research article existence of mild and classical solutions...
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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
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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
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
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
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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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom 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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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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
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