Existence of Mild Solutions and Controllability of...

Research ArticleExistence of Mild Solutions and Controllability ofFractional Impulsive Integrodifferential Systems withNonlocal Conditions

Haiyong Qin12 Zhenyun Gu2 Youliang Fu1 and Tongxing Li34

1School of Control Science and Engineering Shandong University Jinan Shandong 250061 China2School of Mathematics Qilu Normal University Jinan Shandong 250013 China3LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing Linyi University LinyiShandong 276005 China4School of Information Science and Engineering Linyi University Linyi Shandong 276005 China

Correspondence should be addressed to Tongxing Li litongx2007163com

Received 11 May 2017 Accepted 24 July 2017 Published 20 September 2017

Academic Editor Xinguang Zhang

Copyright copy 2017 Haiyong Qin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the existence results of nonlocal problems for a class of fractional impulsive integrodifferentialequations in Banach spaces We define a piecewise continuous control function to obtain the results on controllability of thecorresponding fractional impulsive integrodifferential control systems The results are obtained by means of fixed point methodsAn example to illustrate the applications of our main results is given

1 Introduction

In recent decades existence of mild solutions of nonlocalCauchy problems has been investigated extensively by manyresearchers (see [1ndash15] and the references cited therein) Thestudy of abstract nonlocal semilinear initial value problemswas initiated by Byszewski and Lakshmikantham [11] andByszewski [12] Byszewski [12] considered the existence anduniqueness ofmild strong and classical solutions of nonlocalCauchy problems Lin and Liu [8] studied the existenceand uniqueness of mild and classical solutions of semilinearintegrodifferential equations with nonlocal Cauchy prob-lems Using Krasnoselskiirsquos fixed point theorem Schauderrsquosfixed point theorem and Banach contraction principle Zhouand Jiao [13] obtained several criteria on the existence anduniqueness of mild solutions of nonlocal Cauchy problemsfor fractional evolution equations without impulse

Such analysis on nonlocal Cauchy problems is importantfrom an applied viewpoint since the nonlocal condition hasa better effect in applications than a classical initial one Forinstance the diffusion phenomenon of a small amount of gasin a transparent tube can be given a better description than

using the usual local Cauchy problem On the other handcontrollability of nonlocal problems in Banach spaces hasbecome an active area of investigation we refer the reader tofor example the papers [16ndash29] The most common methodis to transform the controllability problem into a fixed-point problem of solutions for an appropriate operator in afunction space that is the existence problem of differentialand integrodifferential equations Unfortunately by [16] weknow that the concept of mild solutions used in [14 15 17]was not suitable for fractional evolution systems

Chang et al [18] investigated the controllability of a classof first-order semilinear differential systems with nonlocalinitial conditions in a Banach space

1199091015840 (119905) = 119860119909 (119905) + 119891 (119905 119909 (119905)) + 119861119906 (119905) 119905 isin 119869 = [0 119887]

119909 (0) + 119892 (119909) = 1199090 isin X(1)

where 119860 generates a strongly continuous not necessarilycompact semigroup (119879(119905))119905ge0 in the Banach space X Suf-ficient conditions for the controllability of the first-order

HindawiJournal of Function SpacesVolume 2017 Article ID 6979571 11 pageshttpsdoiorg10115520176979571

2 Journal of Function Spaces

semilinear differential systemwith nonlocal initial conditionswere establishedThe approach used is Sadovskiirsquos fixed pointtheorem

Balachandran et al [19] discussed the controllability of aclass of fractional integrodifferential systems with nonlocalconditions in a Banach space

119889119902119909 (119905)119889119905119902 = 119860119909 (119905) + 119891 (119905 119909 (119905) (119867119909) (119905)) + 119861119906 (119905) 119905 isin 119869 = [0 119887]

119909 (0) + 119892 (119909) = 1199090 isin X(2)

Motivated by the work of the above papers and wideapplications of nonlocal Cauchy problems in various fieldsof natural sciences and engineering in this paper we studythe existence of nonlocal problems for a class of fractionalimpulsive integrodifferential systems in a Banach space of thefollowing type

119863119902119905119909 (119905) = 119860119909 (119905) + 119891 (119905 119909 (119905) (119867119909) (119905)) 119905 isin 119868 = [0 119887] 119905 = 119905119896

Δ119909|119905=119905119896 = 119868119896 (119909 (119905minus119896 )) 119896 = 1 2 119898119909 (0) + 119892 (119909) = 1199090 isin X

(3)

where (119867119909)(119905) = int1199050ℎ(119905 119904 119909(119904))119889119904 and 119863119902119905 is the Caputo

fractional derivative (0 lt 119902 lt 1) the state 119909(sdot) takes values inthe Banach space X 119860 119863(119860) sube X rarr X is the infinitesimalgenerator of a strongly continuous semigroup (119879(119905))119905ge0 ofuniformly bounded operators inX and119860 is a bounded linearoperator 119891 119868 times X times X rarr X is a given X-value functionℎ ΔtimesX rarr X is continuous here Δ = (119905 119904) 0 le 119904 le 119905 le 119887119868119896 X rarr X 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119898 lt 119905119898+1 = 119887Δ119909|119905=119905119896 = 119909(119905+119896 )minus119909(119905minus119896 ) 119909(119905+119896 ) = limℎrarr0+119909(119905119896 +ℎ) and 119909(119905minus119896 ) =limℎrarr0minus119909(119905119896 + ℎ) represent the right and left limits of 119909(119905) at119905 = 119905119896 respectively Using the similar method and a piecewisecontinuous control function we consider the controllabilityof a class of fractional impulsive integrodifferential systemswith nonlocal initial conditions

119863119902119905119909 (119905) = 119860119909 (119905) + 119891 (119905 119909 (119905) (119867119909) (119905)) + 119861119906 (119905) 119905 isin 119868 = [0 119887] 119905 = 119905119896

Δ119909|119905=119905119896 = 119868119896 (119909 (119905minus119896 )) 119896 = 1 2 119898119909 (0) + 119892 (119909) = 1199090 isin X

(4)

where 119861 is a bounded linear operator from 119880 to X and thecontrol function 119906(sdot) is given in 1198712[119868 119880] with 119880 as a Banachspace

We study the nonlocal initial problem (3) that describes amore general form than the previous ones reported in [18 19]We introduce a suitable concept of PC-mild solutions fornonlocal initial problem (3) We not only study the existenceand uniqueness of a mild solution for impulsive fractionalsemilinear integrodifferential equation (3) but also definea piecewise continuous control function and present theresults on the controllability of the corresponding fractionalimpulsive integrodifferential system (4) which include some

known results obtained in [14 17] Assumptions in our resultsare less restrictive

2 Preliminaries and Lemmas

Throughout this paper let us consider the set of functionsPC[119868X] = 119909 119868 rarr X | 119909 isin 119862[(119905119896 119905119896+1)X] and thereexist 119909(119905minus119896 ) and 119909(119905+119896 ) 119896 = 0 1 2 119898 with 119909(119905minus119896 ) = 119909(119905119896)Endowed with the norm 119909PC = sup119905isin119868119909(119905) it is easy toverify that (PC[119868X] sdot PC) is a Banach space Let 119871119861(X) bethe Banach space of all linear and bounded operators on XFor a 1198620-semigroup (119879(119905))119905ge0 we set1198721 = sup119905isin119868119879(119905)119871119861(X)For each positive constant 119903 we set 119861119903 = 119909 isin PC[119868X] 119909 le 119903 Obviously 119861119903 is a bounded closed and convexsubset

Definition 1 The fractional integral of order 120574 with the lowerlimit zero for a function 119891 is defined as

119868120574119891 (119905) = 1Γ (120574) int119905

0

119891 (119904)(119905 minus 119904)1minus120574 119889119904 119905 gt 0 120574 gt 0 (5)

provided that the right side is point-wise defined on [0infin)where Γ(sdot) is the gamma function

Definition 2 The Riemann-Liouville derivative of order 120574with the lower limit zero for a function 119891 [0infin) rarr R canbe written as

119871119863120574119891 (119905) = 1Γ (119899 minus 120574) 119889119899

119889119905119899 int119905

0

119891 (119904)(119905 minus 119904)1minus119899+120574 119889119904

119905 gt 0 119899 minus 1 lt 120574 lt 119899(6)

Definition 3 The Caputo derivative of order 120574 for a function119891 [0infin) rarr R can be written as

119863120574119905119891 (119905) = 119871119863120574(119891 (119905) minus 119899minus1sum119896=0

119905119896119896119891(119896) (0)) 119905 gt 0 119899 minus 1 lt 120574 lt 119899

(7)

Remark 4 If 119891 is an abstract function with values inX thenintegrals that appear in Definitions 1ndash3 are taken in Bochnerrsquossense

Definition 5 (see [20]) Let X be a Banach space a one-parameter family 119879(119905) 0 le 119905 lt infin of bounded linearoperators from X to X is a semigroup of bounded linearoperators onX if

(1) 119879(0) = 119868 119868 is the identity operator onX(2) 119879(119905 + 119904) = 119879(119905)119879(119904) for every 119905 119904 ge 0 (the semigroup

property)

A semigroup of bounded linear operators 119879(119905) is uniformlycontinuous if lim119905darr0119879(119905) minus 119868 = 0Definition 6 (see [21]) By a PC-mild solution of system (3)we mean a function 119909 isin PC[119868X] that satisfies the followingintegral equation

Journal of Function Spaces 3

119909 (119905) =

T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 119905 isin [0 1199051] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 +T (119905 minus 1199051) 1198681 (119909 (119905minus1 )) 119905 isin (1199051 1199052] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 + 119898sum

119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119905 isin ( 119905119898 119887] (8)

whereT(sdot) and S(sdot) are called characteristic solution opera-tors and are given by

T (119905) = intinfin0

120585119902 (120579) 119879 (119905119902120579) 119889120579

S (119905) = 119902intinfin0

120579120585119902 (120579) 119879 (119905119902120579) 119889120579(9)

and for 120579 isin (0infin)120585119902 (120579) = 1119902120579minus1minus1119902120603119902 (120579minus1119902) ge 0

120603119902 (120579) = 1120587infinsum119899=1

(minus1)119899minus1 120579minus119902119899minus1 Γ (119899119902 + 1)119899 sin (119902119899120587) (10)

where 120585119902 is a probability density function defined on (0infin)that is

120585119902 (120579) ge 0 120579 isin (0infin) intinfin0

120585119902 (120579) 119889120579 = 1 (11)

Definition 7 (see [21]) By a PC-mild solution of system (4)we mean a function 119909 isin PC[119868X] that satisfies the followingintegral equation

119909 (119905)

=

T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) [119891 (119904 119909 (119904) (119867119909) (119904)) + 119861119906 (119904)] 119889119904 119905 isin [0 1199051] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) [119891 (119904 119909 (119904) (119867119909) (119904)) + 119861119906 (119904)] 119889119904 +T (119905 minus 1199051) 1198681 (119909 (119905minus1 )) 119905 isin ( 1199051 1199052] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) [119891 (119904 119909 (119904) (119867119909) (119904)) + 119861119906 (119904)] 119889119904 + 119898sum

119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119905 isin ( 119905119898 119887]

(12)

Definition 8 System (4) is said to be controllable on theinterval 119868 if for every 1199090 1199091 isin X there exists a control119906 isin 1198712[119868 119880] such that a mild solution 119909 of (4) satisfies119909(119887) + 119892(119909) = 1199091Lemma 9 (see [20]) Linear operator 119860 is the infinitesimalgenerator of a uniformly continuous semigroup if and only if119860 is a bounded linear operator

Lemma 10 (see [13] Krasnoselskiirsquos fixed point theorem) LetX be a Banach space let 119861 be a bounded closed and convexsubset ofX and let 1198651 1198652 be maps of 119861 intoX such that 1198651119909 +1198652119910 isin 119861 for every pair 119909 119910 isin 119861 If 1198651 is a contraction and 1198652 iscompletely continuous then the equation 1198651119909 + 1198652119909 = 119909 has asolution in 119861Lemma 11 (see [22 23]) The operatorsT(119905) andS(119905) definedby (9) have the following properties

(i) For any fixed 119905 ge 0 T(119905) and S(119905) are linear andbounded operators that is for any 119909 isin X

T (119905) 119909 le 1198721 119909 S (119905) 119909 le 1199021198721Γ (1 + 119902) 119909

(13)

(ii) T(119905) 119905 ge 0 and S(119905) 119905 ge 0 are strongly continuous(iii) T(119905) 119905 ge 0 and S(119905) 119905 ge 0 are uniformly

continuous

Remark 12 Since the infinitesimal generator 119860 is a linearbounded operator and thanks to Definition 5 and Lemma 9we can get that (iii) is satisfied

Lemma 13 (see [21]) For 120590 isin (0 1] and 0 lt 119886 le 119887 |119886120590 minus119887120590| le(119887 minus 119886)120590


3 Existence and Uniqueness ofPC-Mild Solutions

In order to prove the existence and uniqueness of mildsolutions of (3) we have the following assumptions

(H1) 119891 119868 timesX timesX rarr X is continuous and there exist twofunctions 1205831 1205832 isin 119871[119868R+] such that

1003817100381710038171003817119891 (119905 1199091 1199101) minus 119891 (119905 1199092 1199102)1003817100381710038171003817le 1205831 (119905) 10038171003817100381710038171199091 minus 11990921003817100381710038171003817 + 1205832 (119905) 10038171003817100381710038171199101 minus 11991021003817100381710038171003817

1199091 1199092 1199101 1199102 isin X(14)

(H2) ℎ 998779 times X rarr X is continuous and there exists afunction ]1 isin 119862[119868R+] such that

1003817100381710038171003817ℎ (119905 119904 1199091) minus ℎ (119905 119904 1199092)1003817100381710038171003817 le ]1 (119905) 10038171003817100381710038171199091 minus 11990921003817100381710038171003817 1199091 1199092 isin X (15)

(H3) 119868119896 X rarr X are continuous and there exist 120596119896 isin119862[119868R+] such that

1003817100381710038171003817119868119896 (1199091) minus 119868119896 (1199092)1003817100381710038171003817 le 120596119896 (119905) 10038171003817100381710038171199091 minus 11990921003817100381710038171003817 1199091 1199092 isin X 119896 = 1 2 119898 (16)

(H4) 119892 is continuous and there exists a function 120601 isin119862[119868R+] such that

1003817100381710038171003817119892 (1199091) minus 119892 (1199092)1003817100381710038171003817 le 120601 (119905) 10038171003817100381710038171199091 minus 11990921003817100381710038171003817 (17)

(H5) The functionΩ119898(119905) 119868 rarr R+ is defined by

Ω119898 (119905) = 11989812059601198721 +1198721120601 (119905) + 1199021198721Γ (1 + 119902)times int1199050(119905 minus 119904)119902minus1 (1205831 (119904) + ]011198871205832 (119904)) 119889119904

(18)

where ]01 = max]1(119905) | 119905 isin 119868 1205960 = max120596119896(119905) | 119905 isin119868 119896 = 1 2 119898 and 0 lt Ω119898(119905) lt 1 119905 isin 119868(H10158405) The constant Ω119906 and function Ω1015840119898(119905) 119868 rarr R+ are

defined by

Ω119906 = 12059601198981198721 + 12060101198721 + 1199021198701198721Γ (1 + 119902)times int1198870(119887 minus 119904)119902minus1 (1205831 (119904) + ]011198871205832 (119904)) 119889119904

Ω1015840119898 (119905) = 12059601198981198721 + 12060101198721 + 1199021198721Γ (1 + 119902)times int1199050(119905 minus 119904)119902minus1 (1205831 (119905) + ]011198871205832 (119905)) 119889119904

+ 1199021198721Ω119906Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1 119889119904

(19)

where 1206010 = max120601(119905) | 119905 isin 119868 and 0 lt Ω1015840119898(119905) lt 1119905 isin 119868Theorem 14 If hypotheses (H1)ndash(H5) are satisfied then (3)has a unique PC-mild solution

Proof Define the operator 119876 on PC[119868X] by(119876119909) (119905)

=

T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 119905 isin [0 1199051] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 +T (119905 minus 1199051) 1198681 (119909 (119905minus1 )) 119905 isin ( 1199051 1199052] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 + 119898sum

119896=1


(20)

For 0 le 120591 lt 119905 le 1199051 by virtue of (20) we conclude that(119876119909) (119905) minus (119876119909) (120591) le T (119905) minusT (120591) 10038171003817100381710038171199090 minus 119892 (119909)1003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int119905

120591(119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

+ 10038171003817100381710038171003817100381710038171003817int120591

0(119905 minus 119904)119902minus1 [S (119905 minus 119904) minusS (120591 minus 119904)]

times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

+ 10038171003817100381710038171003817100381710038171003817int120591

0[(119905 minus 119904)119902minus1 minus (120591 minus 119904)119902minus1]S (120591 minus 119904)

times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817 (21)

It follows from Lemma 11 part (iii) and Lemma 13 that

(119876119909) (119905) minus (119876119909) (120591) 997888rarr 0 as 119905 997888rarr 120591 (22)


Thus we deduce that 119876119909 isin 119862[[0 1199051]X] For 1199051 lt 120591 lt 119905 le 1199052we have

(119876119909) (119905) minus (119876119909) (120591) le T (119905) minusT (120591) 10038171003817100381710038171199090 minus 119892 (119909)1003817100381710038171003817+ 1003817100381710038171003817T (119905 minus 1199051) minusT (120591 minus 1199051)1003817100381710038171003817 10038171003817100381710038171198681 (119909 (119905minus1 ))1003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int119905

120591(119905 minus 119904)119902minus1S (119905 minus 119904)

times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591

0(119905 minus 119904)119902minus1 [S (119905 minus 119904) minus S (120591 minus 119904)]

times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

(23)

From (23) we know that 119876119909 isin 119862[(1199051 1199052]X] Using thesame method we obtain 119876119909 isin 119862[(1199052 1199053]X] 119876119909 isin

119862[(119905119898 119887]X] and therefore 119876119909 isin PC[119868X] For each 119905 isin(119905119894 119905119894+1] 1 le 119894 le 119898 119909 119910 isin PC[119868X]1003817100381710038171003817(119876119909) (119905) minus (119876119910) (119905)1003817100381710038171003817 le 1198721120601 (119905) + 1199021198721Γ (1 + 119902) int

119905

0(119905

minus 119904)119902minus1 times (1205831 (119904) + ]011198871205832 (119904)) 119889119904 1003817100381710038171003817119909 minus 1199101003817100381710038171003817PC+ 1003817100381710038171003817100381710038171003817100381710038171003817119894sum119896=1

T (119905 minus 119905119894) 119868119896 (119909 (119905minus119896 ))

minus 119894sum119896=1

T (119905 minus 119905119894) 119868119896 (119910 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817 le Ω119894 (119905) 1003817100381710038171003817119909 minus 1199101003817100381710038171003817PC

(24)

When 119894 = 119898 we get

1003817100381710038171003817(119876119909) (119905) minus (119876119910) (119905)1003817100381710038171003817 le Ω119898 (119905) 1003817100381710038171003817119909 minus 1199101003817100381710038171003817PC (25)

It follows now fromΩ119894(119905) le Ω119898(119905) (H5) and the contractionmapping principle that 119876 has a unique fixed point 119909 isinPC[119868X] that is

119909 (119905) =


119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119905 isin (119905119898 119887] (26)

is a unique PC-mild solution of (3) The proof is complete

In order to obtain more existence results we have thefollowing assumptions

(H6) 119891 119868 timesXtimesX rarr X is continuous and there exist threefunctions 1205833 1205834 1205835 isin 119871[119868R+] such that1003817100381710038171003817119891 (119905 119909 119910)1003817100381710038171003817 le 1205833 (119905) + 1205834 (119905) 119909 + 1205835 (119905) 10038171003817100381710038171199101003817100381710038171003817

119905 isin 119868 119909 119910 isin X (27)

(H7) ℎ 998779 times X rarr X is continuous and there exist twofunctions ]2 ]3 isin 119862[119868R+] such that

ℎ (119905 119904 119909) le ]2 (119904) + ]3 (119904) 119909 119909 isin X (28)

(H8) 119868119896 X rarr X are continuous and there exist 120595119896 isin119862[119868R+] such that1003817100381710038171003817119868119896 (119909)1003817100381710038171003817 le 120595119896 (119905) 119909 119909 isin X (29)

Define 1205950 = max 120595119896(119905) | 119905 isin 119868 119896 = 1 2 119898

(H9) There exists a function 120581 isin 119862[119868R+] such that1003817100381710038171003817119892 (119909)1003817100381710038171003817 le 120581 (119905) 119909 119909 isin X (30)

Define 1205810 = max120581(119905) | 119905 isin 119868(H10) For all bounded subsets 119861119903 the set

Π119898ℎ120575 (119905) = int119905minusℎ0

(119905 minus 119904)119902minus1S120575 (119905 minus 119904) 119865 (119904) 119889119904

+ 119898sum119896=1

T120575 (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119909 isin 119861119903(31)

is relatively compact in X for arbitrary ℎ isin (0 119905) and120575 gt 0 whereT120575(119905) and S120575(119905) are defined by

T120575 (119905) = intinfin120575

120585119902 (120579) 119879 (119905119902120579) 119889120579S120575 (119905) = 119902intinfin

120575120579120585119902 (120579) 119879 (119905119902120579) 119889120579

(32)


(H101584010) For all bounded subsets 119861119903 the setΠ1015840119898ℎ120575 (119905)

= int119905minusℎ0

(119905 minus 119904)119902minus1S120575 (119905 minus 119904) [119865 (119904) + 119861119906 (119904)] 119889119904

+ 119898sum119896=1


is relatively compact in X for arbitrary ℎ isin (0 119905) and120575 gt 0Theorem 15 Let hypotheses (H4) and (H6)ndash(H10) be satisfiedIf the inequalities

1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(34)

hold where 1205932(119904) = 1205834(119904) + 1205835(119904) int1199040 ]3(120579) 119889120579 and 1206010 is as in(H10158405) then (3) has at least one PC-mild solution

Proof We shall present the results in six steps

Step 1 (Continuity of 119876 defined by (20) on (119905119894 119905119894+1] (119894 =0 1 2 119898)) Let 119909119899 119909 isin PC[119868X] and 119909119899 minus 119909lowastPC rarr0 (119899 rarr infin) Then 119903 = sup119899119909119899PC lt infin and 119909lowastPC lt 119903For 119905 isin (119905119894 119905119894+1] (119894 = 0 1 2 119898) we have

1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817 le 1199021198721Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1

times 1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904))minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817 119889119904+ 12059501198721 119898sum

119896=1

1003817100381710038171003817119868119896 (119909119899 (119905minus119896 )) minus 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817+1198721 1003817100381710038171003817119892 (119909119899) minus 119892 (119909)1003817100381710038171003817

(35)

Since the functions 119891 119868119896 and 119892 are continuous we concludethat

119891 (119904 119909119899 (119904) (119867119909119899) (119904)) 997888rarr 119891 (119904 119909 (119904) (119867119909) (119904)) 119892 (119909119899) 997888rarr 119892 (119909)

119868119896 (119909119899 (119905minus119896 )) 997888rarr 119868119896 (119909 (119905minus119896 )) 119899 997888rarr infin(36)

Applications of (H6) and (H7) yield1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904)) minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817le 21205833 (119904) + 21205835 (119904) int119904

0]2 (120579) 119889120579

+ (21205834 (119904) + 21205835 (119904) int1199040]3 (120579) 119889120579) 119903

(37)

which implies that

(119905 minus 119904)119902minus1sdot 1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904)) minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817isin 1198711 [119868R+]

(38)

By Lebesguersquos dominated convergence theorem we get

int1199050(119905 minus 119904)119902minus1 times 1003817100381710038171003817119891 (119904 119909119899 (119904) 119867119909119899 (119904))minus 119891 (119904 119909 (119904) 119867119909 (119904))1003817100381710038171003817 119889119904 997888rarr 0

(39)

and so

lim119899rarrinfin

1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817PC = 0 (40)

Step 2 (119876maps bounded sets into bounded sets in PC[119868X])From (20) we get

(119876119909) (119905)= 1003817100381710038171003817T (119905) [1199090 minus 119892 (119909)]1003817100381710038171003817+ int1199050(119905 minus 119904)119902minus1 1003817100381710038171003817S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817 119889119904

+ 119898sum119896=1

1003817100381710038171003817T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817

(41)

where 1003817100381710038171003817119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817le 1205833 (119904) + 1205835 (119904) int119904

0]2 (120579) 119889120579

+ (1205834 (119904) + 1205835 (119904) int1199040]3 (120579) 119889120579) 119909

le 1205931 (119904) + 1205932 (119904) 119909

(42)

By Lemma 11 and (42) we obtain

(119876119909) (119905) le 1199021198871199021198721Γ (1 + 119902) int119905

0(1205931 (119904) + 1205932 (119904) 119909) 119889119904

+1198721 100381710038171003817100381711990901003817100381710038171003817 + 11987211205810 119909 + 11989811987211205950 119909 (43)

Thus for any 119909 isin 119861119903 = 119909 isin PC[119868X] 119909PC le 119903 we have(119876119909) (119905)

le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

= 1205741

(44)


Hence we deduce that (119876119909)(119905) le 1205741 that is 119876 mapsbounded sets into bounded sets in PC[119868X]Step 3 (119876(119861119903) is equicontinuous with 119861119903 on (119905119894 119905119894+1] (119894 =0 1 2 119898)) For any 119909 isin 119861119903 1199051015840 11990510158401015840 isin (119905119894 119905119894+1] (119894 =0 1 2 119898) we obtain

10038171003817100381710038171003817(119876119909) (11990510158401015840) minus (119876119909) (1199051015840)10038171003817100381710038171003817 le 10038171003817100381710038171003817T (11990510158401015840) 1199090 minusT (1199051015840) 119909010038171003817100381710038171003817+ 10038171003817100381710038171003817T (11990510158401015840) 119892 (119909) minusT (1199051015840) 119892 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817100381710038171003817100381710038171003817int11990510158401015840

0(11990510158401015840 minus 119904)119902minus1S (11990510158401015840 minus 119904) 119865 (119904) 119889119904

minus int11990510158400(1199051015840 minus 119904)119902minus1S (1199051015840 minus 119904) 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817

+ 1003817100381710038171003817100381710038171003817100381710038171003817119898sum119896=1

T (11990510158401015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))

minus 119898sum119896=1

T (1199051015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817

(45)

Based on a straightforward computation we have

10038171003817100381710038171003817(119876119909) (11990510158401015840) minus (119876119909) (1199051015840)10038171003817100381710038171003817 le 10038171003817100381710038171003817T (11990510158401015840) minusT (1199051015840)10038171003817100381710038171003817 100381710038171003817100381711990901003817100381710038171003817+ 10038171003817100381710038171003817T (11990510158401015840) 119892 (119909) minusT (1199051015840) 119892 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817100381710038171003817100381710038171003817int11990510158401015840


+ 1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840


sdot 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817 +1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840

0(1199051015840 minus 119904)119902minus1

sdot [S (11990510158401015840 minus 119904) minusS (1199051015840 minus 119904)] 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817+ 1198981198721 10038171003817100381710038171003817T (11990510158401015840 minus 1199051015840) minus 11986810038171003817100381710038171003817 1003817100381710038171003817119868119896 (119909 (119905minus119896 ))1003817100381710038171003817

(46)

It follows from Lemma 11 part (iii) and Lemma 13 thatlim11990510158401015840rarr1199051015840(119876119909)(11990510158401015840) minus (119876119909)(1199051015840) = 0 Thus 119876(119861119903) is equicon-tinuous with 119861119903 on (119905119894 119905119894+1] (119894 = 0 1 2 119898)Step 4 (119875119894 map 119861119903 into a precompact set in X (119894 = 1 119898))We define the operator

(119876119909) (119905) = (119875119894119909) (119905) + (119871119909) (119905) (47)

where

(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)

= int1199050(119905 minus 119904)119902minus1S (119905 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904

+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(48)

Define Π = 119875119894119861119903 and Π(119905) = (119875119894119909)(119905) 119909 isin 119861119903 for 119905 isin 119868Set

Π119894ℎ120575 (119905) = (119875119894ℎ120575119909) (119905) 119909 isin 119861119903 (49)

where

Π119894ℎ120575 (119905) = int119905minusℎ0


+ 119894sum119896=1

T120575 (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119909 isin 119861119903 (50)

From hypotheses we imposed and the same method usedin [16 Theorem 32] it is not difficult to verify that the setΠ(119905) can be arbitrary approximated by the relatively compactset Π119894ℎ120575(119905) Thus 119875119894(119861119903)(119905) are relatively compact inX

Step 5 (119871119909 + 119875119894119910 isin 119861119903 for 119909 119910 isin 119861119903 (119894 = 1 119898)) Note that1199021198871199021198721Γ (1 + 119902) int

119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1 (51)

Choose

1198721 100381710038171003817100381711990901003817100381710038171003817 + (1199021198871199021198721Γ (1 + 119902)) int1198870 1205931 (119904) 1198891199041 minus (1199021198871199021198721Γ (1 + 119902)) int1198870 1205932 (119904) 119889119904 minus 11989811987211205950 minus11987211205810

le 119903(52)

and define operators 119871 and 119875119894 on 119861119903 by(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)


+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(53)

It is sufficient to proceed exactly as in step 1 to step 4 of theproof to deduce that 119875119894 are continuous and compact Thus tocomplete this proof it suffices to show that 119871 is a contraction


mapping and that 119871119909+119875119894119910 isin 119861119903 for 119909 119910 isin 119861119903 Indeed for any119909 isin 119861119903 by virtue of (43) and (51) we have

(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)

Consequently if 119909 119910 isin 119861119903 then 119871119909 + 119875119894119910 isin 119861119903Step 6 (119871 is a contraction mapping) For any 1199051015840 11990510158401015840 isin(119905119894 119905119894+1] (119894 = 0 1 2 119898) and 119909 119910 isin PC[119868X] we have

1003817100381710038171003817(119871119909) (119905) minus (119871119910) (119905)1003817100381710038171003817 le 1003817100381710038171003817T (119905) (119892 (119909) minus 119892 (119910))1003817100381710038171003817le T (119905) 1003817100381710038171003817119892 (119909) minus 119892 (119910)1003817100381710038171003817le 12060101198721 1003817100381710038171003817119909 minus 1199101003817100381710038171003817PC

(55)

Since 12060101198721 lt 1 119871 is a contraction mapping Hence byLemma 10 we conclude that (3) has at least one PC-mildsolution on 119868 This completes the proof

4 Controllability Results

In this section we impose the following conditions to provethe results

(H11) Define 119868119894 = (119905119894minus1 119905119894] (119894 = 1 2 119898 + 1) The linearoperator119882119894 from 1198712[119868119894 119880] intoX defined by

119882119894119906 = int1199051198940(119905119894 minus 119904)119902minus1S (119905119894 minus 119904) 119861119906 (119904) 119889119904 (56)

induces an invertible operator minus119894 defined on1198712[119868119894 119880]Ker119882119894 and there exists a positive constant119870 gt 0 such that 119861minus119894 le 119870Theorem 16 If hypotheses (H1)ndash(H4) (H10158405) and (H11) aresatisfied then system (4) is controllable on 119868Proof Using (H11) for an arbitrary function 119909(sdot) we definethe piecewise continuous control 119906 by

119906 (119905)

=

minus1 [1199090 + 1199091 minus 1199090119898 + 1 minusT (1199051) [1199090 minus 119892 (119909)] minus int11990510 (1199051 minus 119904)119902minus1S (1199051 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904] (119905) 119905 isin [0 1199051] minus2 [1199090 + 2 (1199091 minus 1199090)119898 + 1 minusT (1199052) [1199090 minus 119892 (119909)] minus int11990520 (1199052 minus 119904)119902minus1S (1199052 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 minusT (1199052 minus 1199051) 1198681 (119909 (119905minus1 ))] (119905) 119905 isin (1199051 1199052] minus119898+1 [1199091 minusT (119887) [1199090 minus 119892 (119909)] minus int1198870 (119887 minus 119904)119902minus1S (119887 minus 119904) 119891 (119904 119909 (119904) (119867119909) (119904)) 119889119904 minus 119898sum

119896=1

T (119887 minus 119905119896) 119868119896 (119909 (119905minus119896 ))] (119905) 119905 isin (119905119898 119887]

(57)

On the basis of this control with a similar proof toTheorem 14 we can conclude that the operator 119876 defined by

(119876119909) (119905)

=

T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) [119891 (119904 119909 (119904) (119867119909) (119904)) + 119861119906 (119904)] 119889119904 119905 isin [0 1199051] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) [119891 (119904 119909 (119904) (119867119909) (119904)) + 119861119906 (119904)] 119889119904 +T (119905 minus 1199051) 1198681 (119909 (119905minus1 )) 119905 isin (1199051 1199052] T (119905) [1199090 minus 119892 (119909)] + int1199050 (119905 minus 119904)119902minus1S (119905 minus 119904) [119891 (119904 119909 (119904) (119867119909) (119904)) + 119861119906 (119904)] 119889119904 + 119898sum

119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119905 isin (119905119898 119887]

(58)


has a fixed point 119909(sdot) This fixed point is a PC-mild solutionof system (4) which implies that the system is controllable on119868 The proof is complete

Theorem 17 Assume that hypotheses (H4) (H6)ndash(H9) (H101584010)and (H11) are satisfied If the inequalities

1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)

hold where 1198731 = 1199021198701198721 int1198870 (119887 minus 119904)119902minus11205932(119904) 119889119904Γ(1 + 119902) +11989811987011987211205950 and 1205932(119904) and 1206010 are as in Theorem 15 then system(4) is controllable on 119868Proof The proof is similar to that of Theorem 15 and so isomitted

5 Example

Consider the following nonlinear partial integrodifferentialequation of the form

1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)

where 120583 119869 times (0 1) rarr (0 1) is continuous Let us take X =119862([0 1]) Consider the operator 119860 119863(119860) sube X rarr X definedby

(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)

It is not difficult to get

119860119908 = 119908int10

1003816100381610038161003816119910 minus 1199041003816100381610038161003816 119889119904 = (12 minus 119910 (1 minus 119910)) 119908le 12 119908

(62)

and clearly 119860 is the infinitesimal generator of a uniformlycontinuous semigroup (119879(119905))119905ge0 on X Put 119909(119905)(119910) = 119911(119905 119910)and 119906(119905)(119910) = 120583(119905 119910) and take

119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)

where 1198960 and 1198961 are positive constants and 120596(119905) and 120601(119905) arecontinuous functions Then 119891 [0 1] times X times X rarr X and1198681 X rarr X are continuous functions 119891 119892 1198681 and ℎ satisfy(H6)ndash(H9) respectively

For 119910 isin (0 1] we define1198821119906 = int12

0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1

0(1 minus 119904)minus23S (1 minus 119904) 119861119906 (119904) 119889119904

(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)

and for 120579 isin (0infin)12058513 (120579) = 3120579minus412060313 (120579minus3) 12060313 (120579)

= 1120587infinsum119899=1

(minus1)119899minus1 120579minus(119899+3)3 Γ ((119899 + 3) 3)119899 sin(1198991205873 ) (66)

Moreover the linear operator 119882119894 from 1198712[119868119894 119880] (119894 = 1 2)into X induces an invertible operator minus119894 defined on1198712[119868119894 119880]Ker119882119894 and there exists a positive constant 119870 gt 0such that 119861minus119894 le 119870 that is (H11) is satisfied With thechoices of 119860 119891 119892119867 and 119861 = 119868 (the identity operator) wesee that (60) is an abstract formulation of (4) All conditionsof Theorem 17 are able to be fulfilled so we deduce that (60)is controllable on 119868 On the other hand we have1003817100381710038171003817119891 (119905 119909119867119909) minus 119891 (119905 119910119867119910)1003817100381710038171003817

le 1198960 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 1198961 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 1003817100381710038171003817ℎ (119905 119904 119909) minus ℎ (119905 119904 119910)1003817100381710038171003817 le 1198961 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 10038171003817100381710038171198681 (119909) minus 1198681 (119910)1003817100381710038171003817 le 120596 (119905) 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 1003817100381710038171003817119892 (119909) minus 119892 (119910)1003817100381710038171003817 le 120601 (119905) 1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(67)

It is easy to see that all assumptions of Theorem 16 aresatisfiedwhen using the suitable choices of 1198960 1198961 120596 120601 HenceTheorem 16 can also yield controllability of (60) on 119868


6 Conclusions

In this paper we studied the existence and uniqueness resultsfor a class of impulsive fractional semilinear integrodifferen-tial equations with nonlocal initial conditions in a Banachspace Introducing the concept of PC-mild solutions andusing the piecewise continuous control functions and uni-formly continuous semigroup we obtained the controllabilityresults for the corresponding fractional impulsive integrod-ifferential system Assuming that the semigroup is compactand utilizing some additional conditions Hernandez andOrsquoRegan [30] showed that some known results on exactcontrollability (see the references cited therein) are valid ifand only if the Banach space is finite dimensional RecentlyHernandez et al [31] pointed out that some recent results onexact controllability of abstract differential systems with anunbounded linear operator dominated by a sectorial operatorwere not applicable Contrary to those results we do not needin our results conflicting conditions which in a certain senseis a significant improvement compared to the results in thecited papers An illustrative example is given to demonstratethe effectiveness of the results obtained Our future work willfocus on constrained controllability nonlocal problems andtheir applications in nonlinear dynamical systems (see [32ndash36])

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Shandong Provincial Nat-ural Science Foundation (Grants nos ZR2016AB04 andZR2016JL021) a Project of ShandongProvinceHigher Educa-tional Science andTechnologyProgram (Grant no J17KB121)Major International (Regional) Joint Research Project ofNational Natural Science Foundation of China (Grant no61320106011) National Natural Science Foundation of China(Grants nos 61503171 and 61527809) China PostdoctoralScience Foundation (Grant no 2015M582091) Foundationfor Young Teachers of Qilu Normal University (Grants nos2016L0605 2017JX2311 and 2017JX2312) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and Scientific Research Foundation forUniversity Students of Qilu Normal University (Grant noXS2017L05)

References

[1] A Anguraj and K Karthikeyan ldquoExistence of solutions forimpulsive neutral functional differential equations with nonlo-cal conditionsrdquoNonlinear Analysis Theory Methods amp Applica-tions vol 70 no 7 pp 2717ndash2721 2009

[2] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 75 no1 pp 177ndash181 2012

[3] T Cardinali and P Rubbioni ldquoImpulsive mild solutions forsemilinear differential inclusions with nonlocal conditions inBanach spacesrdquo Nonlinear Analysis Theory Methods amp Appli-cations vol 75 no 2 pp 871ndash879 2012

[4] X Xue ldquoNonlocal nonlinear differential equations with a mea-sure of noncompactness in Banach spacesrdquo Nonlinear AnalysisTheory Methods amp Applications vol 70 no 7 pp 2593ndash26012009

[5] J Liang J van Casteren and T-J Xiao ldquoNonlocal Cauchy prob-lems for semilinear evolution equationsrdquo Nonlinear AnalysisTheory Methods amp Applications An International Multidisci-plinary Journal vol 50 no 2 Ser A Theory Methods pp 173ndash189 2002

[6] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications An International Multidisci-plinary Journal vol 57 no 2 pp 183ndash189 2004

[7] K Balachandran J Y Park and M Chandrasekaran ldquoNonlo-cal Cauchy problem for delay integrodifferential equations ofSobolev type in Banach spacesrdquo Applied Mathematics LettersAn International Journal of Rapid Publication vol 15 no 7 pp845ndash854 2002

[8] Y Lin and J H Liu ldquoSemilinear integrodifferential equationswith nonlocal Cauchy problemrdquo Nonlinear Analysis TheoryMethods and Applications vol 26 no 5 pp 1023ndash1033 1996

[9] J Liang and T-J Xiao ldquoSemilinear integrodifferential equationswith nonlocal initial conditionsrdquo Computers amp Mathematicswith Applications An International Journal vol 47 no 6-7 pp863ndash875 2004

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

[11] L Byszewski and V Lakshmikantham ldquoTheorem about theexistence and uniqueness of a solution of a nonlocal abstractCauchy problem in a Banach spacerdquo Applicable Analysis AnInternational Journal vol 40 no 1 pp 11ndash19 1991

[12] L Byszewski ldquoTheorems about the existence and uniqueness ofsolutions of a semilinear evolution nonlocal Cauchy problemrdquoJournal of Mathematical Analysis and Applications vol 162 no2 pp 494ndash505 1991

[13] Y Zhou and F Jiao ldquoNonlocal Cauchy problem for fractionalevolution equationsrdquo Nonlinear Analysis Real World Applica-tions An International Multidisciplinary Journal vol 11 no 5pp 4465ndash4475 2010

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

[15] O K Jaradat A Al-Omari and S Momani ldquoExistence of themild solution for fractional semilinear initial value problemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no9 pp 3153ndash3159 2008

[16] J Wang and Y Zhou ldquoComplete controllability of fractionalevolution systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 11 pp 4346ndash4355 2012

[17] K Balachandran and J Y Park ldquoControllability of fractionalintegrodifferential systems in Banach spacesrdquo Nonlinear Anal-ysis Hybrid Systems vol 3 no 4 pp 363ndash367 2009

[18] Y K Chang J J Nieto and W S Li ldquoControllability ofsemilinear differential systems with nonlocal initial conditionsin Banach spacesrdquo Journal of Optimization Theory and Applica-tions vol 142 no 2 pp 267ndash273 2009


[19] K Balachandran J P Dauer and P Balasubramaniam ldquoCon-trollability of nonlinear integrodifferential systems in Banachspacerdquo Journal of OptimizationTheory and Applications vol 84no 1 pp 83ndash91 1995

[20] A Pazy Semigroups of Linear Operator and Applications toPartial Differential Equations Springer New York NY USA1983

[21] J Wang M Feckan and Y Zhou ldquoOn the new concept of solu-tions and existence results for impulsive fractional evolutionequationsrdquoDynamics of Partial Differential Equations vol 8 no4 pp 345ndash361 2011

[22] J Wang and Y Zhou ldquoA class of fractional evolution equationsand optimal controlsrdquo Nonlinear Analysis Real World Applica-tions An International Multidisciplinary Journal vol 12 no 1pp 262ndash272 2011

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

[24] A Debbouche and D Baleanu ldquoControllability of frac-tional evolution nonlocal impulsive quasilinear delay integro-differential systemsrdquo Computers amp Mathematics with Applica-tions An International Journal vol 62 no 3 pp 1442ndash14502011

[25] J A Machado C Ravichandran M Rivero and J J TrujilloldquoControllability results for impulsive mixed-type functionalintegro-differential evolution equations with nonlocal condi-tionsrdquo Fixed PointTheory and Applications vol 2013 Article ID66 16 pages 2013

[26] H Qin X Zuo and J Liu ldquoExistence and controllability resultsfor fractional impulsive integrodifferential systems in Banachspacesrdquo Abstract and Applied Analysis vol 2013 Article ID295837 12 pages 2013

[27] N I Mahmudov ldquoApproximate controllability of fractionalSobolev-type evolution equations in Banach spacesrdquo Abstractand Applied Analysis vol 2013 Article ID 502839 9 pages 2013

[28] Z Liu and X Li ldquoOn the controllability of impulsive fractionalevolution inclusions in Banach spacesrdquo Journal of OptimizationTheory and Applications vol 156 no 1 pp 167ndash182 2013

[29] A G Ibrahim and N A Al Sarori ldquoMild solutions for nonlocalimpulsive fractional semilinear differential inclusions withdelay in Banach spacesrdquoAppliedMathematics vol 4 pp 40ndash562013

[30] E Hernandez and D OrsquoRegan ldquoControllability of Volterra-Fredholm type systems in Banach spacesrdquo Journal of theFranklin Institute vol 346 no 2 pp 95ndash101 2009

[31] E Hernandez D OrsquoRegan and K Balachandran ldquoCommentson some recent results on controllability of abstract differentialproblemsrdquo Journal of OptimizationTheory andApplications vol159 no 1 pp 292ndash295 2013

[32] J Klamka ldquoConstrained controllability of semilinear delayedsystemsrdquo Bulletin of the Polish Academy of Sciences TechnicalSciences vol 49 no 3 pp 505ndash515 2001

[33] J Klamka ldquoConstrained exact controllability of semilinearsystemsrdquo Systems amp Control Letters vol 47 no 2 pp 139ndash1472002

[34] J Klamka ldquoConstrained controllability of semilinear systemswith delayed controlsrdquoBulletin of the PolishAcademy of SciencesTechnical Sciences vol 56 no 4 pp 333ndash337 2008

[35] HQin C Zhang T Li and Y Chen ldquoControllability of abstractfractional differential evolution equations with nonlocal condi-tionsrdquo Journal of Mathematics and Computer Science vol 17 no2 pp 293ndash300 2017

[36] L Wang B Yang Y Chen X Zhang and J Orchard ldquoImprov-ing neural-network classifiers using nearest neighbor parti-tioningrdquo IEEE Transactions on Neural Networks and LearningSystems vol PP no 99 pp 1ndash13 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


semilinear differential systemwith nonlocal initial conditionswere establishedThe approach used is Sadovskiirsquos fixed pointtheorem

Balachandran et al [19] discussed the controllability of aclass of fractional integrodifferential systems with nonlocalconditions in a Banach space

119889119902119909 (119905)119889119905119902 = 119860119909 (119905) + 119891 (119905 119909 (119905) (119867119909) (119905)) + 119861119906 (119905) 119905 isin 119869 = [0 119887]

119909 (0) + 119892 (119909) = 1199090 isin X(2)

Motivated by the work of the above papers and wideapplications of nonlocal Cauchy problems in various fieldsof natural sciences and engineering in this paper we studythe existence of nonlocal problems for a class of fractionalimpulsive integrodifferential systems in a Banach space of thefollowing type

119863119902119905119909 (119905) = 119860119909 (119905) + 119891 (119905 119909 (119905) (119867119909) (119905)) 119905 isin 119868 = [0 119887] 119905 = 119905119896

Δ119909|119905=119905119896 = 119868119896 (119909 (119905minus119896 )) 119896 = 1 2 119898119909 (0) + 119892 (119909) = 1199090 isin X

(3)

where (119867119909)(119905) = int1199050ℎ(119905 119904 119909(119904))119889119904 and 119863119902119905 is the Caputo

fractional derivative (0 lt 119902 lt 1) the state 119909(sdot) takes values inthe Banach space X 119860 119863(119860) sube X rarr X is the infinitesimalgenerator of a strongly continuous semigroup (119879(119905))119905ge0 ofuniformly bounded operators inX and119860 is a bounded linearoperator 119891 119868 times X times X rarr X is a given X-value functionℎ ΔtimesX rarr X is continuous here Δ = (119905 119904) 0 le 119904 le 119905 le 119887119868119896 X rarr X 0 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119898 lt 119905119898+1 = 119887Δ119909|119905=119905119896 = 119909(119905+119896 )minus119909(119905minus119896 ) 119909(119905+119896 ) = limℎrarr0+119909(119905119896 +ℎ) and 119909(119905minus119896 ) =limℎrarr0minus119909(119905119896 + ℎ) represent the right and left limits of 119909(119905) at119905 = 119905119896 respectively Using the similar method and a piecewisecontinuous control function we consider the controllabilityof a class of fractional impulsive integrodifferential systemswith nonlocal initial conditions

119863119902119905119909 (119905) = 119860119909 (119905) + 119891 (119905 119909 (119905) (119867119909) (119905)) + 119861119906 (119905) 119905 isin 119868 = [0 119887] 119905 = 119905119896

Δ119909|119905=119905119896 = 119868119896 (119909 (119905minus119896 )) 119896 = 1 2 119898119909 (0) + 119892 (119909) = 1199090 isin X

(4)

where 119861 is a bounded linear operator from 119880 to X and thecontrol function 119906(sdot) is given in 1198712[119868 119880] with 119880 as a Banachspace

We study the nonlocal initial problem (3) that describes amore general form than the previous ones reported in [18 19]We introduce a suitable concept of PC-mild solutions fornonlocal initial problem (3) We not only study the existenceand uniqueness of a mild solution for impulsive fractionalsemilinear integrodifferential equation (3) but also definea piecewise continuous control function and present theresults on the controllability of the corresponding fractionalimpulsive integrodifferential system (4) which include some

known results obtained in [14 17] Assumptions in our resultsare less restrictive

2 Preliminaries and Lemmas

Throughout this paper let us consider the set of functionsPC[119868X] = 119909 119868 rarr X | 119909 isin 119862[(119905119896 119905119896+1)X] and thereexist 119909(119905minus119896 ) and 119909(119905+119896 ) 119896 = 0 1 2 119898 with 119909(119905minus119896 ) = 119909(119905119896)Endowed with the norm 119909PC = sup119905isin119868119909(119905) it is easy toverify that (PC[119868X] sdot PC) is a Banach space Let 119871119861(X) bethe Banach space of all linear and bounded operators on XFor a 1198620-semigroup (119879(119905))119905ge0 we set1198721 = sup119905isin119868119879(119905)119871119861(X)For each positive constant 119903 we set 119861119903 = 119909 isin PC[119868X] 119909 le 119903 Obviously 119861119903 is a bounded closed and convexsubset

Definition 1 The fractional integral of order 120574 with the lowerlimit zero for a function 119891 is defined as

119868120574119891 (119905) = 1Γ (120574) int119905

0

119891 (119904)(119905 minus 119904)1minus120574 119889119904 119905 gt 0 120574 gt 0 (5)

provided that the right side is point-wise defined on [0infin)where Γ(sdot) is the gamma function

Definition 2 The Riemann-Liouville derivative of order 120574with the lower limit zero for a function 119891 [0infin) rarr R canbe written as

119871119863120574119891 (119905) = 1Γ (119899 minus 120574) 119889119899

119889119905119899 int119905

0

119891 (119904)(119905 minus 119904)1minus119899+120574 119889119904

119905 gt 0 119899 minus 1 lt 120574 lt 119899(6)

Definition 3 The Caputo derivative of order 120574 for a function119891 [0infin) rarr R can be written as

119863120574119905119891 (119905) = 119871119863120574(119891 (119905) minus 119899minus1sum119896=0

119905119896119896119891(119896) (0)) 119905 gt 0 119899 minus 1 lt 120574 lt 119899

(7)

Remark 4 If 119891 is an abstract function with values inX thenintegrals that appear in Definitions 1ndash3 are taken in Bochnerrsquossense

Definition 5 (see [20]) Let X be a Banach space a one-parameter family 119879(119905) 0 le 119905 lt infin of bounded linearoperators from X to X is a semigroup of bounded linearoperators onX if

(1) 119879(0) = 119868 119868 is the identity operator onX(2) 119879(119905 + 119904) = 119879(119905)119879(119904) for every 119905 119904 ge 0 (the semigroup

property)

A semigroup of bounded linear operators 119879(119905) is uniformlycontinuous if lim119905darr0119879(119905) minus 119868 = 0Definition 6 (see [21]) By a PC-mild solution of system (3)we mean a function 119909 isin PC[119868X] that satisfies the followingintegral equation


119909 (119905) =


119896=1




120585119902 (120579) 119879 (119905119902120579) 119889120579

S (119905) = 119902intinfin0

120579120585119902 (120579) 119879 (119905119902120579) 119889120579(9)


120603119902 (120579) = 1120587infinsum119899=1




120585119902 (120579) 119889120579 = 1 (11)


119909 (119905)

=


119896=1


(12)




T (119905) 119909 le 1198721 119909 S (119905) 119909 le 1199021198721Γ (1 + 119902) 119909

(13)


continuous








1199091 1199092 1199101 1199102 isin X(14)






1003817100381710038171003817119892 (1199091) minus 119892 (1199092)1003817100381710038171003817 le 120601 (119905) 10038171003817100381710038171199091 minus 11990921003817100381710038171003817 (17)



(18)


defined by



+ 1199021198721Ω119906Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1 119889119904

(19)



=


119896=1


(20)



+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817 (21)







times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

(23)



119905

0(119905


T (119905 minus 119905119894) 119868119896 (119909 (119905minus119896 ))

minus 119894sum119896=1


(24)

When 119894 = 119898 we get



119909 (119905) =


119896=1





119905 isin 119868 119909 119910 isin X (27)


ℎ (119905 119904 119909) le ]2 (119904) + ]3 (119904) 119909 119909 isin X (28)


Define 1205950 = max 120595119896(119905) | 119905 isin 119868 119896 = 1 2 119898



Π119898ℎ120575 (119905) = int119905minusℎ0


+ 119898sum119896=1



T120575 (119905) = intinfin120575

120585119902 (120579) 119879 (119905119902120579) 119889120579S120575 (119905) = 119902intinfin

120575120579120585119902 (120579) 119879 (119905119902120579) 119889120579

(32)





+ 119898sum119896=1



1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(34)




1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817 le 1199021198721Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1

times 1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904))minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817 119889119904+ 12059501198721 119898sum

119896=1


(35)


119891 (119904 119909119899 (119904) (119867119909119899) (119904)) 997888rarr 119891 (119904 119909 (119904) (119867119909) (119904)) 119892 (119909119899) 997888rarr 119892 (119909)



0]2 (120579) 119889120579

+ (21205834 (119904) + 21205835 (119904) int1199040]3 (120579) 119889120579) 119903

(37)

which implies that


(38)



(39)

and so

lim119899rarrinfin

1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817PC = 0 (40)



+ 119898sum119896=1

1003817100381710038171003817T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817

(41)

where 1003817100381710038171003817119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817le 1205833 (119904) + 1205835 (119904) int119904

0]2 (120579) 119889120579

+ (1205834 (119904) + 1205835 (119904) int1199040]3 (120579) 119889120579) 119909

le 1205931 (119904) + 1205932 (119904) 119909

(42)


(119876119909) (119905) le 1199021198871199021198721Γ (1 + 119902) int119905

0(1205931 (119904) + 1205932 (119904) 119909) 119889119904

+1198721 100381710038171003817100381711990901003817100381710038171003817 + 11987211205810 119909 + 11989811987211205950 119909 (43)


le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

= 1205741

(44)






+ 1003817100381710038171003817100381710038171003817100381710038171003817119898sum119896=1

T (11990510158401015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))

minus 119898sum119896=1

T (1199051015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817

(45)




+ 1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840


sdot 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817 +1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840

0(1199051015840 minus 119904)119902minus1


(46)


(119876119909) (119905) = (119875119894119909) (119905) + (119871119909) (119905) (47)

where

(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)


+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(48)


Π119894ℎ120575 (119905) = (119875119894ℎ120575119909) (119905) 119909 isin 119861119903 (49)

where

Π119894ℎ120575 (119905) = int119905minusℎ0


+ 119894sum119896=1




119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1 (51)

Choose


le 119903(52)



+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(53)




(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)



(55)







119906 (119905)

=


119896=1


(57)


(119876119909) (119905)

=


119896=1


(58)




1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





119909 (119905) =


119896=1




120585119902 (120579) 119879 (119905119902120579) 119889120579

S (119905) = 119902intinfin0

120579120585119902 (120579) 119879 (119905119902120579) 119889120579(9)


120603119902 (120579) = 1120587infinsum119899=1




120585119902 (120579) 119889120579 = 1 (11)


119909 (119905)

=


119896=1


(12)




T (119905) 119909 le 1198721 119909 S (119905) 119909 le 1199021198721Γ (1 + 119902) 119909

(13)


continuous








1199091 1199092 1199101 1199102 isin X(14)






1003817100381710038171003817119892 (1199091) minus 119892 (1199092)1003817100381710038171003817 le 120601 (119905) 10038171003817100381710038171199091 minus 11990921003817100381710038171003817 (17)



(18)


defined by



+ 1199021198721Ω119906Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1 119889119904

(19)



=


119896=1


(20)



+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817 (21)







times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

(23)



119905

0(119905


T (119905 minus 119905119894) 119868119896 (119909 (119905minus119896 ))

minus 119894sum119896=1


(24)

When 119894 = 119898 we get



119909 (119905) =


119896=1





119905 isin 119868 119909 119910 isin X (27)


ℎ (119905 119904 119909) le ]2 (119904) + ]3 (119904) 119909 119909 isin X (28)


Define 1205950 = max 120595119896(119905) | 119905 isin 119868 119896 = 1 2 119898



Π119898ℎ120575 (119905) = int119905minusℎ0


+ 119898sum119896=1



T120575 (119905) = intinfin120575

120585119902 (120579) 119879 (119905119902120579) 119889120579S120575 (119905) = 119902intinfin

120575120579120585119902 (120579) 119879 (119905119902120579) 119889120579

(32)





+ 119898sum119896=1



1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(34)




1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817 le 1199021198721Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1

times 1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904))minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817 119889119904+ 12059501198721 119898sum

119896=1


(35)


119891 (119904 119909119899 (119904) (119867119909119899) (119904)) 997888rarr 119891 (119904 119909 (119904) (119867119909) (119904)) 119892 (119909119899) 997888rarr 119892 (119909)



0]2 (120579) 119889120579

+ (21205834 (119904) + 21205835 (119904) int1199040]3 (120579) 119889120579) 119903

(37)

which implies that


(38)



(39)

and so

lim119899rarrinfin

1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817PC = 0 (40)



+ 119898sum119896=1

1003817100381710038171003817T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817

(41)

where 1003817100381710038171003817119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817le 1205833 (119904) + 1205835 (119904) int119904

0]2 (120579) 119889120579

+ (1205834 (119904) + 1205835 (119904) int1199040]3 (120579) 119889120579) 119909

le 1205931 (119904) + 1205932 (119904) 119909

(42)


(119876119909) (119905) le 1199021198871199021198721Γ (1 + 119902) int119905

0(1205931 (119904) + 1205932 (119904) 119909) 119889119904

+1198721 100381710038171003817100381711990901003817100381710038171003817 + 11987211205810 119909 + 11989811987211205950 119909 (43)


le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

= 1205741

(44)






+ 1003817100381710038171003817100381710038171003817100381710038171003817119898sum119896=1

T (11990510158401015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))

minus 119898sum119896=1

T (1199051015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817

(45)




+ 1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840


sdot 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817 +1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840

0(1199051015840 minus 119904)119902minus1


(46)


(119876119909) (119905) = (119875119894119909) (119905) + (119871119909) (119905) (47)

where

(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)


+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(48)


Π119894ℎ120575 (119905) = (119875119894ℎ120575119909) (119905) 119909 isin 119861119903 (49)

where

Π119894ℎ120575 (119905) = int119905minusℎ0


+ 119894sum119896=1




119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1 (51)

Choose


le 119903(52)



+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(53)




(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)



(55)







119906 (119905)

=


119896=1


(57)


(119876119909) (119905)

=


119896=1


(58)




1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









1199091 1199092 1199101 1199102 isin X(14)






1003817100381710038171003817119892 (1199091) minus 119892 (1199092)1003817100381710038171003817 le 120601 (119905) 10038171003817100381710038171199091 minus 11990921003817100381710038171003817 (17)



(18)


defined by



+ 1199021198721Ω119906Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1 119889119904

(19)



=


119896=1


(20)



+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817 (21)







times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

(23)



119905

0(119905


T (119905 minus 119905119894) 119868119896 (119909 (119905minus119896 ))

minus 119894sum119896=1


(24)

When 119894 = 119898 we get



119909 (119905) =


119896=1





119905 isin 119868 119909 119910 isin X (27)


ℎ (119905 119904 119909) le ]2 (119904) + ]3 (119904) 119909 119909 isin X (28)


Define 1205950 = max 120595119896(119905) | 119905 isin 119868 119896 = 1 2 119898



Π119898ℎ120575 (119905) = int119905minusℎ0


+ 119898sum119896=1



T120575 (119905) = intinfin120575

120585119902 (120579) 119879 (119905119902120579) 119889120579S120575 (119905) = 119902intinfin

120575120579120585119902 (120579) 119879 (119905119902120579) 119889120579

(32)





+ 119898sum119896=1



1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(34)




1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817 le 1199021198721Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1

times 1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904))minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817 119889119904+ 12059501198721 119898sum

119896=1


(35)


119891 (119904 119909119899 (119904) (119867119909119899) (119904)) 997888rarr 119891 (119904 119909 (119904) (119867119909) (119904)) 119892 (119909119899) 997888rarr 119892 (119909)



0]2 (120579) 119889120579

+ (21205834 (119904) + 21205835 (119904) int1199040]3 (120579) 119889120579) 119903

(37)

which implies that


(38)



(39)

and so

lim119899rarrinfin

1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817PC = 0 (40)



+ 119898sum119896=1

1003817100381710038171003817T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817

(41)

where 1003817100381710038171003817119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817le 1205833 (119904) + 1205835 (119904) int119904

0]2 (120579) 119889120579

+ (1205834 (119904) + 1205835 (119904) int1199040]3 (120579) 119889120579) 119909

le 1205931 (119904) + 1205932 (119904) 119909

(42)


(119876119909) (119905) le 1199021198871199021198721Γ (1 + 119902) int119905

0(1205931 (119904) + 1205932 (119904) 119909) 119889119904

+1198721 100381710038171003817100381711990901003817100381710038171003817 + 11987211205810 119909 + 11989811987211205950 119909 (43)


le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

= 1205741

(44)






+ 1003817100381710038171003817100381710038171003817100381710038171003817119898sum119896=1

T (11990510158401015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))

minus 119898sum119896=1

T (1199051015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817

(45)




+ 1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840


sdot 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817 +1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840

0(1199051015840 minus 119904)119902minus1


(46)


(119876119909) (119905) = (119875119894119909) (119905) + (119871119909) (119905) (47)

where

(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)


+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(48)


Π119894ℎ120575 (119905) = (119875119894ℎ120575119909) (119905) 119909 isin 119861119903 (49)

where

Π119894ℎ120575 (119905) = int119905minusℎ0


+ 119894sum119896=1




119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1 (51)

Choose


le 119903(52)



+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(53)




(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)



(55)







119906 (119905)

=


119896=1


(57)


(119876119909) (119905)

=


119896=1


(58)




1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817+ 10038171003817100381710038171003817100381710038171003817int120591


times 119891 (119904 119909 (119904) (119867119909) (119904)) 11988911990410038171003817100381710038171003817100381710038171003817

(23)



119905

0(119905


T (119905 minus 119905119894) 119868119896 (119909 (119905minus119896 ))

minus 119894sum119896=1


(24)

When 119894 = 119898 we get



119909 (119905) =


119896=1





119905 isin 119868 119909 119910 isin X (27)


ℎ (119905 119904 119909) le ]2 (119904) + ]3 (119904) 119909 119909 isin X (28)


Define 1205950 = max 120595119896(119905) | 119905 isin 119868 119896 = 1 2 119898



Π119898ℎ120575 (119905) = int119905minusℎ0


+ 119898sum119896=1



T120575 (119905) = intinfin120575

120585119902 (120579) 119879 (119905119902120579) 119889120579S120575 (119905) = 119902intinfin

120575120579120585119902 (120579) 119879 (119905119902120579) 119889120579

(32)





+ 119898sum119896=1



1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(34)




1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817 le 1199021198721Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1

times 1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904))minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817 119889119904+ 12059501198721 119898sum

119896=1


(35)


119891 (119904 119909119899 (119904) (119867119909119899) (119904)) 997888rarr 119891 (119904 119909 (119904) (119867119909) (119904)) 119892 (119909119899) 997888rarr 119892 (119909)



0]2 (120579) 119889120579

+ (21205834 (119904) + 21205835 (119904) int1199040]3 (120579) 119889120579) 119903

(37)

which implies that


(38)



(39)

and so

lim119899rarrinfin

1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817PC = 0 (40)



+ 119898sum119896=1

1003817100381710038171003817T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817

(41)

where 1003817100381710038171003817119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817le 1205833 (119904) + 1205835 (119904) int119904

0]2 (120579) 119889120579

+ (1205834 (119904) + 1205835 (119904) int1199040]3 (120579) 119889120579) 119909

le 1205931 (119904) + 1205932 (119904) 119909

(42)


(119876119909) (119905) le 1199021198871199021198721Γ (1 + 119902) int119905

0(1205931 (119904) + 1205932 (119904) 119909) 119889119904

+1198721 100381710038171003817100381711990901003817100381710038171003817 + 11987211205810 119909 + 11989811987211205950 119909 (43)


le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

= 1205741

(44)






+ 1003817100381710038171003817100381710038171003817100381710038171003817119898sum119896=1

T (11990510158401015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))

minus 119898sum119896=1

T (1199051015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817

(45)




+ 1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840


sdot 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817 +1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840

0(1199051015840 minus 119904)119902minus1


(46)


(119876119909) (119905) = (119875119894119909) (119905) + (119871119909) (119905) (47)

where

(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)


+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(48)


Π119894ℎ120575 (119905) = (119875119894ℎ120575119909) (119905) 119909 isin 119861119903 (49)

where

Π119894ℎ120575 (119905) = int119905minusℎ0


+ 119894sum119896=1




119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1 (51)

Choose


le 119903(52)



+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(53)




(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)



(55)







119906 (119905)

=


119896=1


(57)


(119876119909) (119905)

=


119896=1


(58)




1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








+ 119898sum119896=1



1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(34)




1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817 le 1199021198721Γ (1 + 119902) int119905

0(119905 minus 119904)119902minus1

times 1003817100381710038171003817119891 (119904 119909119899 (119904) (119867119909119899) (119904))minus 119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817 119889119904+ 12059501198721 119898sum

119896=1


(35)


119891 (119904 119909119899 (119904) (119867119909119899) (119904)) 997888rarr 119891 (119904 119909 (119904) (119867119909) (119904)) 119892 (119909119899) 997888rarr 119892 (119909)



0]2 (120579) 119889120579

+ (21205834 (119904) + 21205835 (119904) int1199040]3 (120579) 119889120579) 119903

(37)

which implies that


(38)



(39)

and so

lim119899rarrinfin

1003817100381710038171003817119876119909119899 (119905) minus 119876119909 (119905)1003817100381710038171003817PC = 0 (40)



+ 119898sum119896=1

1003817100381710038171003817T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817

(41)

where 1003817100381710038171003817119891 (119904 119909 (119904) (119867119909) (119904))1003817100381710038171003817le 1205833 (119904) + 1205835 (119904) int119904

0]2 (120579) 119889120579

+ (1205834 (119904) + 1205835 (119904) int1199040]3 (120579) 119889120579) 119909

le 1205931 (119904) + 1205932 (119904) 119909

(42)


(119876119909) (119905) le 1199021198871199021198721Γ (1 + 119902) int119905

0(1205931 (119904) + 1205932 (119904) 119909) 119889119904

+1198721 100381710038171003817100381711990901003817100381710038171003817 + 11987211205810 119909 + 11989811987211205950 119909 (43)


le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

= 1205741

(44)






+ 1003817100381710038171003817100381710038171003817100381710038171003817119898sum119896=1

T (11990510158401015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))

minus 119898sum119896=1

T (1199051015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817

(45)




+ 1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840


sdot 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817 +1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840

0(1199051015840 minus 119904)119902minus1


(46)


(119876119909) (119905) = (119875119894119909) (119905) + (119871119909) (119905) (47)

where

(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)


+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(48)


Π119894ℎ120575 (119905) = (119875119894ℎ120575119909) (119905) 119909 isin 119861119903 (49)

where

Π119894ℎ120575 (119905) = int119905minusℎ0


+ 119894sum119896=1




119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1 (51)

Choose


le 119903(52)



+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(53)




(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)



(55)







119906 (119905)

=


119896=1


(57)


(119876119909) (119905)

=


119896=1


(58)




1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









+ 1003817100381710038171003817100381710038171003817100381710038171003817119898sum119896=1

T (11990510158401015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))

minus 119898sum119896=1

T (1199051015840 minus 119905119896) 119868119896 (119909 (119905minus119896 ))1003817100381710038171003817100381710038171003817100381710038171003817

(45)




+ 1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840


sdot 119865 (119904) 1198891199041003817100381710038171003817100381710038171003817100381710038171003817 +1003817100381710038171003817100381710038171003817100381710038171003817int1199051015840

0(1199051015840 minus 119904)119902minus1


(46)


(119876119909) (119905) = (119875119894119909) (119905) + (119871119909) (119905) (47)

where

(119871119909) (119905) = T (119905) [1199090 minus 119892 (119909)] (119875119894119909) (119905)


+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(48)


Π119894ℎ120575 (119905) = (119875119894ℎ120575119909) (119905) 119909 isin 119861119903 (49)

where

Π119894ℎ120575 (119905) = int119905minusℎ0


+ 119894sum119896=1




119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810 lt 1 (51)

Choose


le 119903(52)



+ 119894sum119896=1

T (119905 minus 119905119896) 119868119896 (119909 (119905minus119896 )) 119894 = 1 119898

(53)




(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)



(55)







119906 (119905)

=


119896=1


(57)


(119876119909) (119905)

=


119896=1


(58)




1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






(119876119909) (119905)le 1198721 100381710038171003817100381711990901003817100381710038171003817 + 1199021198871199021198721Γ (1 + 119902) int

119887

01205931 (119904) 119889119904

+ ( 1199021198871199021198721Γ (1 + 119902) int119887

01205932 (119904) 119889119904 + 11989811987211205950 +11987211205810) 119903

le 119903

(54)



(55)







119906 (119905)

=


119896=1


(57)


(119876119909) (119905)

=


119896=1


(58)




1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







1199021198871199021198721Γ (1 + 119902) int119887

0(1205932 (119904) + 1198731) 119889119904 + 11989811987211205950 +11987211205810 lt 1

12060101198721 lt 1(59)


5 Example


1205971312059711990513 119911 (119905 119910)

= int10(119910 minus 119904) 119911 (119904 119910) 119889119904 + 119891 (119905 119911 (119905 119910) 119867119911 (119905 119910))

+ 120583 (119905 119910) 119911 (119905 0) = 119911 (119905 1) = 0 119905 isin 119869 = [0 1] 119911 (0 119910) + 120601 (119905) 119911 (119905 119910) = 1199110 (119910) 0 le 119910 le 1Δ119911|119905=12 = 1198681 (119909(12

minus))

(60)


(119860119908) (119905) = int10(119910 minus 119904)119908 (119904) 119889119904 (61)


119860119908 = 119908int10


(62)


119891 (119905 119909119867119909) = 1198960119909 + 119867119909(119867119909) (119905) = int119905

0ℎ (119905 119904 119909 (119904)) 119889119904

ℎ (119905 119904 119909) = 11989611199091198681 (119909) = 120596 (119905) 119909119892 (119909) = 120601 (119905) 119909

(63)



0(12 minus 119904)

minus23

S(12 minus 119904) 119861119906 (119904) 1198891199041198822119906 = int1


(64)

where

T (119905) 119908 (119904) = intinfin0

12058513 (120579) 119908 (11990513120579 + 119904) 119889120579S (119905) 119908 (119904) = 13 int

infin

012057912058513 (120579) 119908 (11990513120579 + 119904) 119889120579

(65)


= 1120587infinsum119899=1




(67)



6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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Existence of Mild Solutions and Controllability of...

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