Research ArticleExistence of Mild and Classical Solutions for Nonlocal ImpulsiveIntegrodifferential Equations in Banach Spaces with Measure ofNoncompactness

K Karthikeyan1 A Anguraj2 K Malar3 and Juan J Trujillo4

1 Department of Mathematics KSR College of Technology Tiruchengode Tamil Nadu 637215 India2Department of Mathematics PSG College of Arts and Science Coimbatore Tamil Nadu 641014 India3 Department of Mathematics Erode Arts and Science College Erode Tamil Nadu 638 009 India4Universidad de La Laguna Departamento de Analisis Matematico CAstr Fco Snchez sn Tenerife 38271 La Laguna Spain

Correspondence should be addressed to K Karthikeyan karthi phd2010yahoocoin

Received 24 December 2013 Revised 24 May 2014 Accepted 24 May 2014 Published 19 June 2014

Academic Editor Juan J Nieto

Copyright copy 2014 K Karthikeyan et alThis 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

We study the existence of mild and classical solutions are proved for a class of impulsive integrodifferential equations with nonlocalconditions in Banach spacesThemain results are obtained by usingmeasure of noncompactness and semigroup theory An exampleis presented

1 Introduction

In this present paper we are concerned with the existence ofmild and classical solutions are proved for a class of impulsiveintegrodifferential equations with nonlocal conditions

1199061015840

(119905) = 119860119906 (119905)

+ 119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060+ 119892 (119906)

Δ119906 (119905119894) = 119868119894(119906 (119905119894)) 119894 = 1 2 119901

(1)

where119860 is the infinitesimal generator of a1198620-semigroup119879(119905)

in a Banach space119883 and 119891 [0 119886] times 119883 times119883 rarr 119883 119896 [0 119886] times

[0 119886] times 119883 rarr 119883 ℎ [0 119886] times [0 119886] times 119883 rarr 1198830 lt 1199051lt

1199052lt 1199053lt sdot sdot sdot lt 119905

119901lt 119886 and 119868

119894 119883 rarr 119883 119894 = 1 2 119901

are impulsive functions and 119892 PC([0 a]X) rarr X u0 isin X

and119883 is a real Banach space with norm sdot Δ119906(119905119894) = 119906(119905

+

119894) minus

119906(119905minus

119894) 119906(119905+119894) 119906(119905minus

119894) denote the right and left limits of 119906 at 119905

119894

respectivelyIntegrodifferential equations are important for investigat-

ing some problems raised from natural phenomena Theyhave been studied in many different aspects The theory ofsemigroups of bounded linear operators is closely related tothe solution of differential and integrodifferential equationsin Banach spaces In recent years this theory has been appliedto a large class of nonlinear differential equations in Banachspaces We refer to the papers [1ndash5] and the references citedtherein Based on the method of semigroups existence anduniqueness of mild strong and classical solutions of semi-linear evolution equations were discussed by Pazy [6] In [7]Xue studied the semilinear nonlocal differential equationswith measure of noncompactness in Banach spaces Lizamaand Pozo [8] investigated the existence of mild solutions forsemilinear integrodifferential equation with nonlocal initialconditions by using Hausdorff measure of noncompactnessvia a fixed point

In recent years the impulsive differential equations havebeen an object of intensive investigation because of the wide

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2014 Article ID 319250 10 pageshttpdxdoiorg1011552014319250

2 International Journal of Differential Equations

possibilities for their applications in various fields of scienceand technology as theoretical physics population dynamicseconomics and so forth see [9ndash13] The study of semilinearnonlocal initial problem was initiated by Byszewski [14 15]and the importance of the problem lies in the fact that it ismore general and yields better effect than the classical initialconditions Therefore it has been extensively studied undervarious conditions on the operator 119860 and the nonlinearity 119891by several authors [13 16ndash18]

Byszwski and Lakshmikantham [19] prove the existenceand uniqueness of mild solutions and classical solutionswhen 119891 and 119892 satisfy Lipschitz-type conditions Ntouyas andTsamotas [20 21] study the case of compactness conditions of119891 and 119879(119905) Zhu et al [22] studied the existence of mild solu-tions for abstract semilinear evolution equations in Banachspaces In [23] Liu discussed the existence and uniquenessof mild and classical solutions for the impulsive semilineardifferential evolution equation In [24] the authors studiedthe existence of mild solutions to an impulsive differen-tial equation with nonlocal conditions by applying Darbo-Sadovskiirsquos fixed point theorem In recent paper [25] Ahmadet al studied nonlocal problems of impulsive integrodiffer-ential equations with measure of noncompactness For somemore recent results and details see [26ndash29]

Motivated by the above-mentioned works we derivesome sufficient conditions for the solutions of integrodif-ferential equations (1) combining impulsive conditions andnonlocal conditions Our results are achieved by applyingthe Hausdorff measure of noncompactness and fixed pointtheorem In this paper we denote by 119873 = sup119879(119905) 119905 isin[0 119886] Without loss of generality we let 119906

0= 0

2 Preliminaries

Let (119883 sdot ) be a real Banach spaceWe denote by119862([0 119886] 119883)the space of X-valued continuous functions on [0 119886]with thenorm 119909 = sup 119909(119905) 119905 isin [0 119886] and by 1198711(0 119886 119883) the spaceof X-valued Bochner integrable functions on [0 119886] with thenorm 119891

1198711 = int

119886

0119891(119905)119889119905

We put 119869119894= (119905119894 119905119894+1] 119894 = 1 2 119901 In order to define the

mild solution of problem (1) we introduce the following setPC([0 119886] 119883) = 119906 [0 119886] rarr 119883 119906 is continuous

on 119869119894 119894 = 0 1 2 119901 and the right limit 119906(119905

119894) exists 119894 =

1 2 119901

Definition 1 A function 119906 isin PC([0 119886] 119883) is a mild solutionof (1) if

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(2)

The Hausdorff measure of noncompactness 120573Y (119861) is definedby 120573Y (119861) = inf119903 gt 0 119861 can be covered by finite number ofballs with radii 119903 for bounded set 119861 in a Banach space 119884

Lemma 2 (see [30]) Let Y be a real Banach space and 119861 119862 sube

Y be bounded with the following properties(1) 119861 is precompact if and only if 120573X(119861) = 0(2) 120573Y (119861) = 120573Y (119861) = 120573Y (conv119861) where 119861 and conv119861

mean the closure and convex hull of 119861 respectively(3) 120573Y (119861) le 120573Y (119862) where 119861 sube 119862(4) 120573Y (119861 + 119862) le 120573Y (119861) + 120573Y (119862) where 119861 + 119862 = 119909 + 119910

119909 isin 119861 119910 isin 119862(5) 120573Y (119861 cup 119862) le max120573Y (119861) 120573Y (119862)(6) 120573Y (120582119861) le |120582|120573Y (119861) for any 120582 isin R(7) if the map 119876 119863(119876) sube Y rarr Z is Lipschitz

continuous with constant 119896 then 120573Z(119876119861) le 119896120573Y (119861) forany bounded subset 119861 sube 119863(119876) where Z be a Banachspace

(8) 120573Y (119861) = inf119889Y (119861 119862) 119862 sube Y is precompact =

inf119889Y (119861 119862) 119862 sube Y is finite valued where 119889Y (119861 119862)means the nonsymmetric (or symmetric) Hausdorffdistance between 119861 and 119862 in Y

(9) if 119882119899+infin

119899=1is decreasing sequence of bounded closed

nonempty subsets of Y and lim119899rarrinfin

120573Y (119882119899) = 0 then⋂+infin

119899=1119882119899is nonempty and compact in Y

The map 119876 119882 sube Y rarr Y is said to be a 120573Y -contractionif there exists a positive constant 119896 lt 1 such that 120573Y (119876(119861)) le119896120573Y (119861) for any bounded closed subset 119861 sube 119882 where Y is aBanach space

Lemma 3 (Darbo-Sadovskii [30]) If 119882 sube 119884 is boundedclosed and convex the continuous map 119876 119882 rarr 119882 is a120573Y -contraction then the map 119876 has at least one fixed point in119882

Lemma 4 (see [2]) If 119882 sube 119875119862([0 119886] 119883) is bounded then120572(119882(119905)) le 120572

119875119862(119882) for all 119905 isin [0 119886] where 119882(119905) = 119906(119905)

119906 isin 119882 sube 119883 Furthermore if 119882 is equicontinuous on eachinterval 119869

119894of [0a] then 120572(119882(119905)) is continuous on [0a] and

120572119875119862(119882) = sup120572(119882(119905)) 119905 isin [0 119886]

Lemma 5 (see [3]) If 119906119899infin

119899=1sub 1198711(0 119886 119883) is uniformly

integrable then 120572(119906119899(119905)infin

119899=1) is measurable and

120572(int

119905

0

119906119899(119904) 119889119904

infin

119899=1

) le 2int

119905

0

120572 (119906119899(119904)infin

119899=1) 119889119904 (3)

Lemma 6 (see [31]) If the semigroup 119879(119905) is equicontinuousand 120578 isin 1198711(0 119886 119877+) then the set

119905 997888rarr int

119905

0

119879 (119905 minus 119904) 119906 (119904) 119889119904 119906 isin 1198711(0 119870 119877

+)

119906 (119904) le 120578 (119904) for ae 119904 isin [0 119886] (4)

is equicontinuous on [0a]

International Journal of Differential Equations 3

Lemma 7 (see [23]) If 119882 is bounded then for each 120576 gt

0 there is a sequence 119906119899infin

119899=1sube 119882 such that 120572(119882) le

2120572(119906119899infin

119899=1) + 120576

3 119892 Is Compact

In this section we give the existence results of nonlocalintegrodifferential equation (39) Here we list the followinghypotheses(1198671198921) The 119888

0semigroup 119879(119905) 0 le 119905 le 119886 generated by 119860 is

equicontinuous(1198671198922) (i) 119892 PC([0 119886] 119883) rarr 119883 is continuous and

compact(ii) There exists 119872 gt 0 such that 119892(119906) le

119872 for all 119906 isin PC([0 119886] 119883) and 119896(119904) =

max1 int1199040119896(119904 120591)119889120591

(I) Let 119868119894 119883 rarr 119883 be continuous compact map and

there are nondecreasing functions 119897119894 119877+rarr 119877+ satisfying

119868119894(119909) le 119897

119894(119909) 119894 = 1 2 119901

(1198671198911) There exists a continuous function 119886

119896 [0 119886] times

[0 119886] rarr [0infin) and a nondecreasing continuousfunction Ω

119896 119877+

rarr 119877+ such that 119896(119905 119904 119909) le

119886119896(119905 119904)Ω

119896(119909) for all 119909 isin 119883 ae 119905 119904 isin [0 119886] And

there exists at least one mild solution to the followingscalar equation

119898(119905) = 119872119873

+119872int

119905

0

119886119891(119904) Ω119891(119898 (119904) 119886

119896(119905 119904) Ω

119896119898(119904)

119887ℎ(119905 119904) Ω

ℎ119898(119904)) 119889119904

+119872

119901

sum

119894=1

119897119894(119898 (119905)) 119905 isin [0 119886]

(5)

(1198671198912) (i) 119891(sdot sdot 119909 119910) is measurable for 119909 119910 isin 119883 119891(119905 sdot sdot sdot)

is continuous for ae 119905 isin [0 119886](ii) There exist a function 119886

119891(sdot) isin 119871

1(0 119886 119877

+)

and an increasing continuous function Ω119891

119877+

rarr 119877+ such that 119891(119905 119909 119910 119911) le

119886119891(119905)Ω119891(119909 119910 119911) for all 119909 119910 119911 isin 119883 and ae

119905 isin [0 119886](iii) 119891 [0 119886] times 119883 times 119883 rarr 119883 is compact

(1198671198913) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(6)

for ae 119905 isin [0 119886] and for any bounded subset 119863 sub

PC([0 119886] 119883)Here we let 119896(119904) = int

119904

0119896(119904 120591)119889120591 and (1 + 2119896

1(119904) + 2

1198962(119904)) le 119876

Theorem 8 Assume that the hypotheses (1198671198921) (119867119892

2) I

(1198671198911) and (119867119891

2) are satisfied then the nonlocal impulsive

problem (1) has at least one mild solution

Proof Let m(t) be a solution of the scalar equation (5) themap 119870 PC([0 119886] 119883) rarr PC([0 119886] 119883) is defined by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) (7)

with

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

(1198702119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(8)

for all 119905 isin [0 119886]It is easy to see that the fixedpoint of119870 is themild solution

of nonlocal impulsive problem (1)From our hypotheses the continuity of 119870 is proved as

followsFor this purpose we assume that 119906

119899rarr 119906 in

PC([0 119886] 119883) It comes from the continuity of 119896 and ℎ

that 119896(119904 120591 119906119899(120591)) rarr 119896(119904 120591 119906(120591)) and ℎ(119904 120591 119906

119899(120591)) rarr

ℎ(119904 120591 119906(120591)) respectivelyBy Lebesgue convergence theorem

int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591 997888rarr int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591 997888rarr int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591

(9)

Similarly we have

119891(119904 119906119899(119904) int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591)

997888rarr 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

(10)

for all 119904 isin [0 119886] Consider

1003817100381710038171003817119870119906119899minus 119870119906

1003817100381710038171003817

le 1198731003817100381710038171003817119892 (119906119899) minus 119892 (119906)

1003817100381710038171003817

+ 119873int

119905

0

(

10038171003817100381710038171003817100381710038171003817

119891(119904 119906119899(119904) int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591

4 International Journal of Differential Equations

int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

) 119889119904

+

119901

sum

119894=1

1198721003817100381710038171003817119868119894(119906119899(119905119894)) minus 119868119894(119906 (119905119894))1003817100381710038171003817997888rarr 0

(11)

as 119899 rarr infin So 119870119906119899rarr 119870119906 in PC([0 119886] 119883) That is 119870 is

continuousWe denote119882

0= 119906 isin PC([0 a])X) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] then119882 sube PC([119900 119886] 119883) is bounded and convexDefine 119882

1= conv119870(119882

0) where conv means that the

closure of the convex hull in PC([0 119886] 119883)For any 119906 isin 119870(119882

0) we know that

(119906) (119905) le 119872119873

+119873int

119905

0

119886119891(119904) Ω119891(119898 (119904) 119886

119896(119905 119904) Ω

119896119898(119904)

119887ℎ(119905 119904) Ω

ℎ119898(119904)) 119889119904

+119872

119901

sum

119894=1

119897119894(119898 (119905)) 119905 isin [0 119886]

(12)

and by (1198671198912)1198821sub 1198820

From the Arzela-Ascoli theorem to prove the compact-ness of 119870 we can prove that 119870

1119906 119906 isin 119882

0is equicontinuous

and1198701119906(119905) sub 119883 is precompact for 119905 isin [0 119886]

10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119879 (119905 + 120590) 119892 (119906)

+ int

119905+120590

0

119879 (119905 + 120590 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

minus [119879 (119905) 119892 (119906) +int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904]

10038171003817100381710038171003817100381710038171003817

le1003817100381710038171003817[119879 (119905 + 120590) minus 119879 (119906)] 119892 (119906)

1003817100381710038171003817

+ int

119905+120590

119905

119879 (119905 + 120590 minus 119904)

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

+ int

119905

0

119879 (119905 + 120590 minus 119904) minus 119879 (119905 minus 119904)

times

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

le 119872119873119879 (120590) minus 119868

+ 119873int

119905+120590

0

119886119891(119904) Ω119891(119906 (119904) int

119904

0

119886119896(119904 120591)Ω

119896(119906 (119904)) 119889120591

int

119886

0

119887ℎ(119904 120591)Ω

ℎ(119906 (119904)) 119889120591) 119889119904

+ 119873int

119905

0

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868] 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

(13)

Since 119891 is compact 119872119873119879(120590) minus 119868 and [119879(120590) minus

119868]119891(119904 119906(119904) int1199040119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) rarr 0

as 120590 rarr 0 uniformly for 119904 isin [0 119886] and 119906 isin PC([0 119886] 119883) Thisimplies that for any 120576

1gt 0 and 120576

2gt 0 there exist a 120575 gt 0 such

that

[119879 (120590) minus 119868] le 1205761

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891 (119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205762

(14)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) Therefore

[119879 (120590) minus 119868]

+

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205761+ 1205762= 120576

(15)

We know that10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

le 119872119873120576 + 119873Ω119891(119898 (119904)

1003817100381710038171003817119886119896(119886 119886)

1003817100381710038171003817 Ω119896(119898 (119904))

1003817100381710038171003817119887ℎ(119886 119886)

1003817100381710038171003817 Ωℎ(119898 (119904))) + 119873120576

(16)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) So 1198701119906 119906 isin 119882

0 is

equicontinuous

International Journal of Differential Equations 5

The set 119879(119905 minus 119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591int119886

0ℎ(119904

120591 119906(120591))119889120591) 119905 119904 isin [0 119886] 119906 isin PC([0 119886] 119883) is precompact as fis compact and 119879(sdot) is a 119862

0-semigroup

So 1198701119906(119905) sub 119883 is precompact as 119870

1119906(119905) sub 119905 conv119879(119905 minus

119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) 119904 isin [0 119905]

119906 isin PC([0 119886] 119883) for all 119905 isin [0 119886]1198821is equicontinuous on

each interval 119869119894of [0 119886] For 119905

119894le 119905 lt 119905+120590 le 119905

119894+1 119894 = 1 2 119901

we have using the semigroup properties1003817100381710038171003817(1198702119906) (119905 + 120590) minus (119870

2119906) (119905)

1003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905+120590

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

le sum

0lt119905119894lt119905+120590

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

+ sum

0lt119905119894lt119905

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894)) minus 119879 (119905 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

(17)which follows that 119870

2119906 119906 isin 119882

0 is equicontinuous on

each 119869119894due to the equicontinuous of 119879(119905) and hypotheses

(I) Therefore1198821sub PC([0 119886] 119883) is bounded closed convex

nonempty and equicontinuous on each interval 119869119894 119894 =

0 1 2 119901We define 119882

119899+1= conv119870(119882

119899) for 119899 = 1 2 119901 From

above we know that 119882119899infin

119899=1is a decreasing sequence of

bounded closed convex nonempty subsets in PC([0 119886] 119883)and equicontinuous on each 119869

119894 119894 = 1 2 119901

Now for 119899 ge 1 and 119905 isin [0 119886] 119882119899(119905) and 119870(119882

119899(119905)) are

bounded subsets of119883 Hence for any 120576 gt 0 there is a sequence119906119896infin

119896=1sub 119882119899such that (see eg [2 page 125])

120573 (119882119899+1

(119905))

= 120573 (119870119882119899(119905))

le 2120573 (119879 (119905) 119892 (119906119896infin

119896=1))

+2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904)

+ 2120573(

119901

sum

119894=1

119879 (119905 minus 119905119894) 119868119894119906119896(119905119894)infin

119896=1) + 120576

(18)for 119905 isin [0 119886]

From the compactness of119892 and 119868119894 by Lemmas 2 and 5 and

(1198671198913) we have

120573 (119882119899+1

(119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

+2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119882119899(119904)) + 2

1198961(119904) 120573 (119882

119899(119904))

+21198962(119904) 120573 (119882

119899(119904))) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) 120573 (119882119899(119904)) (1 + 2

1198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119905

0

120578 (119904) 120573 (119882119899(119904)) 119889119904 + 120576

(19)

Since 120576 gt 0 is arbitrary it follows from the above inequalitythat

120573 (119882119899+1

(119905)) le 4119873119876int

119905

0

120578 (119904) 120573PC (119882119899 (119904)) 119889119904 (20)

6 International Journal of Differential Equations

for all 119905 isin [0 119886] Since119882119899is decreasing for 119899 we define119891

119899(119905) =

lim119899rarrinfin

120573(119882119899(119905)) for all 119905 isin [0 119886] From (20) we have

119891119899+1

(119905) le 4119873119876int

119905

0

120578 (119904) 119891119899(119904) 119889119904 (21)

for 119905 isin [0 119886] which implies that 119891119899(119905) = 0 for all 119905 isin [0 119886] By

Lemma 4 we know that

lim119899rarrinfin

120573PC (119882119899) = 0 (22)

Using Lemma 2 we also know that

119882 =

infin

⋂

119899=1

119882119899 (23)

is convex compact and nonempty in PC([0 119886] 119883) and119870(119882) sub 119882

By the famous Schauderrsquos fixed point theorem there existsat least one mild solution 119906 of the problem (1) where 119906 isin 119882is a fixed point of the continuous map119870

4 119892 Is Lipschitz

In this section we discuss the problem (1) when g is Lipschitzcontinuous and 119868

119894 119894 = 1 2 119901 is not compact We replace

hypotheses (1198671198912) (119868) by

(1198671198912)1015840There is a constant 119871 isin (0 1119872) such that 119892(119906) minus

119892(V) le 119871119906 minus VPC for all 119906 V isin PC([0 119886] 119883)(I1015840) There exists 119871

119894gt 0 119894 = 1 2 119901 such that 119868

119894(119906) minus

119868119894(V) le 119871

119894119906 minus V for all 119906 V isin 119883

Theorem 9 Assume that the hypotheses (1198671198921) (119867119891

2)1015840 (I1015840)

(1198671198911)ndash(119867119891

3) are satisfied Then the nonlocal impulsive prob-

lem (1) has at least one mild solution on [0a] provided that

119872119871 + 4119873119876int

119905

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894lt 1 (24)

Proof Define the operator 119870 PC([0 119886] 119883) rarr

PC([0 119886] 119883) by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) + (119870

3119906) (119905) (25)

With

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

(1198702119906) (119905) = int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

(1198703119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(26)

for all 119906 isin PC([0 119886] 119883)Define 119882

0= 119906 isin PC([0 119886] 119883) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] and let119882 = 1198701198820

Then from the proof of Theorem 8 we know that 119882is a bounded closed convex and equicontinuous subset ofPC([0 119886] 119883) and 119870119882 sub 119882 We will prove that 119870 is120573PC-contraction on 119882 Then Darbo-Sadovskii fixed pointtheorem can be used to get a fixed point of 119870 in 119882 whichis a mild solution of (1)

We first show that1198701is Lipschitz on PC([0 119886] 119883)

In fact take 119906 V isin PC([0 119886] 119883) arbitraryThen by (1198671198912)1015840

we have

1003817100381710038171003817(1198701119906) (119905) minus (119870

2119906) (119905)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906) minus 119892 (V)100381710038171003817

1003817le 119872119871119906 minus VPC forall119905 isin [0 119886]

(27)

It follows that 1198701119906 minus 119870

1V le 119872119871119906 minus VPC for all 119906 V isin

PC([0 119886] 119883) That is 1198701is Lipschitz with Lipschitz constant

119872119871Next for every bounded subset 119861 sub 119882 for any 120576 gt 0

there is a sequence 119906119896infin

119896=1sub 119861 such that 120573(119870

2119861(119905)) le

2120573(1198702119906119896(119905)infin

119896=1) + 120576 for 119905 isin [0 119886] Since 119861 and 119870

2119861 are

equicontinuous we get from Lemmas 2 and 5 and (1198671198913) that

120573 (1198702119861 (119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+ 120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

International Journal of Differential Equations 7

+ 2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+ 21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) (120573PC (119906119896infin

119896=1) + 2

1198961(119904) 120573PC (119906119896

infin

119896=1)

+ 21198962(119904) 120573PC (119906119896

infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) 120573PC (119861) (1 + 21198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119886

0

120578 (119904) 119889119904120573PC (119861) + 120576

(28)

for 119905 isin [0 119886] Since 120576 gt 0 arbitrary we have

120573PC (1198702119861) le 4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 (29)

for any bounded subset 119861 sub 119882

120573 (1198703119861 (119905)) le 2119873

119901

sum

119894=1

120573 (119868119894119906119896(119905119894)infin

119896=1)

le 2119873

119901

sum

119894=1

119871119894120573 (119906119896(119905119894)infin

119896=1)

120573PC (1198703119861) le 2119873

119901

sum

119894=1

119871119894120573PC (119861)

(30)

for any subset 119861 sub 119882 due to Lemma 2 (29) and (30) wehave

120573PC (119870119861)

= 120573PC (1198701119861) + 120573PC (1198702119861) + 120573PC (1198703119861)

le 119872119871120573PC (119861) +4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 + 2119873

119901

sum

119894=1

119871119894120573PC (119861)

le (119872119871 + 4119873119876int

119886

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894)120573PC (119861)

(31)

From (24) we know that 119870 is 120573PC-contraction on 119882 ByLemma 3 there is a fixed point 119906 of 119870 in119882 which is a mildsolution of problem (1)

5 Classical Solutions

To study the classical solutions let us recall the followingresult

Lemma 10 Assume that 1199060

isin 119863(119860) 119902119894isin 119863(119860) 119894 =

1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883)

Then the impulsive differential equation

1199061015840

(119905) = 119860119906 (119905) + 119891 (119905 119906 (119905)) 119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060

Δ119906 (119905) = 119902119894 119894 = 1 2 119901

(32)

has a unique classical solution u(sdot)which satisfies for 119905 isin [0 119886]

119906 (119905)

= 119879 (119905) 119906 (0) = int

119905

0

119879 (119905 minus 119904) 119891 (119904 119906 (119904)) 119889119904 + sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119902119894

(33)

Now we make the following assumption(1198671) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(34)

or ae 119905 isin [0 119886] and for any bounded subset119863 sub 119875119862([0 119886] 119883)

Theorem 11 Let (1198671) be satisfied and 119906(sdot) a mild solution ofthe problem (1) Assume that 119906(0) isin 119863(119860) 119868

119894(119906(119905119894)) isin 119863(119860)

119894 = 1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883) Then u(0) gives

rise to a classical solution of the problem (1)If 119906(sdot) is a uniquely determined mild solution then it gives

rise to a unique classical solution

Proof We can define 119902119894= 119868119894(119906(119905119894)) 119894 = 1 2 119901

From Lemma 10

V1015840 (119905) = 119860V (119905) + 119891(119905 V (119905) int119905

0

119896 (119905 119904 V (119904)) 119889119904

int

119886

0

ℎ (119905 119904 V (119904)) 119889119904) 119905 isin [0 119886] 119905 = 119905119894

V (0) = 1199060+ 119892 (119906)

ΔV (119905) = 119902119894 119894 = 1 2 119901

(35)

has a unique classical solution V(sdot) which satisfies for 119905 isin

[0 119886]

V (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(36)

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Differential Equations

possibilities for their applications in various fields of scienceand technology as theoretical physics population dynamicseconomics and so forth see [9ndash13] The study of semilinearnonlocal initial problem was initiated by Byszewski [14 15]and the importance of the problem lies in the fact that it ismore general and yields better effect than the classical initialconditions Therefore it has been extensively studied undervarious conditions on the operator 119860 and the nonlinearity 119891by several authors [13 16ndash18]

Byszwski and Lakshmikantham [19] prove the existenceand uniqueness of mild solutions and classical solutionswhen 119891 and 119892 satisfy Lipschitz-type conditions Ntouyas andTsamotas [20 21] study the case of compactness conditions of119891 and 119879(119905) Zhu et al [22] studied the existence of mild solu-tions for abstract semilinear evolution equations in Banachspaces In [23] Liu discussed the existence and uniquenessof mild and classical solutions for the impulsive semilineardifferential evolution equation In [24] the authors studiedthe existence of mild solutions to an impulsive differen-tial equation with nonlocal conditions by applying Darbo-Sadovskiirsquos fixed point theorem In recent paper [25] Ahmadet al studied nonlocal problems of impulsive integrodiffer-ential equations with measure of noncompactness For somemore recent results and details see [26ndash29]

Motivated by the above-mentioned works we derivesome sufficient conditions for the solutions of integrodif-ferential equations (1) combining impulsive conditions andnonlocal conditions Our results are achieved by applyingthe Hausdorff measure of noncompactness and fixed pointtheorem In this paper we denote by 119873 = sup119879(119905) 119905 isin[0 119886] Without loss of generality we let 119906

0= 0

2 Preliminaries

Let (119883 sdot ) be a real Banach spaceWe denote by119862([0 119886] 119883)the space of X-valued continuous functions on [0 119886]with thenorm 119909 = sup 119909(119905) 119905 isin [0 119886] and by 1198711(0 119886 119883) the spaceof X-valued Bochner integrable functions on [0 119886] with thenorm 119891

1198711 = int

119886

0119891(119905)119889119905

We put 119869119894= (119905119894 119905119894+1] 119894 = 1 2 119901 In order to define the

mild solution of problem (1) we introduce the following setPC([0 119886] 119883) = 119906 [0 119886] rarr 119883 119906 is continuous

on 119869119894 119894 = 0 1 2 119901 and the right limit 119906(119905

119894) exists 119894 =

1 2 119901

Definition 1 A function 119906 isin PC([0 119886] 119883) is a mild solutionof (1) if

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(2)

The Hausdorff measure of noncompactness 120573Y (119861) is definedby 120573Y (119861) = inf119903 gt 0 119861 can be covered by finite number ofballs with radii 119903 for bounded set 119861 in a Banach space 119884

Lemma 2 (see [30]) Let Y be a real Banach space and 119861 119862 sube

Y be bounded with the following properties(1) 119861 is precompact if and only if 120573X(119861) = 0(2) 120573Y (119861) = 120573Y (119861) = 120573Y (conv119861) where 119861 and conv119861

mean the closure and convex hull of 119861 respectively(3) 120573Y (119861) le 120573Y (119862) where 119861 sube 119862(4) 120573Y (119861 + 119862) le 120573Y (119861) + 120573Y (119862) where 119861 + 119862 = 119909 + 119910

119909 isin 119861 119910 isin 119862(5) 120573Y (119861 cup 119862) le max120573Y (119861) 120573Y (119862)(6) 120573Y (120582119861) le |120582|120573Y (119861) for any 120582 isin R(7) if the map 119876 119863(119876) sube Y rarr Z is Lipschitz

continuous with constant 119896 then 120573Z(119876119861) le 119896120573Y (119861) forany bounded subset 119861 sube 119863(119876) where Z be a Banachspace

(8) 120573Y (119861) = inf119889Y (119861 119862) 119862 sube Y is precompact =

inf119889Y (119861 119862) 119862 sube Y is finite valued where 119889Y (119861 119862)means the nonsymmetric (or symmetric) Hausdorffdistance between 119861 and 119862 in Y

(9) if 119882119899+infin

119899=1is decreasing sequence of bounded closed

nonempty subsets of Y and lim119899rarrinfin

120573Y (119882119899) = 0 then⋂+infin

119899=1119882119899is nonempty and compact in Y

The map 119876 119882 sube Y rarr Y is said to be a 120573Y -contractionif there exists a positive constant 119896 lt 1 such that 120573Y (119876(119861)) le119896120573Y (119861) for any bounded closed subset 119861 sube 119882 where Y is aBanach space

Lemma 3 (Darbo-Sadovskii [30]) If 119882 sube 119884 is boundedclosed and convex the continuous map 119876 119882 rarr 119882 is a120573Y -contraction then the map 119876 has at least one fixed point in119882

Lemma 4 (see [2]) If 119882 sube 119875119862([0 119886] 119883) is bounded then120572(119882(119905)) le 120572

119875119862(119882) for all 119905 isin [0 119886] where 119882(119905) = 119906(119905)

119906 isin 119882 sube 119883 Furthermore if 119882 is equicontinuous on eachinterval 119869

119894of [0a] then 120572(119882(119905)) is continuous on [0a] and

120572119875119862(119882) = sup120572(119882(119905)) 119905 isin [0 119886]

Lemma 5 (see [3]) If 119906119899infin

119899=1sub 1198711(0 119886 119883) is uniformly

integrable then 120572(119906119899(119905)infin

119899=1) is measurable and

120572(int

119905

0

119906119899(119904) 119889119904

infin

119899=1

) le 2int

119905

0

120572 (119906119899(119904)infin

119899=1) 119889119904 (3)

Lemma 6 (see [31]) If the semigroup 119879(119905) is equicontinuousand 120578 isin 1198711(0 119886 119877+) then the set

119905 997888rarr int

119905

0

119879 (119905 minus 119904) 119906 (119904) 119889119904 119906 isin 1198711(0 119870 119877

+)

119906 (119904) le 120578 (119904) for ae 119904 isin [0 119886] (4)

is equicontinuous on [0a]

International Journal of Differential Equations 3

Lemma 7 (see [23]) If 119882 is bounded then for each 120576 gt

0 there is a sequence 119906119899infin

119899=1sube 119882 such that 120572(119882) le

2120572(119906119899infin

119899=1) + 120576

3 119892 Is Compact

In this section we give the existence results of nonlocalintegrodifferential equation (39) Here we list the followinghypotheses(1198671198921) The 119888

0semigroup 119879(119905) 0 le 119905 le 119886 generated by 119860 is

equicontinuous(1198671198922) (i) 119892 PC([0 119886] 119883) rarr 119883 is continuous and

compact(ii) There exists 119872 gt 0 such that 119892(119906) le

119872 for all 119906 isin PC([0 119886] 119883) and 119896(119904) =

max1 int1199040119896(119904 120591)119889120591

(I) Let 119868119894 119883 rarr 119883 be continuous compact map and

there are nondecreasing functions 119897119894 119877+rarr 119877+ satisfying

119868119894(119909) le 119897

119894(119909) 119894 = 1 2 119901

(1198671198911) There exists a continuous function 119886

119896 [0 119886] times

[0 119886] rarr [0infin) and a nondecreasing continuousfunction Ω

119896 119877+

rarr 119877+ such that 119896(119905 119904 119909) le

119886119896(119905 119904)Ω

119896(119909) for all 119909 isin 119883 ae 119905 119904 isin [0 119886] And

there exists at least one mild solution to the followingscalar equation

119898(119905) = 119872119873

+119872int

119905

0

119886119891(119904) Ω119891(119898 (119904) 119886

119896(119905 119904) Ω

119896119898(119904)

119887ℎ(119905 119904) Ω

ℎ119898(119904)) 119889119904

+119872

119901

sum

119894=1

119897119894(119898 (119905)) 119905 isin [0 119886]

(5)

(1198671198912) (i) 119891(sdot sdot 119909 119910) is measurable for 119909 119910 isin 119883 119891(119905 sdot sdot sdot)

is continuous for ae 119905 isin [0 119886](ii) There exist a function 119886

119891(sdot) isin 119871

1(0 119886 119877

+)

and an increasing continuous function Ω119891

119877+

rarr 119877+ such that 119891(119905 119909 119910 119911) le

119886119891(119905)Ω119891(119909 119910 119911) for all 119909 119910 119911 isin 119883 and ae

119905 isin [0 119886](iii) 119891 [0 119886] times 119883 times 119883 rarr 119883 is compact

(1198671198913) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(6)

for ae 119905 isin [0 119886] and for any bounded subset 119863 sub

PC([0 119886] 119883)Here we let 119896(119904) = int

119904

0119896(119904 120591)119889120591 and (1 + 2119896

1(119904) + 2

1198962(119904)) le 119876

Theorem 8 Assume that the hypotheses (1198671198921) (119867119892

2) I

(1198671198911) and (119867119891

2) are satisfied then the nonlocal impulsive

problem (1) has at least one mild solution

Proof Let m(t) be a solution of the scalar equation (5) themap 119870 PC([0 119886] 119883) rarr PC([0 119886] 119883) is defined by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) (7)

with

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

(1198702119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(8)

for all 119905 isin [0 119886]It is easy to see that the fixedpoint of119870 is themild solution

of nonlocal impulsive problem (1)From our hypotheses the continuity of 119870 is proved as

followsFor this purpose we assume that 119906

119899rarr 119906 in

PC([0 119886] 119883) It comes from the continuity of 119896 and ℎ

that 119896(119904 120591 119906119899(120591)) rarr 119896(119904 120591 119906(120591)) and ℎ(119904 120591 119906

119899(120591)) rarr

ℎ(119904 120591 119906(120591)) respectivelyBy Lebesgue convergence theorem

int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591 997888rarr int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591 997888rarr int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591

(9)

Similarly we have

119891(119904 119906119899(119904) int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591)

997888rarr 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

(10)

for all 119904 isin [0 119886] Consider

1003817100381710038171003817119870119906119899minus 119870119906

1003817100381710038171003817

le 1198731003817100381710038171003817119892 (119906119899) minus 119892 (119906)

1003817100381710038171003817

+ 119873int

119905

0

(

10038171003817100381710038171003817100381710038171003817

119891(119904 119906119899(119904) int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591

4 International Journal of Differential Equations

int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

) 119889119904

+

119901

sum

119894=1

1198721003817100381710038171003817119868119894(119906119899(119905119894)) minus 119868119894(119906 (119905119894))1003817100381710038171003817997888rarr 0

(11)

as 119899 rarr infin So 119870119906119899rarr 119870119906 in PC([0 119886] 119883) That is 119870 is

continuousWe denote119882

0= 119906 isin PC([0 a])X) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] then119882 sube PC([119900 119886] 119883) is bounded and convexDefine 119882

1= conv119870(119882

0) where conv means that the

closure of the convex hull in PC([0 119886] 119883)For any 119906 isin 119870(119882

0) we know that

(119906) (119905) le 119872119873

+119873int

119905

0

119886119891(119904) Ω119891(119898 (119904) 119886

119896(119905 119904) Ω

119896119898(119904)

119887ℎ(119905 119904) Ω

ℎ119898(119904)) 119889119904

+119872

119901

sum

119894=1

119897119894(119898 (119905)) 119905 isin [0 119886]

(12)

and by (1198671198912)1198821sub 1198820

From the Arzela-Ascoli theorem to prove the compact-ness of 119870 we can prove that 119870

1119906 119906 isin 119882

0is equicontinuous

and1198701119906(119905) sub 119883 is precompact for 119905 isin [0 119886]

10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119879 (119905 + 120590) 119892 (119906)

+ int

119905+120590

0

119879 (119905 + 120590 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

minus [119879 (119905) 119892 (119906) +int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904]

10038171003817100381710038171003817100381710038171003817

le1003817100381710038171003817[119879 (119905 + 120590) minus 119879 (119906)] 119892 (119906)

1003817100381710038171003817

+ int

119905+120590

119905

119879 (119905 + 120590 minus 119904)

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

+ int

119905

0

119879 (119905 + 120590 minus 119904) minus 119879 (119905 minus 119904)

times

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

le 119872119873119879 (120590) minus 119868

+ 119873int

119905+120590

0

119886119891(119904) Ω119891(119906 (119904) int

119904

0

119886119896(119904 120591)Ω

119896(119906 (119904)) 119889120591

int

119886

0

119887ℎ(119904 120591)Ω

ℎ(119906 (119904)) 119889120591) 119889119904

+ 119873int

119905

0

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868] 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

(13)

Since 119891 is compact 119872119873119879(120590) minus 119868 and [119879(120590) minus

119868]119891(119904 119906(119904) int1199040119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) rarr 0

as 120590 rarr 0 uniformly for 119904 isin [0 119886] and 119906 isin PC([0 119886] 119883) Thisimplies that for any 120576

1gt 0 and 120576

2gt 0 there exist a 120575 gt 0 such

that

[119879 (120590) minus 119868] le 1205761

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891 (119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205762

(14)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) Therefore

[119879 (120590) minus 119868]

+

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205761+ 1205762= 120576

(15)

We know that10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

le 119872119873120576 + 119873Ω119891(119898 (119904)

1003817100381710038171003817119886119896(119886 119886)

1003817100381710038171003817 Ω119896(119898 (119904))

1003817100381710038171003817119887ℎ(119886 119886)

1003817100381710038171003817 Ωℎ(119898 (119904))) + 119873120576

(16)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) So 1198701119906 119906 isin 119882

0 is

equicontinuous

International Journal of Differential Equations 5

The set 119879(119905 minus 119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591int119886

0ℎ(119904

120591 119906(120591))119889120591) 119905 119904 isin [0 119886] 119906 isin PC([0 119886] 119883) is precompact as fis compact and 119879(sdot) is a 119862

0-semigroup

So 1198701119906(119905) sub 119883 is precompact as 119870

1119906(119905) sub 119905 conv119879(119905 minus

119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) 119904 isin [0 119905]

119906 isin PC([0 119886] 119883) for all 119905 isin [0 119886]1198821is equicontinuous on

each interval 119869119894of [0 119886] For 119905

119894le 119905 lt 119905+120590 le 119905

119894+1 119894 = 1 2 119901

we have using the semigroup properties1003817100381710038171003817(1198702119906) (119905 + 120590) minus (119870

2119906) (119905)

1003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905+120590

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

le sum

0lt119905119894lt119905+120590

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

+ sum

0lt119905119894lt119905

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894)) minus 119879 (119905 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

(17)which follows that 119870

2119906 119906 isin 119882

0 is equicontinuous on

each 119869119894due to the equicontinuous of 119879(119905) and hypotheses

(I) Therefore1198821sub PC([0 119886] 119883) is bounded closed convex

nonempty and equicontinuous on each interval 119869119894 119894 =

0 1 2 119901We define 119882

119899+1= conv119870(119882

119899) for 119899 = 1 2 119901 From

above we know that 119882119899infin

119899=1is a decreasing sequence of

bounded closed convex nonempty subsets in PC([0 119886] 119883)and equicontinuous on each 119869

119894 119894 = 1 2 119901

Now for 119899 ge 1 and 119905 isin [0 119886] 119882119899(119905) and 119870(119882

119899(119905)) are

bounded subsets of119883 Hence for any 120576 gt 0 there is a sequence119906119896infin

119896=1sub 119882119899such that (see eg [2 page 125])

120573 (119882119899+1

(119905))

= 120573 (119870119882119899(119905))

le 2120573 (119879 (119905) 119892 (119906119896infin

119896=1))

+2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904)

+ 2120573(

119901

sum

119894=1

119879 (119905 minus 119905119894) 119868119894119906119896(119905119894)infin

119896=1) + 120576

(18)for 119905 isin [0 119886]

From the compactness of119892 and 119868119894 by Lemmas 2 and 5 and

(1198671198913) we have

120573 (119882119899+1

(119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

+2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119882119899(119904)) + 2

1198961(119904) 120573 (119882

119899(119904))

+21198962(119904) 120573 (119882

119899(119904))) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) 120573 (119882119899(119904)) (1 + 2

1198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119905

0

120578 (119904) 120573 (119882119899(119904)) 119889119904 + 120576

(19)

Since 120576 gt 0 is arbitrary it follows from the above inequalitythat

120573 (119882119899+1

(119905)) le 4119873119876int

119905

0

120578 (119904) 120573PC (119882119899 (119904)) 119889119904 (20)

6 International Journal of Differential Equations

for all 119905 isin [0 119886] Since119882119899is decreasing for 119899 we define119891

119899(119905) =

lim119899rarrinfin

120573(119882119899(119905)) for all 119905 isin [0 119886] From (20) we have

119891119899+1

(119905) le 4119873119876int

119905

0

120578 (119904) 119891119899(119904) 119889119904 (21)

for 119905 isin [0 119886] which implies that 119891119899(119905) = 0 for all 119905 isin [0 119886] By

Lemma 4 we know that

lim119899rarrinfin

120573PC (119882119899) = 0 (22)

Using Lemma 2 we also know that

119882 =

infin

⋂

119899=1

119882119899 (23)

is convex compact and nonempty in PC([0 119886] 119883) and119870(119882) sub 119882

By the famous Schauderrsquos fixed point theorem there existsat least one mild solution 119906 of the problem (1) where 119906 isin 119882is a fixed point of the continuous map119870

4 119892 Is Lipschitz

In this section we discuss the problem (1) when g is Lipschitzcontinuous and 119868

119894 119894 = 1 2 119901 is not compact We replace

hypotheses (1198671198912) (119868) by

(1198671198912)1015840There is a constant 119871 isin (0 1119872) such that 119892(119906) minus

119892(V) le 119871119906 minus VPC for all 119906 V isin PC([0 119886] 119883)(I1015840) There exists 119871

119894gt 0 119894 = 1 2 119901 such that 119868

119894(119906) minus

119868119894(V) le 119871

119894119906 minus V for all 119906 V isin 119883

Theorem 9 Assume that the hypotheses (1198671198921) (119867119891

2)1015840 (I1015840)

(1198671198911)ndash(119867119891

3) are satisfied Then the nonlocal impulsive prob-

lem (1) has at least one mild solution on [0a] provided that

119872119871 + 4119873119876int

119905

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894lt 1 (24)

Proof Define the operator 119870 PC([0 119886] 119883) rarr

PC([0 119886] 119883) by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) + (119870

3119906) (119905) (25)

With

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

(1198702119906) (119905) = int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

(1198703119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(26)

for all 119906 isin PC([0 119886] 119883)Define 119882

0= 119906 isin PC([0 119886] 119883) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] and let119882 = 1198701198820

Then from the proof of Theorem 8 we know that 119882is a bounded closed convex and equicontinuous subset ofPC([0 119886] 119883) and 119870119882 sub 119882 We will prove that 119870 is120573PC-contraction on 119882 Then Darbo-Sadovskii fixed pointtheorem can be used to get a fixed point of 119870 in 119882 whichis a mild solution of (1)

We first show that1198701is Lipschitz on PC([0 119886] 119883)

In fact take 119906 V isin PC([0 119886] 119883) arbitraryThen by (1198671198912)1015840

we have

1003817100381710038171003817(1198701119906) (119905) minus (119870

2119906) (119905)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906) minus 119892 (V)100381710038171003817

1003817le 119872119871119906 minus VPC forall119905 isin [0 119886]

(27)

It follows that 1198701119906 minus 119870

1V le 119872119871119906 minus VPC for all 119906 V isin

PC([0 119886] 119883) That is 1198701is Lipschitz with Lipschitz constant

119872119871Next for every bounded subset 119861 sub 119882 for any 120576 gt 0

there is a sequence 119906119896infin

119896=1sub 119861 such that 120573(119870

2119861(119905)) le

2120573(1198702119906119896(119905)infin

119896=1) + 120576 for 119905 isin [0 119886] Since 119861 and 119870

2119861 are

equicontinuous we get from Lemmas 2 and 5 and (1198671198913) that

120573 (1198702119861 (119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+ 120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

International Journal of Differential Equations 7

+ 2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+ 21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) (120573PC (119906119896infin

119896=1) + 2

1198961(119904) 120573PC (119906119896

infin

119896=1)

+ 21198962(119904) 120573PC (119906119896

infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) 120573PC (119861) (1 + 21198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119886

0

120578 (119904) 119889119904120573PC (119861) + 120576

(28)

for 119905 isin [0 119886] Since 120576 gt 0 arbitrary we have

120573PC (1198702119861) le 4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 (29)

for any bounded subset 119861 sub 119882

120573 (1198703119861 (119905)) le 2119873

119901

sum

119894=1

120573 (119868119894119906119896(119905119894)infin

119896=1)

le 2119873

119901

sum

119894=1

119871119894120573 (119906119896(119905119894)infin

119896=1)

120573PC (1198703119861) le 2119873

119901

sum

119894=1

119871119894120573PC (119861)

(30)

for any subset 119861 sub 119882 due to Lemma 2 (29) and (30) wehave

120573PC (119870119861)

= 120573PC (1198701119861) + 120573PC (1198702119861) + 120573PC (1198703119861)

le 119872119871120573PC (119861) +4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 + 2119873

119901

sum

119894=1

119871119894120573PC (119861)

le (119872119871 + 4119873119876int

119886

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894)120573PC (119861)

(31)

From (24) we know that 119870 is 120573PC-contraction on 119882 ByLemma 3 there is a fixed point 119906 of 119870 in119882 which is a mildsolution of problem (1)

5 Classical Solutions

To study the classical solutions let us recall the followingresult

Lemma 10 Assume that 1199060

isin 119863(119860) 119902119894isin 119863(119860) 119894 =

1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883)

Then the impulsive differential equation

1199061015840

(119905) = 119860119906 (119905) + 119891 (119905 119906 (119905)) 119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060

Δ119906 (119905) = 119902119894 119894 = 1 2 119901

(32)

has a unique classical solution u(sdot)which satisfies for 119905 isin [0 119886]

119906 (119905)

= 119879 (119905) 119906 (0) = int

119905

0

119879 (119905 minus 119904) 119891 (119904 119906 (119904)) 119889119904 + sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119902119894

(33)

Now we make the following assumption(1198671) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(34)

or ae 119905 isin [0 119886] and for any bounded subset119863 sub 119875119862([0 119886] 119883)

Theorem 11 Let (1198671) be satisfied and 119906(sdot) a mild solution ofthe problem (1) Assume that 119906(0) isin 119863(119860) 119868

119894(119906(119905119894)) isin 119863(119860)

119894 = 1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883) Then u(0) gives

rise to a classical solution of the problem (1)If 119906(sdot) is a uniquely determined mild solution then it gives

rise to a unique classical solution

Proof We can define 119902119894= 119868119894(119906(119905119894)) 119894 = 1 2 119901

From Lemma 10

V1015840 (119905) = 119860V (119905) + 119891(119905 V (119905) int119905

0

119896 (119905 119904 V (119904)) 119889119904

int

119886

0

ℎ (119905 119904 V (119904)) 119889119904) 119905 isin [0 119886] 119905 = 119905119894

V (0) = 1199060+ 119892 (119906)

ΔV (119905) = 119902119894 119894 = 1 2 119901

(35)

has a unique classical solution V(sdot) which satisfies for 119905 isin

[0 119886]

V (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(36)

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 3

Lemma 7 (see [23]) If 119882 is bounded then for each 120576 gt

0 there is a sequence 119906119899infin

119899=1sube 119882 such that 120572(119882) le

2120572(119906119899infin

119899=1) + 120576

3 119892 Is Compact

In this section we give the existence results of nonlocalintegrodifferential equation (39) Here we list the followinghypotheses(1198671198921) The 119888

0semigroup 119879(119905) 0 le 119905 le 119886 generated by 119860 is

equicontinuous(1198671198922) (i) 119892 PC([0 119886] 119883) rarr 119883 is continuous and

compact(ii) There exists 119872 gt 0 such that 119892(119906) le

119872 for all 119906 isin PC([0 119886] 119883) and 119896(119904) =

max1 int1199040119896(119904 120591)119889120591

(I) Let 119868119894 119883 rarr 119883 be continuous compact map and

there are nondecreasing functions 119897119894 119877+rarr 119877+ satisfying

119868119894(119909) le 119897

119894(119909) 119894 = 1 2 119901

(1198671198911) There exists a continuous function 119886

119896 [0 119886] times

[0 119886] rarr [0infin) and a nondecreasing continuousfunction Ω

119896 119877+

rarr 119877+ such that 119896(119905 119904 119909) le

119886119896(119905 119904)Ω

119896(119909) for all 119909 isin 119883 ae 119905 119904 isin [0 119886] And

there exists at least one mild solution to the followingscalar equation

119898(119905) = 119872119873

+119872int

119905

0

119886119891(119904) Ω119891(119898 (119904) 119886

119896(119905 119904) Ω

119896119898(119904)

119887ℎ(119905 119904) Ω

ℎ119898(119904)) 119889119904

+119872

119901

sum

119894=1

119897119894(119898 (119905)) 119905 isin [0 119886]

(5)

(1198671198912) (i) 119891(sdot sdot 119909 119910) is measurable for 119909 119910 isin 119883 119891(119905 sdot sdot sdot)

is continuous for ae 119905 isin [0 119886](ii) There exist a function 119886

119891(sdot) isin 119871

1(0 119886 119877

+)

and an increasing continuous function Ω119891

119877+

rarr 119877+ such that 119891(119905 119909 119910 119911) le

119886119891(119905)Ω119891(119909 119910 119911) for all 119909 119910 119911 isin 119883 and ae

119905 isin [0 119886](iii) 119891 [0 119886] times 119883 times 119883 rarr 119883 is compact

(1198671198913) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(6)

for ae 119905 isin [0 119886] and for any bounded subset 119863 sub

PC([0 119886] 119883)Here we let 119896(119904) = int

119904

0119896(119904 120591)119889120591 and (1 + 2119896

1(119904) + 2

1198962(119904)) le 119876

Theorem 8 Assume that the hypotheses (1198671198921) (119867119892

2) I

(1198671198911) and (119867119891

2) are satisfied then the nonlocal impulsive

problem (1) has at least one mild solution

Proof Let m(t) be a solution of the scalar equation (5) themap 119870 PC([0 119886] 119883) rarr PC([0 119886] 119883) is defined by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) (7)

with

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

(1198702119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(8)

for all 119905 isin [0 119886]It is easy to see that the fixedpoint of119870 is themild solution

of nonlocal impulsive problem (1)From our hypotheses the continuity of 119870 is proved as

followsFor this purpose we assume that 119906

119899rarr 119906 in

PC([0 119886] 119883) It comes from the continuity of 119896 and ℎ

that 119896(119904 120591 119906119899(120591)) rarr 119896(119904 120591 119906(120591)) and ℎ(119904 120591 119906

119899(120591)) rarr

ℎ(119904 120591 119906(120591)) respectivelyBy Lebesgue convergence theorem

int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591 997888rarr int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591 997888rarr int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591

(9)

Similarly we have

119891(119904 119906119899(119904) int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591)

997888rarr 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

(10)

for all 119904 isin [0 119886] Consider

1003817100381710038171003817119870119906119899minus 119870119906

1003817100381710038171003817

le 1198731003817100381710038171003817119892 (119906119899) minus 119892 (119906)

1003817100381710038171003817

+ 119873int

119905

0

(

10038171003817100381710038171003817100381710038171003817

119891(119904 119906119899(119904) int

119904

0

119896 (119904 120591 119906119899(120591)) 119889120591

4 International Journal of Differential Equations

int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

) 119889119904

+

119901

sum

119894=1

1198721003817100381710038171003817119868119894(119906119899(119905119894)) minus 119868119894(119906 (119905119894))1003817100381710038171003817997888rarr 0

(11)

as 119899 rarr infin So 119870119906119899rarr 119870119906 in PC([0 119886] 119883) That is 119870 is

continuousWe denote119882

0= 119906 isin PC([0 a])X) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] then119882 sube PC([119900 119886] 119883) is bounded and convexDefine 119882

1= conv119870(119882

0) where conv means that the

closure of the convex hull in PC([0 119886] 119883)For any 119906 isin 119870(119882

0) we know that

(119906) (119905) le 119872119873

+119873int

119905

0

119886119891(119904) Ω119891(119898 (119904) 119886

119896(119905 119904) Ω

119896119898(119904)

119887ℎ(119905 119904) Ω

ℎ119898(119904)) 119889119904

+119872

119901

sum

119894=1

119897119894(119898 (119905)) 119905 isin [0 119886]

(12)

and by (1198671198912)1198821sub 1198820

From the Arzela-Ascoli theorem to prove the compact-ness of 119870 we can prove that 119870

1119906 119906 isin 119882

0is equicontinuous

and1198701119906(119905) sub 119883 is precompact for 119905 isin [0 119886]

10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119879 (119905 + 120590) 119892 (119906)

+ int

119905+120590

0

119879 (119905 + 120590 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

minus [119879 (119905) 119892 (119906) +int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904]

10038171003817100381710038171003817100381710038171003817

le1003817100381710038171003817[119879 (119905 + 120590) minus 119879 (119906)] 119892 (119906)

1003817100381710038171003817

+ int

119905+120590

119905

119879 (119905 + 120590 minus 119904)

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

+ int

119905

0

119879 (119905 + 120590 minus 119904) minus 119879 (119905 minus 119904)

times

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

le 119872119873119879 (120590) minus 119868

+ 119873int

119905+120590

0

119886119891(119904) Ω119891(119906 (119904) int

119904

0

119886119896(119904 120591)Ω

119896(119906 (119904)) 119889120591

int

119886

0

119887ℎ(119904 120591)Ω

ℎ(119906 (119904)) 119889120591) 119889119904

+ 119873int

119905

0

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868] 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

(13)

Since 119891 is compact 119872119873119879(120590) minus 119868 and [119879(120590) minus

119868]119891(119904 119906(119904) int1199040119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) rarr 0

as 120590 rarr 0 uniformly for 119904 isin [0 119886] and 119906 isin PC([0 119886] 119883) Thisimplies that for any 120576

1gt 0 and 120576

2gt 0 there exist a 120575 gt 0 such

that

[119879 (120590) minus 119868] le 1205761

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891 (119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205762

(14)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) Therefore

[119879 (120590) minus 119868]

+

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205761+ 1205762= 120576

(15)

We know that10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

le 119872119873120576 + 119873Ω119891(119898 (119904)

1003817100381710038171003817119886119896(119886 119886)

1003817100381710038171003817 Ω119896(119898 (119904))

1003817100381710038171003817119887ℎ(119886 119886)

1003817100381710038171003817 Ωℎ(119898 (119904))) + 119873120576

(16)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) So 1198701119906 119906 isin 119882

0 is

equicontinuous

International Journal of Differential Equations 5

The set 119879(119905 minus 119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591int119886

0ℎ(119904

120591 119906(120591))119889120591) 119905 119904 isin [0 119886] 119906 isin PC([0 119886] 119883) is precompact as fis compact and 119879(sdot) is a 119862

0-semigroup

So 1198701119906(119905) sub 119883 is precompact as 119870

1119906(119905) sub 119905 conv119879(119905 minus

119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) 119904 isin [0 119905]

119906 isin PC([0 119886] 119883) for all 119905 isin [0 119886]1198821is equicontinuous on

each interval 119869119894of [0 119886] For 119905

119894le 119905 lt 119905+120590 le 119905

119894+1 119894 = 1 2 119901

we have using the semigroup properties1003817100381710038171003817(1198702119906) (119905 + 120590) minus (119870

2119906) (119905)

1003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905+120590

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

le sum

0lt119905119894lt119905+120590

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

+ sum

0lt119905119894lt119905

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894)) minus 119879 (119905 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

(17)which follows that 119870

2119906 119906 isin 119882

0 is equicontinuous on

each 119869119894due to the equicontinuous of 119879(119905) and hypotheses

(I) Therefore1198821sub PC([0 119886] 119883) is bounded closed convex

nonempty and equicontinuous on each interval 119869119894 119894 =

0 1 2 119901We define 119882

119899+1= conv119870(119882

119899) for 119899 = 1 2 119901 From

above we know that 119882119899infin

119899=1is a decreasing sequence of

bounded closed convex nonempty subsets in PC([0 119886] 119883)and equicontinuous on each 119869

119894 119894 = 1 2 119901

Now for 119899 ge 1 and 119905 isin [0 119886] 119882119899(119905) and 119870(119882

119899(119905)) are

bounded subsets of119883 Hence for any 120576 gt 0 there is a sequence119906119896infin

119896=1sub 119882119899such that (see eg [2 page 125])

120573 (119882119899+1

(119905))

= 120573 (119870119882119899(119905))

le 2120573 (119879 (119905) 119892 (119906119896infin

119896=1))

+2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904)

+ 2120573(

119901

sum

119894=1

119879 (119905 minus 119905119894) 119868119894119906119896(119905119894)infin

119896=1) + 120576

(18)for 119905 isin [0 119886]

From the compactness of119892 and 119868119894 by Lemmas 2 and 5 and

(1198671198913) we have

120573 (119882119899+1

(119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

+2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119882119899(119904)) + 2

1198961(119904) 120573 (119882

119899(119904))

+21198962(119904) 120573 (119882

119899(119904))) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) 120573 (119882119899(119904)) (1 + 2

1198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119905

0

120578 (119904) 120573 (119882119899(119904)) 119889119904 + 120576

(19)

Since 120576 gt 0 is arbitrary it follows from the above inequalitythat

120573 (119882119899+1

(119905)) le 4119873119876int

119905

0

120578 (119904) 120573PC (119882119899 (119904)) 119889119904 (20)

6 International Journal of Differential Equations

for all 119905 isin [0 119886] Since119882119899is decreasing for 119899 we define119891

119899(119905) =

lim119899rarrinfin

120573(119882119899(119905)) for all 119905 isin [0 119886] From (20) we have

119891119899+1

(119905) le 4119873119876int

119905

0

120578 (119904) 119891119899(119904) 119889119904 (21)

for 119905 isin [0 119886] which implies that 119891119899(119905) = 0 for all 119905 isin [0 119886] By

Lemma 4 we know that

lim119899rarrinfin

120573PC (119882119899) = 0 (22)

Using Lemma 2 we also know that

119882 =

infin

⋂

119899=1

119882119899 (23)

is convex compact and nonempty in PC([0 119886] 119883) and119870(119882) sub 119882

By the famous Schauderrsquos fixed point theorem there existsat least one mild solution 119906 of the problem (1) where 119906 isin 119882is a fixed point of the continuous map119870

4 119892 Is Lipschitz

In this section we discuss the problem (1) when g is Lipschitzcontinuous and 119868

119894 119894 = 1 2 119901 is not compact We replace

hypotheses (1198671198912) (119868) by

(1198671198912)1015840There is a constant 119871 isin (0 1119872) such that 119892(119906) minus

119892(V) le 119871119906 minus VPC for all 119906 V isin PC([0 119886] 119883)(I1015840) There exists 119871

119894gt 0 119894 = 1 2 119901 such that 119868

119894(119906) minus

119868119894(V) le 119871

119894119906 minus V for all 119906 V isin 119883

Theorem 9 Assume that the hypotheses (1198671198921) (119867119891

2)1015840 (I1015840)

(1198671198911)ndash(119867119891

3) are satisfied Then the nonlocal impulsive prob-

lem (1) has at least one mild solution on [0a] provided that

119872119871 + 4119873119876int

119905

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894lt 1 (24)

Proof Define the operator 119870 PC([0 119886] 119883) rarr

PC([0 119886] 119883) by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) + (119870

3119906) (119905) (25)

With

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

(1198702119906) (119905) = int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

(1198703119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(26)

for all 119906 isin PC([0 119886] 119883)Define 119882

0= 119906 isin PC([0 119886] 119883) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] and let119882 = 1198701198820

Then from the proof of Theorem 8 we know that 119882is a bounded closed convex and equicontinuous subset ofPC([0 119886] 119883) and 119870119882 sub 119882 We will prove that 119870 is120573PC-contraction on 119882 Then Darbo-Sadovskii fixed pointtheorem can be used to get a fixed point of 119870 in 119882 whichis a mild solution of (1)

We first show that1198701is Lipschitz on PC([0 119886] 119883)

In fact take 119906 V isin PC([0 119886] 119883) arbitraryThen by (1198671198912)1015840

we have

1003817100381710038171003817(1198701119906) (119905) minus (119870

2119906) (119905)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906) minus 119892 (V)100381710038171003817

1003817le 119872119871119906 minus VPC forall119905 isin [0 119886]

(27)

It follows that 1198701119906 minus 119870

1V le 119872119871119906 minus VPC for all 119906 V isin

PC([0 119886] 119883) That is 1198701is Lipschitz with Lipschitz constant

119872119871Next for every bounded subset 119861 sub 119882 for any 120576 gt 0

there is a sequence 119906119896infin

119896=1sub 119861 such that 120573(119870

2119861(119905)) le

2120573(1198702119906119896(119905)infin

119896=1) + 120576 for 119905 isin [0 119886] Since 119861 and 119870

2119861 are

equicontinuous we get from Lemmas 2 and 5 and (1198671198913) that

120573 (1198702119861 (119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+ 120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

International Journal of Differential Equations 7

+ 2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+ 21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) (120573PC (119906119896infin

119896=1) + 2

1198961(119904) 120573PC (119906119896

infin

119896=1)

+ 21198962(119904) 120573PC (119906119896

infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) 120573PC (119861) (1 + 21198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119886

0

120578 (119904) 119889119904120573PC (119861) + 120576

(28)

for 119905 isin [0 119886] Since 120576 gt 0 arbitrary we have

120573PC (1198702119861) le 4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 (29)

for any bounded subset 119861 sub 119882

120573 (1198703119861 (119905)) le 2119873

119901

sum

119894=1

120573 (119868119894119906119896(119905119894)infin

119896=1)

le 2119873

119901

sum

119894=1

119871119894120573 (119906119896(119905119894)infin

119896=1)

120573PC (1198703119861) le 2119873

119901

sum

119894=1

119871119894120573PC (119861)

(30)

for any subset 119861 sub 119882 due to Lemma 2 (29) and (30) wehave

120573PC (119870119861)

= 120573PC (1198701119861) + 120573PC (1198702119861) + 120573PC (1198703119861)

le 119872119871120573PC (119861) +4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 + 2119873

119901

sum

119894=1

119871119894120573PC (119861)

le (119872119871 + 4119873119876int

119886

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894)120573PC (119861)

(31)

From (24) we know that 119870 is 120573PC-contraction on 119882 ByLemma 3 there is a fixed point 119906 of 119870 in119882 which is a mildsolution of problem (1)

5 Classical Solutions

To study the classical solutions let us recall the followingresult

Lemma 10 Assume that 1199060

isin 119863(119860) 119902119894isin 119863(119860) 119894 =

1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883)

Then the impulsive differential equation

1199061015840

(119905) = 119860119906 (119905) + 119891 (119905 119906 (119905)) 119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060

Δ119906 (119905) = 119902119894 119894 = 1 2 119901

(32)

has a unique classical solution u(sdot)which satisfies for 119905 isin [0 119886]

119906 (119905)

= 119879 (119905) 119906 (0) = int

119905

0

119879 (119905 minus 119904) 119891 (119904 119906 (119904)) 119889119904 + sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119902119894

(33)

Now we make the following assumption(1198671) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(34)

or ae 119905 isin [0 119886] and for any bounded subset119863 sub 119875119862([0 119886] 119883)

Theorem 11 Let (1198671) be satisfied and 119906(sdot) a mild solution ofthe problem (1) Assume that 119906(0) isin 119863(119860) 119868

119894(119906(119905119894)) isin 119863(119860)

119894 = 1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883) Then u(0) gives

rise to a classical solution of the problem (1)If 119906(sdot) is a uniquely determined mild solution then it gives

rise to a unique classical solution

Proof We can define 119902119894= 119868119894(119906(119905119894)) 119894 = 1 2 119901

From Lemma 10

V1015840 (119905) = 119860V (119905) + 119891(119905 V (119905) int119905

0

119896 (119905 119904 V (119904)) 119889119904

int

119886

0

ℎ (119905 119904 V (119904)) 119889119904) 119905 isin [0 119886] 119905 = 119905119894

V (0) = 1199060+ 119892 (119906)

ΔV (119905) = 119902119894 119894 = 1 2 119901

(35)

has a unique classical solution V(sdot) which satisfies for 119905 isin

[0 119886]

V (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(36)

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Differential Equations

int

119886

0

ℎ (119904 120591 119906119899(120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

) 119889119904

+

119901

sum

119894=1

1198721003817100381710038171003817119868119894(119906119899(119905119894)) minus 119868119894(119906 (119905119894))1003817100381710038171003817997888rarr 0

(11)

as 119899 rarr infin So 119870119906119899rarr 119870119906 in PC([0 119886] 119883) That is 119870 is

continuousWe denote119882

0= 119906 isin PC([0 a])X) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] then119882 sube PC([119900 119886] 119883) is bounded and convexDefine 119882

1= conv119870(119882

0) where conv means that the

closure of the convex hull in PC([0 119886] 119883)For any 119906 isin 119870(119882

0) we know that

(119906) (119905) le 119872119873

+119873int

119905

0

119886119891(119904) Ω119891(119898 (119904) 119886

119896(119905 119904) Ω

119896119898(119904)

119887ℎ(119905 119904) Ω

ℎ119898(119904)) 119889119904

+119872

119901

sum

119894=1

119897119894(119898 (119905)) 119905 isin [0 119886]

(12)

and by (1198671198912)1198821sub 1198820

From the Arzela-Ascoli theorem to prove the compact-ness of 119870 we can prove that 119870

1119906 119906 isin 119882

0is equicontinuous

and1198701119906(119905) sub 119883 is precompact for 119905 isin [0 119886]

10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

=

10038171003817100381710038171003817100381710038171003817

119879 (119905 + 120590) 119892 (119906)

+ int

119905+120590

0

119879 (119905 + 120590 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

minus [119879 (119905) 119892 (119906) +int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904]

10038171003817100381710038171003817100381710038171003817

le1003817100381710038171003817[119879 (119905 + 120590) minus 119879 (119906)] 119892 (119906)

1003817100381710038171003817

+ int

119905+120590

119905

119879 (119905 + 120590 minus 119904)

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

+ int

119905

0

119879 (119905 + 120590 minus 119904) minus 119879 (119905 minus 119904)

times

10038171003817100381710038171003817100381710038171003817

119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

le 119872119873119879 (120590) minus 119868

+ 119873int

119905+120590

0

119886119891(119904) Ω119891(119906 (119904) int

119904

0

119886119896(119904 120591)Ω

119896(119906 (119904)) 119889120591

int

119886

0

119887ℎ(119904 120591)Ω

ℎ(119906 (119904)) 119889120591) 119889119904

+ 119873int

119905

0

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868] 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

119889119904

(13)

Since 119891 is compact 119872119873119879(120590) minus 119868 and [119879(120590) minus

119868]119891(119904 119906(119904) int1199040119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) rarr 0

as 120590 rarr 0 uniformly for 119904 isin [0 119886] and 119906 isin PC([0 119886] 119883) Thisimplies that for any 120576

1gt 0 and 120576

2gt 0 there exist a 120575 gt 0 such

that

[119879 (120590) minus 119868] le 1205761

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891 (119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205762

(14)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) Therefore

[119879 (120590) minus 119868]

+

10038171003817100381710038171003817100381710038171003817

[119879 (120590) minus 119868]

times 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591 int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591)

10038171003817100381710038171003817100381710038171003817

le 1205761+ 1205762= 120576

(15)

We know that10038171003817100381710038171198701119906 (119905 + 120590) minus 119870

1119906 (119905)

1003817100381710038171003817

le 119872119873120576 + 119873Ω119891(119898 (119904)

1003817100381710038171003817119886119896(119886 119886)

1003817100381710038171003817 Ω119896(119898 (119904))

1003817100381710038171003817119887ℎ(119886 119886)

1003817100381710038171003817 Ωℎ(119898 (119904))) + 119873120576

(16)

for 0 le 120590 lt 120575 and all 119906 isin PC([0 119886] 119883) So 1198701119906 119906 isin 119882

0 is

equicontinuous

International Journal of Differential Equations 5

The set 119879(119905 minus 119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591int119886

0ℎ(119904

120591 119906(120591))119889120591) 119905 119904 isin [0 119886] 119906 isin PC([0 119886] 119883) is precompact as fis compact and 119879(sdot) is a 119862

0-semigroup

So 1198701119906(119905) sub 119883 is precompact as 119870

1119906(119905) sub 119905 conv119879(119905 minus

119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) 119904 isin [0 119905]

119906 isin PC([0 119886] 119883) for all 119905 isin [0 119886]1198821is equicontinuous on

each interval 119869119894of [0 119886] For 119905

119894le 119905 lt 119905+120590 le 119905

119894+1 119894 = 1 2 119901

we have using the semigroup properties1003817100381710038171003817(1198702119906) (119905 + 120590) minus (119870

2119906) (119905)

1003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905+120590

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

le sum

0lt119905119894lt119905+120590

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

+ sum

0lt119905119894lt119905

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894)) minus 119879 (119905 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

(17)which follows that 119870

2119906 119906 isin 119882

0 is equicontinuous on

each 119869119894due to the equicontinuous of 119879(119905) and hypotheses

(I) Therefore1198821sub PC([0 119886] 119883) is bounded closed convex

nonempty and equicontinuous on each interval 119869119894 119894 =

0 1 2 119901We define 119882

119899+1= conv119870(119882

119899) for 119899 = 1 2 119901 From

above we know that 119882119899infin

119899=1is a decreasing sequence of

bounded closed convex nonempty subsets in PC([0 119886] 119883)and equicontinuous on each 119869

119894 119894 = 1 2 119901

Now for 119899 ge 1 and 119905 isin [0 119886] 119882119899(119905) and 119870(119882

119899(119905)) are

bounded subsets of119883 Hence for any 120576 gt 0 there is a sequence119906119896infin

119896=1sub 119882119899such that (see eg [2 page 125])

120573 (119882119899+1

(119905))

= 120573 (119870119882119899(119905))

le 2120573 (119879 (119905) 119892 (119906119896infin

119896=1))

+2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904)

+ 2120573(

119901

sum

119894=1

119879 (119905 minus 119905119894) 119868119894119906119896(119905119894)infin

119896=1) + 120576

(18)for 119905 isin [0 119886]

From the compactness of119892 and 119868119894 by Lemmas 2 and 5 and

(1198671198913) we have

120573 (119882119899+1

(119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

+2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119882119899(119904)) + 2

1198961(119904) 120573 (119882

119899(119904))

+21198962(119904) 120573 (119882

119899(119904))) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) 120573 (119882119899(119904)) (1 + 2

1198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119905

0

120578 (119904) 120573 (119882119899(119904)) 119889119904 + 120576

(19)

Since 120576 gt 0 is arbitrary it follows from the above inequalitythat

120573 (119882119899+1

(119905)) le 4119873119876int

119905

0

120578 (119904) 120573PC (119882119899 (119904)) 119889119904 (20)

6 International Journal of Differential Equations

for all 119905 isin [0 119886] Since119882119899is decreasing for 119899 we define119891

119899(119905) =

lim119899rarrinfin

120573(119882119899(119905)) for all 119905 isin [0 119886] From (20) we have

119891119899+1

(119905) le 4119873119876int

119905

0

120578 (119904) 119891119899(119904) 119889119904 (21)

for 119905 isin [0 119886] which implies that 119891119899(119905) = 0 for all 119905 isin [0 119886] By

Lemma 4 we know that

lim119899rarrinfin

120573PC (119882119899) = 0 (22)

Using Lemma 2 we also know that

119882 =

infin

⋂

119899=1

119882119899 (23)

is convex compact and nonempty in PC([0 119886] 119883) and119870(119882) sub 119882

By the famous Schauderrsquos fixed point theorem there existsat least one mild solution 119906 of the problem (1) where 119906 isin 119882is a fixed point of the continuous map119870

4 119892 Is Lipschitz

In this section we discuss the problem (1) when g is Lipschitzcontinuous and 119868

119894 119894 = 1 2 119901 is not compact We replace

hypotheses (1198671198912) (119868) by

(1198671198912)1015840There is a constant 119871 isin (0 1119872) such that 119892(119906) minus

119892(V) le 119871119906 minus VPC for all 119906 V isin PC([0 119886] 119883)(I1015840) There exists 119871

119894gt 0 119894 = 1 2 119901 such that 119868

119894(119906) minus

119868119894(V) le 119871

119894119906 minus V for all 119906 V isin 119883

Theorem 9 Assume that the hypotheses (1198671198921) (119867119891

2)1015840 (I1015840)

(1198671198911)ndash(119867119891

3) are satisfied Then the nonlocal impulsive prob-

lem (1) has at least one mild solution on [0a] provided that

119872119871 + 4119873119876int

119905

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894lt 1 (24)

Proof Define the operator 119870 PC([0 119886] 119883) rarr

PC([0 119886] 119883) by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) + (119870

3119906) (119905) (25)

With

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

(1198702119906) (119905) = int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

(1198703119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(26)

for all 119906 isin PC([0 119886] 119883)Define 119882

0= 119906 isin PC([0 119886] 119883) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] and let119882 = 1198701198820

Then from the proof of Theorem 8 we know that 119882is a bounded closed convex and equicontinuous subset ofPC([0 119886] 119883) and 119870119882 sub 119882 We will prove that 119870 is120573PC-contraction on 119882 Then Darbo-Sadovskii fixed pointtheorem can be used to get a fixed point of 119870 in 119882 whichis a mild solution of (1)

We first show that1198701is Lipschitz on PC([0 119886] 119883)

In fact take 119906 V isin PC([0 119886] 119883) arbitraryThen by (1198671198912)1015840

we have

1003817100381710038171003817(1198701119906) (119905) minus (119870

2119906) (119905)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906) minus 119892 (V)100381710038171003817

1003817le 119872119871119906 minus VPC forall119905 isin [0 119886]

(27)

It follows that 1198701119906 minus 119870

1V le 119872119871119906 minus VPC for all 119906 V isin

PC([0 119886] 119883) That is 1198701is Lipschitz with Lipschitz constant

119872119871Next for every bounded subset 119861 sub 119882 for any 120576 gt 0

there is a sequence 119906119896infin

119896=1sub 119861 such that 120573(119870

2119861(119905)) le

2120573(1198702119906119896(119905)infin

119896=1) + 120576 for 119905 isin [0 119886] Since 119861 and 119870

2119861 are

equicontinuous we get from Lemmas 2 and 5 and (1198671198913) that

120573 (1198702119861 (119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+ 120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

International Journal of Differential Equations 7

+ 2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+ 21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) (120573PC (119906119896infin

119896=1) + 2

1198961(119904) 120573PC (119906119896

infin

119896=1)

+ 21198962(119904) 120573PC (119906119896

infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) 120573PC (119861) (1 + 21198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119886

0

120578 (119904) 119889119904120573PC (119861) + 120576

(28)

for 119905 isin [0 119886] Since 120576 gt 0 arbitrary we have

120573PC (1198702119861) le 4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 (29)

for any bounded subset 119861 sub 119882

120573 (1198703119861 (119905)) le 2119873

119901

sum

119894=1

120573 (119868119894119906119896(119905119894)infin

119896=1)

le 2119873

119901

sum

119894=1

119871119894120573 (119906119896(119905119894)infin

119896=1)

120573PC (1198703119861) le 2119873

119901

sum

119894=1

119871119894120573PC (119861)

(30)

for any subset 119861 sub 119882 due to Lemma 2 (29) and (30) wehave

120573PC (119870119861)

= 120573PC (1198701119861) + 120573PC (1198702119861) + 120573PC (1198703119861)

le 119872119871120573PC (119861) +4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 + 2119873

119901

sum

119894=1

119871119894120573PC (119861)

le (119872119871 + 4119873119876int

119886

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894)120573PC (119861)

(31)

From (24) we know that 119870 is 120573PC-contraction on 119882 ByLemma 3 there is a fixed point 119906 of 119870 in119882 which is a mildsolution of problem (1)

5 Classical Solutions

To study the classical solutions let us recall the followingresult

Lemma 10 Assume that 1199060

isin 119863(119860) 119902119894isin 119863(119860) 119894 =

1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883)

Then the impulsive differential equation

1199061015840

(119905) = 119860119906 (119905) + 119891 (119905 119906 (119905)) 119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060

Δ119906 (119905) = 119902119894 119894 = 1 2 119901

(32)

has a unique classical solution u(sdot)which satisfies for 119905 isin [0 119886]

119906 (119905)

= 119879 (119905) 119906 (0) = int

119905

0

119879 (119905 minus 119904) 119891 (119904 119906 (119904)) 119889119904 + sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119902119894

(33)

Now we make the following assumption(1198671) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(34)

or ae 119905 isin [0 119886] and for any bounded subset119863 sub 119875119862([0 119886] 119883)

Theorem 11 Let (1198671) be satisfied and 119906(sdot) a mild solution ofthe problem (1) Assume that 119906(0) isin 119863(119860) 119868

119894(119906(119905119894)) isin 119863(119860)

119894 = 1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883) Then u(0) gives

rise to a classical solution of the problem (1)If 119906(sdot) is a uniquely determined mild solution then it gives

rise to a unique classical solution

Proof We can define 119902119894= 119868119894(119906(119905119894)) 119894 = 1 2 119901

From Lemma 10

V1015840 (119905) = 119860V (119905) + 119891(119905 V (119905) int119905

0

119896 (119905 119904 V (119904)) 119889119904

int

119886

0

ℎ (119905 119904 V (119904)) 119889119904) 119905 isin [0 119886] 119905 = 119905119894

V (0) = 1199060+ 119892 (119906)

ΔV (119905) = 119902119894 119894 = 1 2 119901

(35)

has a unique classical solution V(sdot) which satisfies for 119905 isin

[0 119886]

V (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(36)

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 5

The set 119879(119905 minus 119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591int119886

0ℎ(119904

120591 119906(120591))119889120591) 119905 119904 isin [0 119886] 119906 isin PC([0 119886] 119883) is precompact as fis compact and 119879(sdot) is a 119862

0-semigroup

So 1198701119906(119905) sub 119883 is precompact as 119870

1119906(119905) sub 119905 conv119879(119905 minus

119904)119891(119904 119906(119904) int

119904

0119896(119904 120591 119906(120591))119889120591 int

119886

0ℎ(119904 120591 119906(120591))119889120591) 119904 isin [0 119905]

119906 isin PC([0 119886] 119883) for all 119905 isin [0 119886]1198821is equicontinuous on

each interval 119869119894of [0 119886] For 119905

119894le 119905 lt 119905+120590 le 119905

119894+1 119894 = 1 2 119901

we have using the semigroup properties1003817100381710038171003817(1198702119906) (119905 + 120590) minus (119870

2119906) (119905)

1003817100381710038171003817

le

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905+120590

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

0lt119905119894lt119905

119879 (119905 + 120590 minus 119905119894) 119868119894(119909 (119905119894))

minus sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119909 (119905119894))

10038171003817100381710038171003817100381710038171003817100381710038171003817

le sum

0lt119905119894lt119905+120590

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

+ sum

0lt119905119894lt119905

1003817100381710038171003817119879 (119905 + 120590 minus 119905

119894) 119868119894(119909 (119905119894)) minus 119879 (119905 minus 119905

119894) 119868119894(119909 (119905119894))1003817100381710038171003817

(17)which follows that 119870

2119906 119906 isin 119882

0 is equicontinuous on

each 119869119894due to the equicontinuous of 119879(119905) and hypotheses

(I) Therefore1198821sub PC([0 119886] 119883) is bounded closed convex

nonempty and equicontinuous on each interval 119869119894 119894 =

0 1 2 119901We define 119882

119899+1= conv119870(119882

119899) for 119899 = 1 2 119901 From

above we know that 119882119899infin

119899=1is a decreasing sequence of

bounded closed convex nonempty subsets in PC([0 119886] 119883)and equicontinuous on each 119869

119894 119894 = 1 2 119901

Now for 119899 ge 1 and 119905 isin [0 119886] 119882119899(119905) and 119870(119882

119899(119905)) are

bounded subsets of119883 Hence for any 120576 gt 0 there is a sequence119906119896infin

119896=1sub 119882119899such that (see eg [2 page 125])

120573 (119882119899+1

(119905))

= 120573 (119870119882119899(119905))

le 2120573 (119879 (119905) 119892 (119906119896infin

119896=1))

+2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904)

+ 2120573(

119901

sum

119894=1

119879 (119905 minus 119905119894) 119868119894119906119896(119905119894)infin

119896=1) + 120576

(18)for 119905 isin [0 119886]

From the compactness of119892 and 119868119894 by Lemmas 2 and 5 and

(1198671198913) we have

120573 (119882119899+1

(119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

+2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119882119899(119904)) + 2

1198961(119904) 120573 (119882

119899(119904))

+21198962(119904) 120573 (119882

119899(119904))) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) 120573 (119882119899(119904)) (1 + 2

1198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119905

0

120578 (119904) 120573 (119882119899(119904)) 119889119904 + 120576

(19)

Since 120576 gt 0 is arbitrary it follows from the above inequalitythat

120573 (119882119899+1

(119905)) le 4119873119876int

119905

0

120578 (119904) 120573PC (119882119899 (119904)) 119889119904 (20)

6 International Journal of Differential Equations

for all 119905 isin [0 119886] Since119882119899is decreasing for 119899 we define119891

119899(119905) =

lim119899rarrinfin

120573(119882119899(119905)) for all 119905 isin [0 119886] From (20) we have

119891119899+1

(119905) le 4119873119876int

119905

0

120578 (119904) 119891119899(119904) 119889119904 (21)

for 119905 isin [0 119886] which implies that 119891119899(119905) = 0 for all 119905 isin [0 119886] By

Lemma 4 we know that

lim119899rarrinfin

120573PC (119882119899) = 0 (22)

Using Lemma 2 we also know that

119882 =

infin

⋂

119899=1

119882119899 (23)

is convex compact and nonempty in PC([0 119886] 119883) and119870(119882) sub 119882

By the famous Schauderrsquos fixed point theorem there existsat least one mild solution 119906 of the problem (1) where 119906 isin 119882is a fixed point of the continuous map119870

4 119892 Is Lipschitz

In this section we discuss the problem (1) when g is Lipschitzcontinuous and 119868

119894 119894 = 1 2 119901 is not compact We replace

hypotheses (1198671198912) (119868) by

(1198671198912)1015840There is a constant 119871 isin (0 1119872) such that 119892(119906) minus

119892(V) le 119871119906 minus VPC for all 119906 V isin PC([0 119886] 119883)(I1015840) There exists 119871

119894gt 0 119894 = 1 2 119901 such that 119868

119894(119906) minus

119868119894(V) le 119871

119894119906 minus V for all 119906 V isin 119883

Theorem 9 Assume that the hypotheses (1198671198921) (119867119891

2)1015840 (I1015840)

(1198671198911)ndash(119867119891

3) are satisfied Then the nonlocal impulsive prob-

lem (1) has at least one mild solution on [0a] provided that

119872119871 + 4119873119876int

119905

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894lt 1 (24)

Proof Define the operator 119870 PC([0 119886] 119883) rarr

PC([0 119886] 119883) by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) + (119870

3119906) (119905) (25)

With

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

(1198702119906) (119905) = int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

(1198703119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(26)

for all 119906 isin PC([0 119886] 119883)Define 119882

0= 119906 isin PC([0 119886] 119883) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] and let119882 = 1198701198820

Then from the proof of Theorem 8 we know that 119882is a bounded closed convex and equicontinuous subset ofPC([0 119886] 119883) and 119870119882 sub 119882 We will prove that 119870 is120573PC-contraction on 119882 Then Darbo-Sadovskii fixed pointtheorem can be used to get a fixed point of 119870 in 119882 whichis a mild solution of (1)

We first show that1198701is Lipschitz on PC([0 119886] 119883)

In fact take 119906 V isin PC([0 119886] 119883) arbitraryThen by (1198671198912)1015840

we have

1003817100381710038171003817(1198701119906) (119905) minus (119870

2119906) (119905)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906) minus 119892 (V)100381710038171003817

1003817le 119872119871119906 minus VPC forall119905 isin [0 119886]

(27)

It follows that 1198701119906 minus 119870

1V le 119872119871119906 minus VPC for all 119906 V isin

PC([0 119886] 119883) That is 1198701is Lipschitz with Lipschitz constant

119872119871Next for every bounded subset 119861 sub 119882 for any 120576 gt 0

there is a sequence 119906119896infin

119896=1sub 119861 such that 120573(119870

2119861(119905)) le

2120573(1198702119906119896(119905)infin

119896=1) + 120576 for 119905 isin [0 119886] Since 119861 and 119870

2119861 are

equicontinuous we get from Lemmas 2 and 5 and (1198671198913) that

120573 (1198702119861 (119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+ 120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

International Journal of Differential Equations 7

+ 2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+ 21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) (120573PC (119906119896infin

119896=1) + 2

1198961(119904) 120573PC (119906119896

infin

119896=1)

+ 21198962(119904) 120573PC (119906119896

infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) 120573PC (119861) (1 + 21198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119886

0

120578 (119904) 119889119904120573PC (119861) + 120576

(28)

for 119905 isin [0 119886] Since 120576 gt 0 arbitrary we have

120573PC (1198702119861) le 4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 (29)

for any bounded subset 119861 sub 119882

120573 (1198703119861 (119905)) le 2119873

119901

sum

119894=1

120573 (119868119894119906119896(119905119894)infin

119896=1)

le 2119873

119901

sum

119894=1

119871119894120573 (119906119896(119905119894)infin

119896=1)

120573PC (1198703119861) le 2119873

119901

sum

119894=1

119871119894120573PC (119861)

(30)

for any subset 119861 sub 119882 due to Lemma 2 (29) and (30) wehave

120573PC (119870119861)

= 120573PC (1198701119861) + 120573PC (1198702119861) + 120573PC (1198703119861)

le 119872119871120573PC (119861) +4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 + 2119873

119901

sum

119894=1

119871119894120573PC (119861)

le (119872119871 + 4119873119876int

119886

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894)120573PC (119861)

(31)

From (24) we know that 119870 is 120573PC-contraction on 119882 ByLemma 3 there is a fixed point 119906 of 119870 in119882 which is a mildsolution of problem (1)

5 Classical Solutions

To study the classical solutions let us recall the followingresult

Lemma 10 Assume that 1199060

isin 119863(119860) 119902119894isin 119863(119860) 119894 =

1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883)

Then the impulsive differential equation

1199061015840

(119905) = 119860119906 (119905) + 119891 (119905 119906 (119905)) 119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060

Δ119906 (119905) = 119902119894 119894 = 1 2 119901

(32)

has a unique classical solution u(sdot)which satisfies for 119905 isin [0 119886]

119906 (119905)

= 119879 (119905) 119906 (0) = int

119905

0

119879 (119905 minus 119904) 119891 (119904 119906 (119904)) 119889119904 + sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119902119894

(33)

Now we make the following assumption(1198671) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(34)

or ae 119905 isin [0 119886] and for any bounded subset119863 sub 119875119862([0 119886] 119883)

Theorem 11 Let (1198671) be satisfied and 119906(sdot) a mild solution ofthe problem (1) Assume that 119906(0) isin 119863(119860) 119868

119894(119906(119905119894)) isin 119863(119860)

119894 = 1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883) Then u(0) gives

rise to a classical solution of the problem (1)If 119906(sdot) is a uniquely determined mild solution then it gives

rise to a unique classical solution

Proof We can define 119902119894= 119868119894(119906(119905119894)) 119894 = 1 2 119901

From Lemma 10

V1015840 (119905) = 119860V (119905) + 119891(119905 V (119905) int119905

0

119896 (119905 119904 V (119904)) 119889119904

int

119886

0

ℎ (119905 119904 V (119904)) 119889119904) 119905 isin [0 119886] 119905 = 119905119894

V (0) = 1199060+ 119892 (119906)

ΔV (119905) = 119902119894 119894 = 1 2 119901

(35)

has a unique classical solution V(sdot) which satisfies for 119905 isin

[0 119886]

V (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(36)

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Differential Equations

for all 119905 isin [0 119886] Since119882119899is decreasing for 119899 we define119891

119899(119905) =

lim119899rarrinfin

120573(119882119899(119905)) for all 119905 isin [0 119886] From (20) we have

119891119899+1

(119905) le 4119873119876int

119905

0

120578 (119904) 119891119899(119904) 119889119904 (21)

for 119905 isin [0 119886] which implies that 119891119899(119905) = 0 for all 119905 isin [0 119886] By

Lemma 4 we know that

lim119899rarrinfin

120573PC (119882119899) = 0 (22)

Using Lemma 2 we also know that

119882 =

infin

⋂

119899=1

119882119899 (23)

is convex compact and nonempty in PC([0 119886] 119883) and119870(119882) sub 119882

By the famous Schauderrsquos fixed point theorem there existsat least one mild solution 119906 of the problem (1) where 119906 isin 119882is a fixed point of the continuous map119870

4 119892 Is Lipschitz

In this section we discuss the problem (1) when g is Lipschitzcontinuous and 119868

119894 119894 = 1 2 119901 is not compact We replace

hypotheses (1198671198912) (119868) by

(1198671198912)1015840There is a constant 119871 isin (0 1119872) such that 119892(119906) minus

119892(V) le 119871119906 minus VPC for all 119906 V isin PC([0 119886] 119883)(I1015840) There exists 119871

119894gt 0 119894 = 1 2 119901 such that 119868

119894(119906) minus

119868119894(V) le 119871

119894119906 minus V for all 119906 V isin 119883

Theorem 9 Assume that the hypotheses (1198671198921) (119867119891

2)1015840 (I1015840)

(1198671198911)ndash(119867119891

3) are satisfied Then the nonlocal impulsive prob-

lem (1) has at least one mild solution on [0a] provided that

119872119871 + 4119873119876int

119905

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894lt 1 (24)

Proof Define the operator 119870 PC([0 119886] 119883) rarr

PC([0 119886] 119883) by

(119870119906) (119905) = (1198701119906) (119905) + (119870

2119906) (119905) + (119870

3119906) (119905) (25)

With

(1198701119906) (119905) = 119879 (119905) 119892 (119906)

(1198702119906) (119905) = int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

(1198703119906) (119905) = sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894))

(26)

for all 119906 isin PC([0 119886] 119883)Define 119882

0= 119906 isin PC([0 119886] 119883) 119906(119905) le 119898(119905) for all

119905 isin [0 119886] and let119882 = 1198701198820

Then from the proof of Theorem 8 we know that 119882is a bounded closed convex and equicontinuous subset ofPC([0 119886] 119883) and 119870119882 sub 119882 We will prove that 119870 is120573PC-contraction on 119882 Then Darbo-Sadovskii fixed pointtheorem can be used to get a fixed point of 119870 in 119882 whichis a mild solution of (1)

We first show that1198701is Lipschitz on PC([0 119886] 119883)

In fact take 119906 V isin PC([0 119886] 119883) arbitraryThen by (1198671198912)1015840

we have

1003817100381710038171003817(1198701119906) (119905) minus (119870

2119906) (119905)

1003817100381710038171003817

le 1198721003817100381710038171003817119892 (119906) minus 119892 (V)100381710038171003817

1003817le 119872119871119906 minus VPC forall119905 isin [0 119886]

(27)

It follows that 1198701119906 minus 119870

1V le 119872119871119906 minus VPC for all 119906 V isin

PC([0 119886] 119883) That is 1198701is Lipschitz with Lipschitz constant

119872119871Next for every bounded subset 119861 sub 119882 for any 120576 gt 0

there is a sequence 119906119896infin

119896=1sub 119861 such that 120573(119870

2119861(119905)) le

2120573(1198702119906119896(119905)infin

119896=1) + 120576 for 119905 isin [0 119886] Since 119861 and 119870

2119861 are

equicontinuous we get from Lemmas 2 and 5 and (1198671198913) that

120573 (1198702119861 (119905))

le 2120573(int

119905

0

119879 (119905 minus 119904) 119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591) 119889119904) + 120576

le 4119873int

119905

0

120573(119891(119904 119906119896(119904)infin

119896=1 int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591

int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+120573(int

119904

0

119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+ 120573(int

119886

0

ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591))119889119904 + 120576

le 4119873int

119905

0

120578 (119904)(120573 (119906119896(119904)infin

119896=1)+ 2int

119904

0

120573 (119896 (119904 120591 119906119896(120591)infin

119896=1) 119889120591)

+2int

119886

0

120573 (ℎ (119904 120591 119906119896(120591)infin

119896=1) 119889120591)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1)

+ 2(int

119904

0

1205781(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)

International Journal of Differential Equations 7

+ 2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+ 21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) (120573PC (119906119896infin

119896=1) + 2

1198961(119904) 120573PC (119906119896

infin

119896=1)

+ 21198962(119904) 120573PC (119906119896

infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) 120573PC (119861) (1 + 21198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119886

0

120578 (119904) 119889119904120573PC (119861) + 120576

(28)

for 119905 isin [0 119886] Since 120576 gt 0 arbitrary we have

120573PC (1198702119861) le 4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 (29)

for any bounded subset 119861 sub 119882

120573 (1198703119861 (119905)) le 2119873

119901

sum

119894=1

120573 (119868119894119906119896(119905119894)infin

119896=1)

le 2119873

119901

sum

119894=1

119871119894120573 (119906119896(119905119894)infin

119896=1)

120573PC (1198703119861) le 2119873

119901

sum

119894=1

119871119894120573PC (119861)

(30)

for any subset 119861 sub 119882 due to Lemma 2 (29) and (30) wehave

120573PC (119870119861)

= 120573PC (1198701119861) + 120573PC (1198702119861) + 120573PC (1198703119861)

le 119872119871120573PC (119861) +4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 + 2119873

119901

sum

119894=1

119871119894120573PC (119861)

le (119872119871 + 4119873119876int

119886

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894)120573PC (119861)

(31)

From (24) we know that 119870 is 120573PC-contraction on 119882 ByLemma 3 there is a fixed point 119906 of 119870 in119882 which is a mildsolution of problem (1)

5 Classical Solutions

To study the classical solutions let us recall the followingresult

Lemma 10 Assume that 1199060

isin 119863(119860) 119902119894isin 119863(119860) 119894 =

1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883)

Then the impulsive differential equation

1199061015840

(119905) = 119860119906 (119905) + 119891 (119905 119906 (119905)) 119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060

Δ119906 (119905) = 119902119894 119894 = 1 2 119901

(32)

has a unique classical solution u(sdot)which satisfies for 119905 isin [0 119886]

119906 (119905)

= 119879 (119905) 119906 (0) = int

119905

0

119879 (119905 minus 119904) 119891 (119904 119906 (119904)) 119889119904 + sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119902119894

(33)

Now we make the following assumption(1198671) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(34)

or ae 119905 isin [0 119886] and for any bounded subset119863 sub 119875119862([0 119886] 119883)

Theorem 11 Let (1198671) be satisfied and 119906(sdot) a mild solution ofthe problem (1) Assume that 119906(0) isin 119863(119860) 119868

119894(119906(119905119894)) isin 119863(119860)

119894 = 1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883) Then u(0) gives

rise to a classical solution of the problem (1)If 119906(sdot) is a uniquely determined mild solution then it gives

rise to a unique classical solution

Proof We can define 119902119894= 119868119894(119906(119905119894)) 119894 = 1 2 119901

From Lemma 10

V1015840 (119905) = 119860V (119905) + 119891(119905 V (119905) int119905

0

119896 (119905 119904 V (119904)) 119889119904

int

119886

0

ℎ (119905 119904 V (119904)) 119889119904) 119905 isin [0 119886] 119905 = 119905119894

V (0) = 1199060+ 119892 (119906)

ΔV (119905) = 119902119894 119894 = 1 2 119901

(35)

has a unique classical solution V(sdot) which satisfies for 119905 isin

[0 119886]

V (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(36)

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 7

+ 2(int

119886

0

1205782(119904 120591) 119889120591)120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119905

0

120578 (119904) (120573 (119906119896(119904)infin

119896=1) + 2

1198961(119904) 120573 (119906

119896(119904)infin

119896=1)

+ 21198962(119904) 120573 (119906

119896(119904)infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) (120573PC (119906119896infin

119896=1) + 2

1198961(119904) 120573PC (119906119896

infin

119896=1)

+ 21198962(119904) 120573PC (119906119896

infin

119896=1)) 119889119904 + 120576

le 4119873int

119886

0

120578 (119904) 120573PC (119861) (1 + 21198961(119904) + 2

1198962(119904)) 119889119904 + 120576

le 4119873119876int

119886

0

120578 (119904) 119889119904120573PC (119861) + 120576

(28)

for 119905 isin [0 119886] Since 120576 gt 0 arbitrary we have

120573PC (1198702119861) le 4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 (29)

for any bounded subset 119861 sub 119882

120573 (1198703119861 (119905)) le 2119873

119901

sum

119894=1

120573 (119868119894119906119896(119905119894)infin

119896=1)

le 2119873

119901

sum

119894=1

119871119894120573 (119906119896(119905119894)infin

119896=1)

120573PC (1198703119861) le 2119873

119901

sum

119894=1

119871119894120573PC (119861)

(30)

for any subset 119861 sub 119882 due to Lemma 2 (29) and (30) wehave

120573PC (119870119861)

= 120573PC (1198701119861) + 120573PC (1198702119861) + 120573PC (1198703119861)

le 119872119871120573PC (119861) +4119873119876int

119886

0

120578 (119904) 120573PC (119861) 119889119904 + 2119873

119901

sum

119894=1

119871119894120573PC (119861)

le (119872119871 + 4119873119876int

119886

0

120578 (119904) 119889119904 + 2119873

119901

sum

119894=1

119871119894)120573PC (119861)

(31)

From (24) we know that 119870 is 120573PC-contraction on 119882 ByLemma 3 there is a fixed point 119906 of 119870 in119882 which is a mildsolution of problem (1)

5 Classical Solutions

To study the classical solutions let us recall the followingresult

Lemma 10 Assume that 1199060

isin 119863(119860) 119902119894isin 119863(119860) 119894 =

1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883)

Then the impulsive differential equation

1199061015840

(119905) = 119860119906 (119905) + 119891 (119905 119906 (119905)) 119905 isin [0 119886] 119905 = 119905119894

119906 (0) = 1199060

Δ119906 (119905) = 119902119894 119894 = 1 2 119901

(32)

has a unique classical solution u(sdot)which satisfies for 119905 isin [0 119886]

119906 (119905)

= 119879 (119905) 119906 (0) = int

119905

0

119879 (119905 minus 119904) 119891 (119904 119906 (119904)) 119889119904 + sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119902119894

(33)

Now we make the following assumption(1198671) There exists a function 120578 isin 119871

1(0 119886 119877

+) such that for

any bounded119863 sub 119883

120573 (119891 (119905 1198631 1198632 1198633)) le 120578 (119905) (120573 (119863

1) + 120573 (119863

2) + 120573 (119863

3))

(34)

or ae 119905 isin [0 119886] and for any bounded subset119863 sub 119875119862([0 119886] 119883)

Theorem 11 Let (1198671) be satisfied and 119906(sdot) a mild solution ofthe problem (1) Assume that 119906(0) isin 119863(119860) 119868

119894(119906(119905119894)) isin 119863(119860)

119894 = 1 2 119901 and that 119891 isin 1198621([0 119886] times 119883119883) Then u(0) gives

rise to a classical solution of the problem (1)If 119906(sdot) is a uniquely determined mild solution then it gives

rise to a unique classical solution

Proof We can define 119902119894= 119868119894(119906(119905119894)) 119894 = 1 2 119901

From Lemma 10

V1015840 (119905) = 119860V (119905) + 119891(119905 V (119905) int119905

0

119896 (119905 119904 V (119904)) 119889119904

int

119886

0

ℎ (119905 119904 V (119904)) 119889119904) 119905 isin [0 119886] 119905 = 119905119894

V (0) = 1199060+ 119892 (119906)

ΔV (119905) = 119902119894 119894 = 1 2 119901

(35)

has a unique classical solution V(sdot) which satisfies for 119905 isin

[0 119886]

V (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(36)

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Differential Equations

Now 119906(sdot) is a mild solution of the problem (1) so that we getfor 119905 isin [0 119886]

119906 (119905) = 119879 (119905) [1199060minus 119892 (119906)]

+ int

119905

0

119879 (119905 minus 119904) 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591) 119889119904

+ sum

0lt119905119894lt119905

119879 (119905 minus 119905119894) 119868119894(119906 (119905119894)) 0 le 119905 le 119886

(37)

Thus we get

V (119905) minus 119906 (119905)

= int

119905

0

119879 (119905 minus 119904) (119891(119904 V (119904) int119904

0

119896 (119904 120591 V (120591)) 119889120591

int

119886

0

ℎ (119904 120591 V (120591)) 119889120591)

minus 119891(119904 119906 (119904) int

119904

0

119896 (119904 120591 119906 (120591)) 119889120591

int

119886

0

ℎ (119904 120591 119906 (120591)) 119889120591))119889119904

(38)

which gives by (1198671) and an application of Gronwallrsquos

inequality 119906 minus VPC = 0This implies that 119906(sdot) gives rise to a classical solution and

completes the proof

6 Example

Let Ω be a bounded domain in 119877119899with smooth boundary

120597Ω and 119883 = 1198712(Ω) Consider the following nonlinear

integrodifferential equation in119883

120597119906 (119905 119910)

120597119905

= Δ119906 (119905 119910)

+

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(39)

119906 (119905+

119894 119910) minus 119906 (119905

minus

119894 119910) = 119868

119894(119906 (119905119894 119910)) 119894 = 1 2 119901 (40)

with nonlocal conditions

119906 (0) = 1199060(119910) +int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

119910 isin Ω

(41)

or

119906 (0) = 1199060(119910) + 120574

4119906 (119905 119910) 119910 isin Ω (42)

where 1205741 1205742 1205743 1205744 isin 119877 ℎ

1(119905 119910) isin PC([0 119886] Ω) Set 119860 =

Δ119863(119860) = 11988222(Ω)⋂119882

12

0(Ω)

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904) (119910)

=

1205741119906 (119905 119910)

(1 + 119905) (1 + 1199052)

sin (119906 (119905 119910))

+ int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

+ int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

(43)

Define nonlocal conditions

119892 (119906) (119910)

= 1199060(119910) + int

Ω

int

119886

0

ℎ1(119905 119910) log (1 + |119906 (119905 119904)|12) 119889119905 119889119904

(44)

or

119892 (119906) (119910) = 1199060(119910) + 120574

4119906 (119905 119910) (45)

It is easy to see that 119860 generates a compact 1198620-semigroup in

119883 and10038171003817100381710038171003817100381710038171003817

119891(119905 119906 (119905) int

119905

0

119896 (119905 119904 119906 (119904)) 119889119904 int

119886

0

ℎ (119905 119904 119906 (119904)) 119889119904)

10038171003817100381710038171003817100381710038171003817

le10038161003816100381610038161205741003816100381610038161003816(1 119906

1003817100381710038171003817119895100381710038171003817100381710038171003817100381710038171199021003817100381710038171003817)

10038161003816100381610038161205741003816100381610038161003816= max 100381610038161003816

10038161205741

100381610038161003816100381610038161003816100381610038161205742

100381610038161003816100381610038161003816100381610038161205743

1003816100381610038161003816

(46)

where

119895 = int

119905

0

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2119889119904

119896 (119905 119904 119906 (119904)) =

1205742119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2

119902 = int

119886

0

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2119889119904

ℎ (119905 119904 119906 (119904)) =

1205743119906 (119904 119910)

(1 + 119905) (1 + 1199052) (1 + 119904)

2(1 + 119904

2)2

(47)

and 119896(119905 119904 119906(119904)) le |1205742|(1+119906) ℎ(119905 119904 119906(119904)) le |120574

3|(1+119906)

119868119894(119906) le 119897

119894(119906) 119894 = 1 2 119901

For nonlocal conditions (44) 119892(119906) le 119886(mes(Ω))max119905isin[0119886]119910isinΩ

|ℎ1(119905 119910)|[119906 + (mes(Ω))12] 119906 isin PC([0 119886] 119883)

and 119892 is compact example of [18]

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 9

For nonlocal conditions (45)

1003817100381710038171003817119892 (1199061) minus 119892 (119906

2)1003817100381710038171003817le10038161003816100381610038161205744

1003816100381610038161003816

10038171003817100381710038171199061minus 1199062

1003817100381710038171003817 (48)

Hence 119892 is Lipschitz Furthermore 1205741 1205742 1205743 1205744and 119886 can be

chosen such that (24) is also satisfied Obviously it satisfies allthe assumptions given in our Theorem 9 the problem has atleast one mild solution in PC([0 119886] 1198712(Ω))

Conflict of Interests

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

Acknowledgment

The authors are thankful for Project MTM2010-16499 fromMEC of Spain

References

[1] Q Dong and G Li ldquoExistence of solutions for semilinear dif-ferential equations with nonlocal conditions in Banach spacesrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 47 p 13 2009

[2] T Kato ldquoQuasi-linear equations of evolution with applicationsto partial differential equationsrdquo in Spectral Theory and Differ-ential Equations vol 448 of Lectures Notes in Mathematics pp25ndash70 Springer Berlin Germany 1975

[3] T Kato ldquoAbstract evolution equations linear and quasilinearrevisitedrdquo in Functional Analysis and Related Topics 1991(Kyoto) vol 1540 of Lectures Notes in Mathematics pp 103ndash125Springer Berlin Germany 1993

[4] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsecond order semilinear ordinary and delay integrodifferentialequationswith nonlocal conditionsrdquoApplicable Analysis vol 67no 3-4 pp 245ndash257 1997

[5] Y Yang and J Wang ldquoOn some existence results of mildsolutions for nonlocal integrodifferential Cauchy problems inBanach spacesrdquo Opuscula Mathematica vol 31 no 3 pp 443ndash455 2011

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

[7] X Xue ldquoSemilinear nonlocal differential equations with mea-sure of noncompactness in Banach spacesrdquo Nanjing UniversityJournal Mathematical Biquarterly vol 24 no 2 pp 264ndash2762007

[8] C Lizama and J C Pozo ldquoExistence of mild solutions fora semilinear integrodifferential equation with nonlocal initialconditionsrdquo Abstract and Applied Analysis vol 2012 Article ID647103 15 pages 2012

[9] N U Ahmed ldquoSystems governed by impulsive differentialinclusions onHilbert spacesrdquoNonlinear AnalysisTheory Meth-ods amp Applications A vol 45 no 6 pp 693ndash706 2001

[10] M Benchohra J Henderson and S Ntouyas Impulsive Differ-ential Equations and Inclusions vol 2 of Contemporary Math-ematics and Its Applications Hindawi Publishing CorporationNew York NY USA 2006

[11] K Malar and A Tamilarasi ldquoExistence of mild solutionsfor nonlocal impulsive integrodifferential equations with themeasure of noncompactnessrdquo Far East Journal of AppliedMathematics vol 57 no 1 pp 15ndash31 2011

[12] S Migorski and A Ochal ldquoNonlinear impulsive evolutioninclusions of second orderrdquo Dynamic Systems and Applicationsvol 16 no 1 pp 155ndash173 2007

[13] S Ji and S Wen ldquoNonlocal Cauchy problem for impulsivedifferential equations in Banach spacesrdquo International Journalof Nonlinear Science vol 10 no 1 pp 88ndash95 2010

[14] L Byszewski and H Akca ldquoExistence of solutions of asemilinear functional-differential evolution nonlocal problemrdquoNonlinear Analysis Theory Methods amp Applications A vol 34no 1 pp 65ndash72 1998

[15] 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

[16] S Aizicovici and M McKibben ldquoExistence results for a classof abstract nonlocal Cauchy problemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 39 no 5 pp 649ndash6682000

[17] J Liang J Liu and T-J Xiao ldquoNonlocal Cauchy problemsgoverned by compact operator familiesrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 57 no 2 pp 183ndash1892004

[18] J Liang J H Liu and T-J Xiao ldquoNonlocal problems forintegrodifferential equationsrdquoDynamics of Continuous Discreteamp Impulsive Systems A vol 15 no 6 pp 815ndash824 2008

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

[20] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence for semi-linear evolution equations with nonlocal conditionsrdquo Journal ofMathematical Analysis and Applications vol 210 no 2 pp 679ndash687 1997

[21] S K Ntouyas and P Ch Tsamatos ldquoGlobal existence forsemilinear evolution integrodifferential equations with delayand nonlocal conditionsrdquo Applicable Analysis vol 64 no 1-2pp 99ndash105 1997

[22] T Zhu C Song and G Li ldquoExistence of mild solutionsfor abstract semilinear evolution equations in Banach spacesrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 1 pp 177ndash181 2012

[23] J H Liu ldquoNonlinear impulsive evolution equationsrdquo Dynamicsof Continuous Discrete and Impulsive Systems vol 6 no 1 pp77ndash85 1999

[24] S Ji and G Li ldquoA unified approach to nonlocal impulsivedifferential equations with the measure of noncompactnessrdquoAdvances in Difference Equations vol 2012 article 182 2012

[25] B Ahmad K Malar and K Karthikeyan ldquoA study of nonlocalproblems of impulsive integrodifferential equations with mea-sure of noncompactnessrdquoAdvances in Difference Equations vol2013 article 205 2013

[26] S Ji andG Li ldquoExistence results for impulsive differential inclu-sionswith nonlocal conditionsrdquoComputersampMathematics withApplications vol 62 no 4 pp 1908ndash1915 2011

[27] L Zhu Q Dong and G Li ldquoImpulsive differential equationswith nonlocal conditionsin general Banach spacesrdquoAdvances inDifference Equations vol 2012 article 10 2012

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Differential Equations

[28] A Debbouche D Baleanu and R P Agarwal ldquoNonlocalnonlinear integrodifferential equations of fractional ordersrdquoBoundary Value Problems vol 2012 article 78 2012

[29] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods vol 3 of Series onComplexity Nonlinearity and Chaos World Scientific Singa-pore 2012

[30] J Banas and K GoebelMeasures of Noncompactness in BanachApaces vol 60 of Lecture Notes in Pure and Applied Mathemat-ics Marcel Dekker New York NY USA 1980

[31] D Both ldquoMultivalued perturbation of m-accretive differentialinclusionsrdquo Israel Journal of Mathematics vol 108 pp 109ndash1381998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of