905 isin [0 119887] and10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887sup0le119904le119887

E119909 (119904)2

(35)


E119909 (119905)2

le 119872lowast+ [6

2

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896))]


E119909 (119905)2+ 6

2

119878119872

119891119870

2

119887

1198872120572

2120572

times int119905

0

(119905 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+ 62

119878119870

2

119887Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904)


E119909 (120591)2119889119904

(36)

where119872

lowast= 119872

lowast+ 6

2

119879[119862

3+ (120576

119896+ 119886

119896) 119862

4]

+ 32

119878119872

1198911198624

1198873120572

3120572+ 3

2

1198781198624Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904

1198624= 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

(37)

Since 119897 = max1le119896le119898

62

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 we have

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897+62

119878119872

1198911198702

1198871198872120572

(1 minus 119897) 2120572

times int119887

0

(119887 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+62

1198781198702

119887Tr (119876) 119888

1

1 minus 119897

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) sup

0le120591le119904

E119909 (120591)2119889119904

(38)


sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)

(39)

Therefore

1199092

PC

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0


(40)


BPC 1199092



119899isin Φ119909(119899) 119909(119899) isin 119881 = 119861

119903lowast(0BPC)

and 119911119899rarr 119911



119904uniformly



119899isin S

Σ119911(119899)

120588


1]

119911119899(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(41)


Σ119909lowast

120588


1]

119911lowast(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(42)

SetΘ119899(119905) = 119911

119899(119905) minus 119879

120572120601(0) minus int

119905

0119878120572(119905 minus 119904)119891(119904 (119911(119899))

120588(119904119911(119899)

119904))119889119904 minus

int119905

0119878120572(119905 minus 119904)120590

119899(119904)119889119908(119904) andΘ

lowast(119905) = 119911


120572120601(0) minus int

119905

0119878120572(119905 minus

119904)119891(119904 (119911lowast)120588(119904119911lowast

119904))119889119904 minus int

119905

0119878120572(119905 minus 119904)120590

lowast(119904)119889119908(119904)


1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172




Γ (120590) (119905) = int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (44)




119899(119905) isin

Γ(SΣ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)


Σ119911lowast

120588


119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911


120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)


119896+1] 119896 = 1 119898 we consider


119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)



119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)




Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)


2is a completely

continuous operator



119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(i) 120588(119860) sub Ψ120579120596




120572 R

+rarr 119871(119883) where 119871(119883) is a


120572 Re 120582 gt 120596 sub 120588(119860) and

(120582120572119868 minus 119860)

minus1

119909 = intinfin

0

119890120582119905119878120572(119905) 119909 119889119905 Re 120582 gt 120596 119909 isin 119883

(9)




(i) 119878120572(119905)D(119860) sub D(119860) and 119860119878

120572(119905)119909 = 119878

120572(119905)119860119909 for all


119905119878120572(119905)119909 isin D(119860) and 119878

120572(119905)119909 = 119909 +

119860119868120572119905119878120572(119905)119909 119905 ge 0


119868120572119905119860119878

120572(119905)119909 119905 ge 0



120572 R

+rarr


120582120572minus1

(120582120572119868 minus 119860)

minus1

119909 = intinfin

0

119890120582119905119878120572(119905) 119909 119889119905

Re 120582 gt 120596 119909 isin 119883(10)









119896 119909(119905

119896) = 119909(119905minus

119896) and 119909(119905+

119896) exists for 119896 = 1 119898


0le119905le119887E119909(119905)



119896isin C([119905

119896 119905

119896+1] 119871

2(ΩH)) 119896 = 0 119898


119909119896(119905) =

119909 (119905) for 119905 isin (119905119896 119905

119896+1]

119909 (119905+119896) for 119905 = 119905

119896

(11)


set 119861119896= 119909



compact in C([119905119896 119905

119896+1] 119871

2(ΩH)) for every 119896 = 0 119898








(a) 119909119905is inB

(b) 119909(119905) le 119867119909119905B









0= 120601(119905) isin

11987102(ΩB) and 119909|


10038171003817100381710038171199091199041003817100381710038171003817B le 119873

119887E10038171003817100381710038171206011003817100381710038171003817119861 + 119870119887

sup0le119904le119887

E 119909 (119904) (12)


119887= 119873(119905) 0 le 119905 le

119887








(13)





sup119910 isin 119866(119909) lt


119887119889119888119897(H) we denote by 119889(119909 119884) =



119889 P

119887119889119888119897(H) times P

119887119889119888119897(H) rarr R

+by



0) is






119899rarr 119909

lowast 119910

119899rarr 119910

lowast 119910

119899isin 119866(119909

119899) imply

119910lowastisin 119866(119909

lowast)





119867119889(119866 (119909) minus 119866 (119910)) le 120599

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (14)








120582Φ(119910) + (1 minus 120582)1199100



Let SΣ120595


119896) =

119909(119905minus

119896) + 119868

119896(119909

119905119896

) 119896 = 1 2 119898




119896) = 119868

119896(119909

119905119896


119896 119905

119896+1] (119896 = 0 1 119898) is



119909 (119905) =

119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119909 (119905minus

1) + 119868

1(119909

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119909 (119905minus

119898) + 119868

119898(119909

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(15)

where 119879120572(119905) = (12120587119894) int

119861119903


120572(119905) =

(12120587119894) int119861119903

119890120582119905(120582120572 minus 119860)minus1119889120582 and 119861



120572(119905) is the



is proved in [31]


0) is a sectorial


1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 119905 gt 0 120596 gt 120596

0 (16)




0) is a sectorial


1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119872119890

120596119905

1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 120596 gt 120596

0

(17)


If 119879= sup

0le119905le119887119879

120572(119905) and

119878= sup

0le119905le119887119862119890120596119905(1 + 1199051minus120572)

we have

1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119879

1003817100381710038171003817119878120572 (119905)

1003817100381710038171003817 le 119905120572minus1

119878












lim sup1205952

Brarrinfin

1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

] (119905) 100381710038171003817100381712059510038171003817100381710038172

B

= Λ (19)


1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

= sup E1205902 120590 isin Σ (119905 120595) (20)



E1003817100381710038171003817119891 (119905 1205951

) minus 119891 (119905 1205952)10038171003817100381710038172

le 119872119891

10038171003817100381710038171205951minus 120595

2

10038171003817100381710038172

B

119905 isin 119869 1205951 120595

2isin B

E1003817100381710038171003817119891 (119905 120595)

10038171003817100381710038172

le 119872119891(1 +

100381710038171003817100381712059510038171003817100381710038172

B) 119905 isin 119869 120595 isin B

(21)



119896such that

lim sup1205952

Brarrinfin

1003817100381710038171003817119868119896 (120595)10038171003817100381710038172

100381710038171003817100381712059510038171003817100381710038172

B

= 120576119896 (22)

for every 120595 isin B 119896 = 1 2 119898





10038171003817100381710038171199091199041003817100381710038171003817B le (119873

119887+W

120601

0)10038171003817100381710038171206011003817100381710038171003817B

+ 119870119887sup 119909 (120579) 120579 isin [0max 0 119904]

119904 isin Z (120588minus) cup 119869

(23)

whereW120601

0= sup

119905isinZ(120588minus)W120601(119905)


120590 C(119869H) rarr P



2(Ω


max1le119896le119898

62

119879(1 + 2119870

2

119887120576119896) lt 1 (24)


1199090= 0 119909|




119911 (119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119911 (119905

minus

1) + 119868

1(119911

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(25)

where 120590 isin SΣ119911120588

= 120590 isin 1198712(119871(KH)) 120590(119905) isin Σ(119905


0= 120601







such that

119909 (119905) =

120582119879120572(119905) 120601 (0)

+120582int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

120582119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)]

+120582int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

120582119879

120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+120582int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(26)


1and 119888


119905 isin 1198691003817100381710038171003817Σ (119905 120595)

10038171003817100381710038172

le 1198881] (119905) + 119888

2] (119905) 1003817100381710038171003817120595

10038171003817100381710038172

B (27)


119896(119896 = 1 119898) 119888


3

E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

le (120576119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B

max1le119896le119898

62

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896)) lt 1

(28)

Let

1198621= 120595

100381710038171003817100381712059510038171003817100381710038172

Ble 119888

3 119862

2= 120595

100381710038171003817100381712059510038171003817100381710038172

Bgt 119888

3

1198623= max E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

120595 isin 1198621

(29)

We have

E1003817100381710038171003817119868119896(120595)

10038171003817100381710038172

le 1198623+ (120576

119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B (30)


E119909 (119905)2

le 3E1003817100381710038171003817119879120572 (119905) 120601 (0)

10038171003817100381710038172

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 32

119879E1003817100381710038171003817120601 (0)

10038171003817100381710038172

+ 32

119878int119905

0

(119905 minus 119904)120572minus1

119889119904

times int119905

0

(119905 minus 119904)120572minus1

119872119891(1 +

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) int

119905

0

(119905 minus 119904)2(120572minus1)

[1198881] (119904) + 119888

2] (119904)

times10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B] 119889119904

le 32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878119872

119891

119887120572

120572

times int119905

0

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(31)


119896+1] 119896 = 1 119898 we have

E119909(119905)2

le 3E10038171003817100381710038171003817119879120572(119905 minus 119905

119896) [119911 (119905

minus

119896) + 119868

119896(119911

119905119896

)]10038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119879[E1003817100381710038171003817119911(119905

minus

119896)10038171003817100381710038172

+ 1198623+ (120576

119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

119887120572

120572int119905

119905119896

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(32)


E119909 (119905)2

le 119872lowast+ 6

2

119879[E1003817100381710038171003817119911 (119905

minus

119896)10038171003817100381710038172

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

1198872120572

2120572int119905

0

(119905 minus 119904)120572minus110038171003817100381710038171003817

119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(33)


where

119872lowast= max32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878Tr (119876) 119888

1

times int119905

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904 62

1198791198623

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(34)


119905 isin [0 119887] and10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887sup0le119904le119887

E119909 (119904)2

(35)


E119909 (119905)2

le 119872lowast+ [6

2

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896))]


E119909 (119905)2+ 6

2

119878119872

119891119870

2

119887

1198872120572

2120572

times int119905

0

(119905 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+ 62

119878119870

2

119887Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904)


E119909 (120591)2119889119904

(36)

where119872

lowast= 119872

lowast+ 6

2

119879[119862

3+ (120576

119896+ 119886

119896) 119862

4]

+ 32

119878119872

1198911198624

1198873120572

3120572+ 3

2

1198781198624Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904

1198624= 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

(37)

Since 119897 = max1le119896le119898

62

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 we have

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897+62

119878119872

1198911198702

1198871198872120572

(1 minus 119897) 2120572

times int119887

0

(119887 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+62

1198781198702

119887Tr (119876) 119888

1

1 minus 119897

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) sup

0le120591le119904

E119909 (120591)2119889119904

(38)


sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)

(39)

Therefore

1199092

PC

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0


(40)


BPC 1199092



119899isin Φ119909(119899) 119909(119899) isin 119881 = 119861

119903lowast(0BPC)

and 119911119899rarr 119911



119904uniformly



119899isin S

Σ119911(119899)

120588


1]

119911119899(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(41)


Σ119909lowast

120588


1]

119911lowast(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(42)

SetΘ119899(119905) = 119911

119899(119905) minus 119879

120572120601(0) minus int

119905

0119878120572(119905 minus 119904)119891(119904 (119911(119899))

120588(119904119911(119899)

119904))119889119904 minus

int119905

0119878120572(119905 minus 119904)120590

119899(119904)119889119908(119904) andΘ

lowast(119905) = 119911


120572120601(0) minus int

119905

0119878120572(119905 minus

119904)119891(119904 (119911lowast)120588(119904119911lowast

119904))119889119904 minus int

119905

0119878120572(119905 minus 119904)120590

lowast(119904)119889119908(119904)


1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172




Γ (120590) (119905) = int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (44)




119899(119905) isin

Γ(SΣ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)


Σ119911lowast

120588


119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911


120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)


119896+1] 119896 = 1 119898 we consider


119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)



119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)




Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)


2is a completely

continuous operator



119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(13)





sup119910 isin 119866(119909) lt


119887119889119888119897(H) we denote by 119889(119909 119884) =



119889 P

119887119889119888119897(H) times P

119887119889119888119897(H) rarr R

+by



0) is






119899rarr 119909

lowast 119910

119899rarr 119910

lowast 119910

119899isin 119866(119909

119899) imply

119910lowastisin 119866(119909

lowast)





119867119889(119866 (119909) minus 119866 (119910)) le 120599

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (14)








120582Φ(119910) + (1 minus 120582)1199100



Let SΣ120595


119896) =

119909(119905minus

119896) + 119868

119896(119909

119905119896

) 119896 = 1 2 119898




119896) = 119868

119896(119909

119905119896


119896 119905

119896+1] (119896 = 0 1 119898) is



119909 (119905) =

119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119909 (119905minus

1) + 119868

1(119909

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119909 (119905minus

119898) + 119868

119898(119909

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119909

120588(119904119909119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(15)

where 119879120572(119905) = (12120587119894) int

119861119903


120572(119905) =

(12120587119894) int119861119903

119890120582119905(120582120572 minus 119860)minus1119889120582 and 119861



120572(119905) is the



is proved in [31]


0) is a sectorial


1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 119905 gt 0 120596 gt 120596

0 (16)




0) is a sectorial


1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119872119890

120596119905

1003817100381710038171003817119878120572 (119905)1003817100381710038171003817 le 119862119890

120596119905(1 + 119905

120572minus1) 120596 gt 120596

0

(17)


If 119879= sup

0le119905le119887119879

120572(119905) and

119878= sup

0le119905le119887119862119890120596119905(1 + 1199051minus120572)

we have

1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119879

1003817100381710038171003817119878120572 (119905)

1003817100381710038171003817 le 119905120572minus1

119878












lim sup1205952

Brarrinfin

1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

] (119905) 100381710038171003817100381712059510038171003817100381710038172

B

= Λ (19)


1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

= sup E1205902 120590 isin Σ (119905 120595) (20)



E1003817100381710038171003817119891 (119905 1205951

) minus 119891 (119905 1205952)10038171003817100381710038172

le 119872119891

10038171003817100381710038171205951minus 120595

2

10038171003817100381710038172

B

119905 isin 119869 1205951 120595

2isin B

E1003817100381710038171003817119891 (119905 120595)

10038171003817100381710038172

le 119872119891(1 +

100381710038171003817100381712059510038171003817100381710038172

B) 119905 isin 119869 120595 isin B

(21)



119896such that

lim sup1205952

Brarrinfin

1003817100381710038171003817119868119896 (120595)10038171003817100381710038172

100381710038171003817100381712059510038171003817100381710038172

B

= 120576119896 (22)

for every 120595 isin B 119896 = 1 2 119898





10038171003817100381710038171199091199041003817100381710038171003817B le (119873

119887+W

120601

0)10038171003817100381710038171206011003817100381710038171003817B

+ 119870119887sup 119909 (120579) 120579 isin [0max 0 119904]

119904 isin Z (120588minus) cup 119869

(23)

whereW120601

0= sup

119905isinZ(120588minus)W120601(119905)


120590 C(119869H) rarr P



2(Ω


max1le119896le119898

62

119879(1 + 2119870

2

119887120576119896) lt 1 (24)


1199090= 0 119909|




119911 (119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119911 (119905

minus

1) + 119868

1(119911

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(25)

where 120590 isin SΣ119911120588

= 120590 isin 1198712(119871(KH)) 120590(119905) isin Σ(119905


0= 120601







such that

119909 (119905) =

120582119879120572(119905) 120601 (0)

+120582int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

120582119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)]

+120582int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

120582119879

120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+120582int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(26)


1and 119888


119905 isin 1198691003817100381710038171003817Σ (119905 120595)

10038171003817100381710038172

le 1198881] (119905) + 119888

2] (119905) 1003817100381710038171003817120595

10038171003817100381710038172

B (27)


119896(119896 = 1 119898) 119888


3

E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

le (120576119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B

max1le119896le119898

62

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896)) lt 1

(28)

Let

1198621= 120595

100381710038171003817100381712059510038171003817100381710038172

Ble 119888

3 119862

2= 120595

100381710038171003817100381712059510038171003817100381710038172

Bgt 119888

3

1198623= max E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

120595 isin 1198621

(29)

We have

E1003817100381710038171003817119868119896(120595)

10038171003817100381710038172

le 1198623+ (120576

119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B (30)


E119909 (119905)2

le 3E1003817100381710038171003817119879120572 (119905) 120601 (0)

10038171003817100381710038172

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 32

119879E1003817100381710038171003817120601 (0)

10038171003817100381710038172

+ 32

119878int119905

0

(119905 minus 119904)120572minus1

119889119904

times int119905

0

(119905 minus 119904)120572minus1

119872119891(1 +

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) int

119905

0

(119905 minus 119904)2(120572minus1)

[1198881] (119904) + 119888

2] (119904)

times10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B] 119889119904

le 32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878119872

119891

119887120572

120572

times int119905

0

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(31)


119896+1] 119896 = 1 119898 we have

E119909(119905)2

le 3E10038171003817100381710038171003817119879120572(119905 minus 119905

119896) [119911 (119905

minus

119896) + 119868

119896(119911

119905119896

)]10038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119879[E1003817100381710038171003817119911(119905

minus

119896)10038171003817100381710038172

+ 1198623+ (120576

119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

119887120572

120572int119905

119905119896

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(32)


E119909 (119905)2

le 119872lowast+ 6

2

119879[E1003817100381710038171003817119911 (119905

minus

119896)10038171003817100381710038172

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

1198872120572

2120572int119905

0

(119905 minus 119904)120572minus110038171003817100381710038171003817

119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(33)


where

119872lowast= max32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878Tr (119876) 119888

1

times int119905

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904 62

1198791198623

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(34)


119905 isin [0 119887] and10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887sup0le119904le119887

E119909 (119904)2

(35)


E119909 (119905)2

le 119872lowast+ [6

2

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896))]


E119909 (119905)2+ 6

2

119878119872

119891119870

2

119887

1198872120572

2120572

times int119905

0

(119905 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+ 62

119878119870

2

119887Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904)


E119909 (120591)2119889119904

(36)

where119872

lowast= 119872

lowast+ 6

2

119879[119862

3+ (120576

119896+ 119886

119896) 119862

4]

+ 32

119878119872

1198911198624

1198873120572

3120572+ 3

2

1198781198624Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904

1198624= 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

(37)

Since 119897 = max1le119896le119898

62

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 we have

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897+62

119878119872

1198911198702

1198871198872120572

(1 minus 119897) 2120572

times int119887

0

(119887 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+62

1198781198702

119887Tr (119876) 119888

1

1 minus 119897

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) sup

0le120591le119904

E119909 (120591)2119889119904

(38)


sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)

(39)

Therefore

1199092

PC

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0


(40)


BPC 1199092



119899isin Φ119909(119899) 119909(119899) isin 119881 = 119861

119903lowast(0BPC)

and 119911119899rarr 119911



119904uniformly



119899isin S

Σ119911(119899)

120588


1]

119911119899(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(41)


Σ119909lowast

120588


1]

119911lowast(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(42)

SetΘ119899(119905) = 119911

119899(119905) minus 119879

120572120601(0) minus int

119905

0119878120572(119905 minus 119904)119891(119904 (119911(119899))

120588(119904119911(119899)

119904))119889119904 minus

int119905

0119878120572(119905 minus 119904)120590

119899(119904)119889119908(119904) andΘ

lowast(119905) = 119911


120572120601(0) minus int

119905

0119878120572(119905 minus

119904)119891(119904 (119911lowast)120588(119904119911lowast

119904))119889119904 minus int

119905

0119878120572(119905 minus 119904)120590

lowast(119904)119889119908(119904)


1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172




Γ (120590) (119905) = int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (44)




119899(119905) isin

Γ(SΣ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)


Σ119911lowast

120588


119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911


120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)


119896+1] 119896 = 1 119898 we consider


119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)



119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)




Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)


2is a completely

continuous operator



119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










If 119879= sup

0le119905le119887119879

120572(119905) and

119878= sup

0le119905le119887119862119890120596119905(1 + 1199051minus120572)

we have

1003817100381710038171003817119879120572 (119905)1003817100381710038171003817 le 119879

1003817100381710038171003817119878120572 (119905)

1003817100381710038171003817 le 119905120572minus1

119878












lim sup1205952

Brarrinfin

1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

] (119905) 100381710038171003817100381712059510038171003817100381710038172

B

= Λ (19)


1003817100381710038171003817Σ (119905 120595)10038171003817100381710038172

= sup E1205902 120590 isin Σ (119905 120595) (20)



E1003817100381710038171003817119891 (119905 1205951

) minus 119891 (119905 1205952)10038171003817100381710038172

le 119872119891

10038171003817100381710038171205951minus 120595

2

10038171003817100381710038172

B

119905 isin 119869 1205951 120595

2isin B

E1003817100381710038171003817119891 (119905 120595)

10038171003817100381710038172

le 119872119891(1 +

100381710038171003817100381712059510038171003817100381710038172

B) 119905 isin 119869 120595 isin B

(21)



119896such that

lim sup1205952

Brarrinfin

1003817100381710038171003817119868119896 (120595)10038171003817100381710038172

100381710038171003817100381712059510038171003817100381710038172

B

= 120576119896 (22)

for every 120595 isin B 119896 = 1 2 119898





10038171003817100381710038171199091199041003817100381710038171003817B le (119873

119887+W

120601

0)10038171003817100381710038171206011003817100381710038171003817B

+ 119870119887sup 119909 (120579) 120579 isin [0max 0 119904]

119904 isin Z (120588minus) cup 119869

(23)

whereW120601

0= sup

119905isinZ(120588minus)W120601(119905)


120590 C(119869H) rarr P



2(Ω


max1le119896le119898

62

119879(1 + 2119870

2

119887120576119896) lt 1 (24)


1199090= 0 119909|




119911 (119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

119879120572(119905 minus 119905

1) [119911 (119905

minus

1) + 119868

1(119911

1199051

)]

+int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(25)

where 120590 isin SΣ119911120588

= 120590 isin 1198712(119871(KH)) 120590(119905) isin Σ(119905


0= 120601







such that

119909 (119905) =

120582119879120572(119905) 120601 (0)

+120582int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

120582119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)]

+120582int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

120582119879

120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+120582int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(26)


1and 119888


119905 isin 1198691003817100381710038171003817Σ (119905 120595)

10038171003817100381710038172

le 1198881] (119905) + 119888

2] (119905) 1003817100381710038171003817120595

10038171003817100381710038172

B (27)


119896(119896 = 1 119898) 119888


3

E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

le (120576119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B

max1le119896le119898

62

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896)) lt 1

(28)

Let

1198621= 120595

100381710038171003817100381712059510038171003817100381710038172

Ble 119888

3 119862

2= 120595

100381710038171003817100381712059510038171003817100381710038172

Bgt 119888

3

1198623= max E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

120595 isin 1198621

(29)

We have

E1003817100381710038171003817119868119896(120595)

10038171003817100381710038172

le 1198623+ (120576

119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B (30)


E119909 (119905)2

le 3E1003817100381710038171003817119879120572 (119905) 120601 (0)

10038171003817100381710038172

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 32

119879E1003817100381710038171003817120601 (0)

10038171003817100381710038172

+ 32

119878int119905

0

(119905 minus 119904)120572minus1

119889119904

times int119905

0

(119905 minus 119904)120572minus1

119872119891(1 +

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) int

119905

0

(119905 minus 119904)2(120572minus1)

[1198881] (119904) + 119888

2] (119904)

times10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B] 119889119904

le 32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878119872

119891

119887120572

120572

times int119905

0

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(31)


119896+1] 119896 = 1 119898 we have

E119909(119905)2

le 3E10038171003817100381710038171003817119879120572(119905 minus 119905

119896) [119911 (119905

minus

119896) + 119868

119896(119911

119905119896

)]10038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119879[E1003817100381710038171003817119911(119905

minus

119896)10038171003817100381710038172

+ 1198623+ (120576

119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

119887120572

120572int119905

119905119896

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(32)


E119909 (119905)2

le 119872lowast+ 6

2

119879[E1003817100381710038171003817119911 (119905

minus

119896)10038171003817100381710038172

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

1198872120572

2120572int119905

0

(119905 minus 119904)120572minus110038171003817100381710038171003817

119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(33)


where

119872lowast= max32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878Tr (119876) 119888

1

times int119905

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904 62

1198791198623

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(34)


119905 isin [0 119887] and10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887sup0le119904le119887

E119909 (119904)2

(35)


E119909 (119905)2

le 119872lowast+ [6

2

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896))]


E119909 (119905)2+ 6

2

119878119872

119891119870

2

119887

1198872120572

2120572

times int119905

0

(119905 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+ 62

119878119870

2

119887Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904)


E119909 (120591)2119889119904

(36)

where119872

lowast= 119872

lowast+ 6

2

119879[119862

3+ (120576

119896+ 119886

119896) 119862

4]

+ 32

119878119872

1198911198624

1198873120572

3120572+ 3

2

1198781198624Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904

1198624= 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

(37)

Since 119897 = max1le119896le119898

62

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 we have

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897+62

119878119872

1198911198702

1198871198872120572

(1 minus 119897) 2120572

times int119887

0

(119887 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+62

1198781198702

119887Tr (119876) 119888

1

1 minus 119897

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) sup

0le120591le119904

E119909 (120591)2119889119904

(38)


sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)

(39)

Therefore

1199092

PC

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0


(40)


BPC 1199092



119899isin Φ119909(119899) 119909(119899) isin 119881 = 119861

119903lowast(0BPC)

and 119911119899rarr 119911



119904uniformly



119899isin S

Σ119911(119899)

120588


1]

119911119899(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(41)


Σ119909lowast

120588


1]

119911lowast(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(42)

SetΘ119899(119905) = 119911

119899(119905) minus 119879

120572120601(0) minus int

119905

0119878120572(119905 minus 119904)119891(119904 (119911(119899))

120588(119904119911(119899)

119904))119889119904 minus

int119905

0119878120572(119905 minus 119904)120590

119899(119904)119889119908(119904) andΘ

lowast(119905) = 119911


120572120601(0) minus int

119905

0119878120572(119905 minus

119904)119891(119904 (119911lowast)120588(119904119911lowast

119904))119889119904 minus int

119905

0119878120572(119905 minus 119904)120590

lowast(119904)119889119908(119904)


1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172




Γ (120590) (119905) = int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (44)




119899(119905) isin

Γ(SΣ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)


Σ119911lowast

120588


119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911


120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)


119896+1] 119896 = 1 119898 we consider


119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)



119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)




Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)


2is a completely

continuous operator



119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













such that

119909 (119905) =

120582119879120572(119905) 120601 (0)

+120582int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

120582119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)]

+120582int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

120582119879

120572(119905 minus 119905

119898) [119911 (119905minus

119898) + 119868

119898(119911

119905119898

)]

+120582int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+120582int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(26)


1and 119888


119905 isin 1198691003817100381710038171003817Σ (119905 120595)

10038171003817100381710038172

le 1198881] (119905) + 119888

2] (119905) 1003817100381710038171003817120595

10038171003817100381710038172

B (27)


119896(119896 = 1 119898) 119888


3

E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

le (120576119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B

max1le119896le119898

62

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896)) lt 1

(28)

Let

1198621= 120595

100381710038171003817100381712059510038171003817100381710038172

Ble 119888

3 119862

2= 120595

100381710038171003817100381712059510038171003817100381710038172

Bgt 119888

3

1198623= max E1003817100381710038171003817119868119896 (120595)

10038171003817100381710038172

120595 isin 1198621

(29)

We have

E1003817100381710038171003817119868119896(120595)

10038171003817100381710038172

le 1198623+ (120576

119896+ 119886

119896)100381710038171003817100381712059510038171003817100381710038172

B (30)


E119909 (119905)2

le 3E1003817100381710038171003817119879120572 (119905) 120601 (0)

10038171003817100381710038172

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 3E10038171003817100381710038171003817100381710038171003817int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 32

119879E1003817100381710038171003817120601 (0)

10038171003817100381710038172

+ 32

119878int119905

0

(119905 minus 119904)120572minus1

119889119904

times int119905

0

(119905 minus 119904)120572minus1

119872119891(1 +

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) int

119905

0

(119905 minus 119904)2(120572minus1)

[1198881] (119904) + 119888

2] (119904)

times10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B] 119889119904

le 32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878119872

119891

119887120572

120572

times int119905

0

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(31)


119896+1] 119896 = 1 119898 we have

E119909(119905)2

le 3E10038171003817100381710038171003817119879120572(119905 minus 119905

119896) [119911 (119905

minus

119896) + 119868

119896(119911

119905119896

)]10038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 3E

100381710038171003817100381710038171003817100381710038171003817int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119879[E1003817100381710038171003817119911(119905

minus

119896)10038171003817100381710038172

+ 1198623+ (120576

119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

119887120572

120572int119905

119905119896

(119905 minus 119904)120572minus1

(1 +10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B) 119889119904

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

+ 32

119878Tr (119876) 119888

2int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(32)


E119909 (119905)2

le 119872lowast+ 6

2

119879[E1003817100381710038171003817119911 (119905

minus

119896)10038171003817100381710038172

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119911119905119896

10038171003817100381710038171003817

2

B]

+ 32

119878119872

119891

1198872120572

2120572int119905

0

(119905 minus 119904)120572minus110038171003817100381710038171003817

119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

+ 32

119878Tr (119876) 119888

2int119905

0

(119905 minus 119904)2(120572minus1)] (119904)

10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

B119889119904

(33)


where

119872lowast= max32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878Tr (119876) 119888

1

times int119905

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904 62

1198791198623

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(34)


119905 isin [0 119887] and10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887sup0le119904le119887

E119909 (119904)2

(35)


E119909 (119905)2

le 119872lowast+ [6

2

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896))]


E119909 (119905)2+ 6

2

119878119872

119891119870

2

119887

1198872120572

2120572

times int119905

0

(119905 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+ 62

119878119870

2

119887Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904)


E119909 (120591)2119889119904

(36)

where119872

lowast= 119872

lowast+ 6

2

119879[119862

3+ (120576

119896+ 119886

119896) 119862

4]

+ 32

119878119872

1198911198624

1198873120572

3120572+ 3

2

1198781198624Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904

1198624= 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

(37)

Since 119897 = max1le119896le119898

62

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 we have

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897+62

119878119872

1198911198702

1198871198872120572

(1 minus 119897) 2120572

times int119887

0

(119887 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+62

1198781198702

119887Tr (119876) 119888

1

1 minus 119897

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) sup

0le120591le119904

E119909 (120591)2119889119904

(38)


sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)

(39)

Therefore

1199092

PC

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0


(40)


BPC 1199092



119899isin Φ119909(119899) 119909(119899) isin 119881 = 119861

119903lowast(0BPC)

and 119911119899rarr 119911



119904uniformly



119899isin S

Σ119911(119899)

120588


1]

119911119899(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(41)


Σ119909lowast

120588


1]

119911lowast(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(42)

SetΘ119899(119905) = 119911

119899(119905) minus 119879

120572120601(0) minus int

119905

0119878120572(119905 minus 119904)119891(119904 (119911(119899))

120588(119904119911(119899)

119904))119889119904 minus

int119905

0119878120572(119905 minus 119904)120590

119899(119904)119889119908(119904) andΘ

lowast(119905) = 119911


120572120601(0) minus int

119905

0119878120572(119905 minus

119904)119891(119904 (119911lowast)120588(119904119911lowast

119904))119889119904 minus int

119905

0119878120572(119905 minus 119904)120590

lowast(119904)119889119908(119904)


1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172




Γ (120590) (119905) = int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (44)




119899(119905) isin

Γ(SΣ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)


Σ119911lowast

120588


119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911


120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)


119896+1] 119896 = 1 119898 we consider


119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)



119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)




Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)


2is a completely

continuous operator



119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119872lowast= max32

119879119867

2E100381710038171003817100381712060110038171003817100381710038172

B+ 3

2

119878Tr (119876) 119888

1

times int119905

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904 62

1198791198623

+ 32

119878Tr (119876) 119888

1int119905

119905119896

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(34)


119905 isin [0 119887] and10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887sup0le119904le119887

E119909 (119904)2

(35)


E119909 (119905)2

le 119872lowast+ [6

2

119879(1 + 2119870

2

119887(120576

119896+ 119886

119896))]


E119909 (119905)2+ 6

2

119878119872

119891119870

2

119887

1198872120572

2120572

times int119905

0

(119905 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+ 62

119878119870

2

119887Tr (119876) 119888

1int119905

0

(119905 minus 119904)2(120572minus1)] (119904)


E119909 (120591)2119889119904

(36)

where119872

lowast= 119872

lowast+ 6

2

119879[119862

3+ (120576

119896+ 119886

119896) 119862

4]

+ 32

119878119872

1198911198624

1198873120572

3120572+ 3

2

1198781198624Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904

1198624= 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

(37)

Since 119897 = max1le119896le119898

62

119879(1 + 21198702

119887(120576

119896+ 119886

119896)) lt 1 we have

sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897+62

119878119872

1198911198702

1198871198872120572

(1 minus 119897) 2120572

times int119887

0

(119887 minus 119904)120572minus1 sup

0le120591le119904

E119909 (120591)2119889119904

+62

1198781198702

119887Tr (119876) 119888

1

1 minus 119897

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) sup

0le120591le119904

E119909 (120591)2119889119904

(38)


sup0le119905le119887

E119909 (119905)2

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0

(119887 minus 119904)2(120572minus1)] (119904) 119889119904)

(39)

Therefore

1199092

PC

le119872

lowast

1 minus 119897exp

62

1198781198702

119887

(1 minus 119897) 3120572(119872

1198911198873120572+ 3120572Tr (119876) 119888

1

times int119887

0


(40)


BPC 1199092



119899isin Φ119909(119899) 119909(119899) isin 119881 = 119861

119903lowast(0BPC)

and 119911119899rarr 119911



119904uniformly



119899isin S

Σ119911(119899)

120588


1]

119911119899(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(41)


Σ119909lowast

120588


1]

119911lowast(119905) = 119879

120572120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(42)

SetΘ119899(119905) = 119911

119899(119905) minus 119879

120572120601(0) minus int

119905

0119878120572(119905 minus 119904)119891(119904 (119911(119899))

120588(119904119911(119899)

119904))119889119904 minus

int119905

0119878120572(119905 minus 119904)120590

119899(119904)119889119908(119904) andΘ

lowast(119905) = 119911


120572120601(0) minus int

119905

0119878120572(119905 minus

119904)119891(119904 (119911lowast)120588(119904119911lowast

119904))119889119904 minus int

119905

0119878120572(119905 minus 119904)120590

lowast(119904)119889119908(119904)


1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172




Γ (120590) (119905) = int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (44)




119899(119905) isin

Γ(SΣ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)


Σ119911lowast

120588


119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911


120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)


119896+1] 119896 = 1 119898 we consider


119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)



119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)




Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)


2is a completely

continuous operator



119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899(119905) isin

Γ(SΣ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119911lowast(119905) minus 119879

120572120601 (0) minus int

119905

0

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

0

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(45)


119896+1] 119896 = 1 119898 we have

119911119899(119905) = 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

(46)


Σ119911lowast

120588


119896 119905

119896+1]

119911lowast(119905) = 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

+ int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

+ int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(47)

For every 119905 isin (119905119896 119905

119896+1] 119896 = 1 119898 we have

1003817100381710038171003817Θ119899(119905) minus Θ

lowast(119905)10038171003817100381710038172


where

Θ119899(119905) = 119911

119899(119905) minus 119879

120572(119905 minus 119905

119896) [119911(119899) (119905

minus

119896) + 119868

119896(119911(119899)

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911(119899))

120588(119904119911(119899)

119904)

)119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

119899(119904) 119889119908 (119904)

Θlowast(119905) = 119911


120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

minus int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(49)


119896+1] 119896 = 1 119898 we consider


119896+1]H) rarr

C((119905119896 119905

119896+1]H)

Γ (120590) (119905) = int119905

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) (50)



119896+1] Θ

119899(119905) isin Γ(S

Σ119911(119899))


119904and Γ ∘S



Σ119911lowast

120588


119896+1]

119911lowast(119905) minus 119879

120572(119905 minus 119905

119896) [119911lowast (119905

minus

119896) + 119868

119896(119911lowast

119905119896

)]

minus int119905

119905119896

119878120572(119905 minus 119904) 119891 (119904 (119911lowast)

120588(119904119911lowast

119904)) 119889119904

= int119905

119905119896

119878120572(119905 minus 119904) 120590

lowast(119904) 119889119908 (119904)

(51)




Φ1119909 = 119911

1isin BPC and Φ

2119909 = 119911

2isin BPC such that

1199111(119905) =

0 119905 isin [0 1199051]

119879120572(119905 minus 119905

1) [119911 (119905minus

1) + 119868

1(119911

1199051

)] 119905 isin (1199051 119905

2]

119879120572(119905 minus 119905

119898) [119911 (119905

minus

119898) + 119868

119898(119911

119905119898

)] 119905 isin (119905119898 119887]

1199112(119905) =


119879120572(119905) 120601 (0)

+int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin [0 119905

1]

int119905

1199051

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

1199051

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

1 119905

2]

int119905

119905119898

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+int119905

119905119898

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904) 119905 isin (119905

119898 119887]

(52)


2is a completely

continuous operator



119896+1] 119896 =

1 119898

E1003817100381710038171003817(Φ1

119906) (119905) minus (Φ1V) (119905)1003817100381710038171003817

2

le 21003817100381710038171003817119879120572 (119905 minus 119905119896)

10038171003817100381710038172

[E1003817100381710038171003817119906 (119905

minus

119896) minus V (119905minus

119896)10038171003817100381710038172

+E10038171003817100381710038171003817119868119896(119906

119905119896

) minus 119868119896(V

119905119896

)10038171003817100381710038171003817

2

]

le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2

+ (120576119896+ 119886

119896)10038171003817100381710038171003817119906119905119896

minus V119905119896

10038171003817100381710038171003817

2

B]


le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


le 22

119879[ sup0le119905le119887

E119906 (119905) minus V (119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887


E119906 (119904) minus V (119904)2]

= 22

119879[ sup0le119905le119887

E119906(119905) minus V(119905)2 + 2 (120576119896+ 119886

119896)119870

2

119887



= 22

119879[1 + 2 (120576

119896+ 119886

119896)119870

2

119887] 119906 minus V2PC

(53)


1003817100381710038171003817(Φ1119906)(119905) minus (Φ

1V)(119905)1003817100381710038171003817

2

PCle 119897

0119906 minus V2PC (54)

where 1198970= max

1le119896le11989822

119879(1 + 21198702

119887(120576

119896+ 119886


1



2 1199112

2belong


1 120590

2isin S

Σ119911120588

such that

119911119894

2(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119905 isin [0 119905

1] 119894 = 1 2

(55)


(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= 119879120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(56)


119896+1] 119896 = 1 119898

we have

(1205821199111

2+ (1 minus 120582) 119911

2

2) (119905)

= int119905

0

119878120572(119905 minus 119904) [120582120590

1(119904) + (1 minus 120582) 120590

2(119904)] 119889119908 (119904)

(57)

where

119911119894

2(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590

119894(119904) 119889119908 (119904) 119894 = 1 2

(58)

Since SΣ119911120588


2+ (1 minus 120582)1199112

2) isin Φ

2119909




10038171003817100381710038171003817119911120588(119904119911119904)

10038171003817100381710038171003817

2

Ble 2[(119873

119887+W

120601

0)E

10038171003817100381710038171206011003817100381710038171003817B]

2

+ 21198702

119887119903lowast= 119903

1015840

(59)

Let 0 lt 1205911lt 120591

2le 119905

1 For each 119909 isin 119881 ℎ

2isin Φ


120590 isin SΣ119911120588

such that

1199112(119905) = 119879

120572(119905) 120601 (0) + int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(60)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 7E1003817100381710038171003817(119879120572 (1205912) minus 119879120572 (1205911)) 120601 (0)

10038171003817100381710038172

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 7E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2


le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 72

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


0


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 72

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


0



2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 72

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(61)


2isin (119905

119896 119905

119896+1] 120591

1lt 120591

2 119896 = 1 119898 we

have

1199112(119905) = int

119905

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(62)

Then

E10038171003817100381710038171199112 (1205912) minus 1199112 (1205911)

10038171003817100381710038172

le 6E

100381710038171003817100381710038171003817100381710038171003817int1205911minus120598

119905119896

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 119891 (119904 119911

120588(119904119911119904)) 119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904))119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904)119891(119904 119911

120588(119904119911119904))119889119904

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E10038171003817100381710038171003817100381710038171003817int1205911minus120598

0

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205911

1205911minus120598

(119878120572(120591

2minus 119904) minus 119878

120572(120591

1minus 119904)) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

+ 6E

100381710038171003817100381710038171003817100381710038171003817int1205912

1205911

119878120572(120591

2minus 119904) 120590 (119904) 119889119908 (119904)

100381710038171003817100381710038171003817100381710038171003817

2

le 62

119878119872

119891

119887120572

120572(1 + 119903

1015840) 119887

1minus120572


119905119896


(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205911

1205911minus120598

(1205911minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878119872

119891

119887120572

120572(1 + 119903

1015840)int

1205912

1205911

(1205912minus 120598 minus 119904)

120572minus1

119889119904

+ 62

119878(119888

1+ 119888

21199031015840) 119887

2(1minus120572) Tr (119876)


119905119896



2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205912minus 120598 minus 119904)

120572minus1] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205911

1205911minus120576

(1205911minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

+ 62

119878(119888

1+ 119888

21199031015840)Tr (119876) int

1205912

1205911

(1205912minus 120598 minus 119904)

2(120572minus1)] (119904) 119889119904

(63)


2(120591

2) minus 119911

2(120591

1)



119896 119896 = 1 119898 are completely


120572


2119909 119909 isin 119881



119905 isin [0 119887]Let 0 lt 119905 le 119904 le 119905



1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

0

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

0

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(64)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112(119905) minus 1199112120598(119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904)120590(119904)119889119908(119904)

10038171003817100381710038171003817100381710038171003817

2


le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(65)


119896+1] 119896 = 1 119898 let 119905

119896lt 119905 le 119904 le

119905119896+1


1199112120598(119905) = 119879

120572(119905) 120601 (0) + int

119905minus120598

119905119896

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

+ int119905minus120598

119905119896

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

(66)

where 120590 isin SΣ119911120588


120572(119905)


2120598(119905) 119909 isin 119881 is


E10038171003817100381710038171199112 (119905) minus 1199112120598 (119905)

10038171003817100381710038172

le 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 119891 (119904 119911

120588(119904119911119904)) 119889119904

10038171003817100381710038171003817100381710038171003817

2

+ 2E10038171003817100381710038171003817100381710038171003817int119905

119905minus120598

119878120572(119905 minus 119904) 120590 (119904) 119889119908 (119904)

10038171003817100381710038171003817100381710038171003817

2

le 22

119878

1205982120572

1205722119872

119891(1 + 119903

1015840) + 2

2

119878(119888

1+ 119888

21199031015840)Tr (119876)

times int119905

119905minus120598

(119905 minus 119904)2(120572minus1)] (119904) 119889119904

(67)


1(119905) 119909 isin 119881 Hence




4 An Example


119863120572

119905119906 (119905 119909) minus

1205972

1205971199092119906 (119909 119905)

isin int119905

minusinfin

1205831(119905 119909 119904 minus 119905) 119906 (119904 minus 120588

1(119905) 120588

2(119906 (119905)) 119909) 119889119904

+ [int119905

minusinfin

1205832(119905 119909 119904 minus 119905)

times 119906 (119904 minus 1205881(119905) 120588

2(119906 (119905)) 119909) 119889119904]

119889120573 (119905)

119889119905

0 le 119905 le 119887 0 le 119909 le 120587

119906 (119905 0) = 119906 (119905 120587) = 0 0 le 119905 le 119887

119906 (120591 119909) = 120601 (120591 119909) 120591 le 0 0 le 119909 le 120587

Δ119906 (119905119896 119909) = int

119905119896

minusinfin

]119896(119904 minus 119905

119896) 119906 (119904 119909) 119889119904 119896 = 1 119898

(68)


119905P) 119863120572

119905is the


1lt 119905







11991010158401015840isin H 119910 (0) = 119910 (120587) = 0

(69)


119899=11198992(119910 119910

119899)119910

119899 119910 isin D(119860) where 119910

119899(119909) =



119879 (119905) 119910 =

infin

sum119899=1

119890minus1198992119905(119910 119910

119899) 119910

119899 forall119910 isin H 119905 gt 0 (70)






120585 where





10038171003817100381710038171206011003817100381710038171003817B = sup


1003817100381710038171003817120601 (120591)1003817100381710038171003817 + (int

minus119903

minusinfin

119911 (120591)10038171003817100381710038171206011003817100381710038171003817119901

119889120591)

1119901

(71)



119873(119905) = ℎ(minus119905)12 and 119896(119905) = 1 + (int0

minus119905119911(120591)119889120591)

12

for 119905 ge 0 (see[34])


Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Here we assume that




with 119897119894= (int

0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12


119896 R rarr R 119896 = 1 2 119898 are con-


0

minus119905((120583

119894(119904))

2119911(119904))119889119904)

12




P(H) by

119891 (119905 120601) (119909) = int0

minusinfin

1205831(119905 120579 119909) 120601 (120579) (119909) 119889120579

Σ (119905 120601) (119909) = int0

minusinfin

1205832(119905 120579 119909) 120601 (120579) (119909) 119889120579

120588 (119905 120601) = 1205881(119905) 120588

2(1003817100381710038171003817120601 (0)

1003817100381710038171003817)

(72)


1198972and 119868

119896 le 119871





References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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