Existence of Mild and Classical Solutions for Nonlocal Impulsive Integrodifferential Equations in Banach Spaces with Measure of Noncompactness

Abstract

We study the existence of mild and classical solutions are proved for a class of impulsive integrodifferential equations with nonlocal conditions in Banach spaces. The main results are obtained by using measure of noncompactness and semigroup theory. An example is presented.

(ProQuest: ... denotes non-US-ASCII text omitted.)

K. Karthikeyan 1 and A. Anguraj 2 and K. Malar 3 and Juan J. Trujillo 4

Academic Editor:Juan J. Nieto

1, Department of Mathematics, K.S.R. College of Technology, Tiruchengode, Tamil Nadu 637215, India
2, Department of Mathematics, PSG College of Arts and Science, Coimbatore, Tamil Nadu 641014, India
3, Department of Mathematics, Erode Arts and Science College, Erode, Tamil Nadu 638 009, India
4, Universidad de La Laguna Departamento de Análisis Matemático, C/Astr. Fco. Snchez s/n, Tenerife, 38271 La Laguna, Spain

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

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this present paper, we are concerned with the existence of mild and classical solutions are proved for a class of impulsive integrodifferential equations with nonlocal conditions: [figure omitted; refer to PDF] where A is the infinitesimal generator of a _{C 0} -semigroup T ( t ) in a Banach space X and f : [ 0 , a ] × X × X [arrow right] X , k : [ 0 , a ] × [ 0 , a ] × X [arrow right] X , h : [ 0 , a ] × [ 0 , a ] × X [arrow right] X , 0 < _{t 1} < _{t 2} < _{t 3} < ... < _{t p} < a and _{I i} : X [arrow right] X , i = 1,2 , ... , p are impulsive functions and g : PC ( [ 0 , a ] ; X ) [arrow right] X , _{u 0} ∈ X , and X is a real Banach space with norm || · || . Δ u ( _{t i} ) = u ( ^{t i +} ) - u ( ^{t i -} ) , u ( ^{t i +} ) , u ( ^{t i -} ) denote the right and left limits of u at _{t i} , respectively.

Integrodifferential equations are important for investigating some problems raised from natural phenomena. They have been studied in many different aspects. The theory of semigroups of bounded linear operators is closely related to the solution of differential and integrodifferential equations in Banach spaces. In recent years, this theory has been applied to a large class of nonlinear differential equations in Banach spaces. We refer to the papers [1-5] and the references cited therein. Based on the method of semigroups, existence, and uniqueness of mild, strong, and classical solutions of semilinear evolution equations were discussed by Pazy [6]. In [7], Xue studied the semilinear nonlocal differential equations with measure of noncompactness in Banach spaces. Lizama and Pozo [8] investigated the existence of mild solutions for semilinear integrodifferential equation with nonlocal initial conditions by using Hausdorff measure of noncompactness via a fixed point.

In recent years, the impulsive differential equations have been an object of intensive investigation because of the wide possibilities for their applications in various fields of science and technology as theoretical physics, population dynamics, economics, and so forth; see [9-13]. The study of semilinear nonlocal initial problem was initiated by Byszewski [14, 15] and the importance of the problem lies in the fact that it is more general and yields better effect than the classical initial conditions. Therefore it has been extensively studied under various conditions on the operator A and the nonlinearity f by several authors [13, 16-18].

Byszwski and Lakshmikantham [19] prove the existence and uniqueness of mild solutions and classical solutions when f and g satisfy Lipschitz-type conditions. Ntouyas and Tsamotas [20, 21] study the case of compactness conditions of f and T ( t ) . Zhu et al. [22] studied the existence of mild solutions for abstract semilinear evolution equations in Banach spaces. In [23], Liu discussed the existence and uniqueness of mild and classical solutions for the impulsive semilinear differential evolution equation. In [24], the authors studied the existence of mild solutions to an impulsive differential equation with nonlocal conditions by applying Darbo-Sadovskii's fixed point theorem. In recent paper [25], Ahmad et al. studied nonlocal problems of impulsive integrodifferential equations with measure of noncompactness. For some more recent results and details, see [26-29].

Motivated by the above-mentioned works, we derive some sufficient conditions for the solutions of integrodifferential equations (1) combining impulsive conditions and nonlocal conditions. Our results are achieved by applying the Hausdorff measure of noncompactness and fixed point theorem. In this paper, we denote by N = sup ... { || T ( t ) || : t ∈ [ 0 , a ] } . Without loss of generality, we let _{u 0} = 0 .

2. Preliminaries

Let ( X , || · || ) be a real Banach space. We denote by C ( [ 0 , a ] ; X ) the space of X -valued continuous functions on [ 0 , a ] with the norm || x || = sup ... || x ( t ) || , t ∈ [ 0 , a ] and by ^{L 1} ( 0 , a ; X ) the space of X -valued Bochner integrable functions on [ 0 , a ] with the norm _{|| f || ^{L 1}} = ^{∫ 0 a} ... || f ( t ) || d t .

We put _{J i} = ( _{t i} , _{t i + 1} ] , i = 1,2 , ... , p . In order to define the mild solution of problem (1), we introduce the following set.

PC ( [ 0 , a ] ; X ) = { u : [ 0 , a ] [arrow right] X : u is continuous on _{J i} , i = 0,1 , 2 , ... , p and the right limit u ( _{t i} ) exists , i = 1,2 , ... , p } .

Definition 1.

A function u ∈ PC ( [ 0 , a ] ; X ) is a mild solution of (1) if [figure omitted; refer to PDF] The Hausdorff measure of noncompactness _{β Y} ( B ) is defined by _{β Y} ( B ) = inf ... { r > 0 , B can be covered by finite number of balls with radii r } for bounded set B in a Banach space Y .

Lemma 2 (see [30]).

Let Y be a real Banach space and B ; C ⊆ Y be bounded, with the following properties:

(1) B is precompact if and only if _{β X} ( B ) = 0 ;

(2) _{β Y} ( B ) = _{β Y} ( B ¯ ) = _{β Y} ( c o n v B ) , where B ¯ and _{c o n v} B mean the closure and convex hull of B , respectively;

(3) _{β Y} ( B ) ...4; _{β Y} ( C ) , where B ⊆ C ;

(4) _{β Y} ( B + C ) ...4; _{β Y} ( B ) + _{β Y} ( C ) , where B + C = { x + y : x ∈ B , y ∈ C } ;

(5) _{β Y} ( B ∪ C ) ...4; max ... { _{β Y} ( B ) , _{β Y} ( C ) } ;

(6) _{β Y} ( λ B ) ...4; | λ | _{β Y} ( B ) for any λ ∈ R ;

(7) if the map Q : D ( Q ) ⊆ Y [arrow right] Z is Lipschitz continuous with constant k , then _{β Z} ( Q B ) ...4; k _{β Y} ( B ) for any bounded subset B ⊆ D ( Q ) , where Z be a Banach space;

(8) _{β Y} ( B ) = inf ... { _{d Y} ( B , C ) ; C ⊆ Y i s p r e c o m p a c t } = inf ... { _{d Y} ( B , C ) ; C ⊆ Y i s f i n i t e v a l u e d } , where _{d Y} ( B , C ) means the nonsymmetric (or symmetric) Hausdorff distance between B and C in Y ;

(9) if ^{{ _{W n} } n = 1 + ∞} is decreasing sequence of bounded closed nonempty subsets of Y and _{lim ... n [arrow right] ∞} ... _{β Y} ( _{W n} ) = 0 , then ^{... n = 1 + ∞} _{W n} is nonempty and compact in Y .

The map Q : W ⊆ Y [arrow right] Y is said to be a _{β Y} -contraction if there exists a positive constant k < 1 such that _{β Y} ( Q ( B ) ) ...4; k _{β Y} ( B ) for any bounded closed subset B ⊆ W , where Y is a Banach space.

Lemma 3 (Darbo-Sadovskii [30]).

If W ⊆ Y is bounded closed and convex, the continuous map Q : W [arrow right] W is a _{β Y} -contraction, then the map Q has at least one fixed point in W .

Lemma 4 (see [2]).

If W ⊆ P C ( [ 0 , a ] ; X ) is bounded, then α ( W ( t ) ) ...4; _{α P C} ( W ) for all t ∈ [ 0 , a ] , where W ( t ) = { u ( t ) : u ∈ W } ⊆ X . Furthermore if W is equicontinuous on each interval _{J i} of [0,a], then α ( W ( t ) ) is continuous on [0,a], and _{α P C} ( W ) = sup ... { α ( W ( t ) ) : t ∈ [ 0 , a ] } .

Lemma 5 (see [3]).

If ^{{ _{u n} } n = 1 ∞} ⊂ ^{L 1} ( 0 , a ; X ) is uniformly integrable, then α ( ^{{ _{u n} ( t ) } n = 1 ∞} ) is measurable and [figure omitted; refer to PDF]

Lemma 6 (see [31]).

If the semigroup T ( t ) is equicontinuous and η ∈ ^{L 1} ( 0 , a ; ^{R +} ) , then the set [figure omitted; refer to PDF] is equicontinuous on [0,a].

Lemma 7 (see [23]).

If W is bounded, then for each [varepsilon] > 0 , there is a sequence ^{{ _{u n} } n = 1 ∞} ⊆ W , such that α ( W ) ...4; 2 α ( ^{{ _{u n} } n = 1 ∞} ) + [varepsilon] .

3. g Is Compact

In this section, we give the existence results of nonlocal integrodifferential equation (39). Here we list the following hypotheses.

( H _{g 1} ): The _{c 0} semigroup T ( t ) , 0 ...4; t ...4; a , generated by A is equicontinuous.

( H _{g 2} ):

(i) g : PC ( [ 0 , a ] ; X ) [arrow right] X is continuous and compact.

(ii) There exists M > 0 such that || g ( u ) || ...4; M , for all u ∈ PC ( [ 0 , a ] ; X ) and k ^ ( s ) = max ... { 1 , ^{∫ 0 s} ... k ( s , τ ) d τ } .

(I) Let _{I i} : X [arrow right] X be continuous, compact map and there are nondecreasing functions _{l i} : ^{R +} [arrow right] ^{R +} , satisfying || _{I i} ( x ) || ...4; _{l i} ( || x || ) , i = 1,2 , ... , p .

( H _{f 1} ): There exists a continuous function _{a k} : [ 0 , a ] × [ 0 , a ] [arrow right] [ 0 , ∞ ) and a nondecreasing continuous function _{Ω k} : ^{R +} [arrow right] ^{R +} such that || k ( t , s , x ) || ...4; _{a k} ( t , s ) _{Ω k} ( || x || ) for all x ∈ X a.e. t , s ∈ [ 0 , a ] . And there exists at least one mild solution to the following scalar equation: [figure omitted; refer to PDF]

( H _{f 2} ):

(i) f ( · , · , x , y ) is measurable for x , y ∈ X , f ( t , · , · , · ) is continuous for a.e. t ∈ [ 0 , a ] .

(ii) There exist a function _{a f} ( · ) ∈ ^{L 1} ( 0 , a , ^{R +} ) and an increasing continuous function _{Ω f} : ^{R +} [arrow right] ^{R +} such that || f ( t , x , y , z ) || ...4; _{a f} ( t ) _{Ω f} ( || x || , || y || , || z || ) for all x , y , z ∈ X and a.e. t ∈ [ 0 , a ] .

(iii): f : [ 0 , a ] × X × X [arrow right] X is compact.

( H _{f 3} ): There exists a function η ∈ ^{L 1} ( 0 , a ; ^{R +} ) such that for any bounded D ⊂ X , [figure omitted; refer to PDF] for a.e. t ∈ [ 0 , a ] and for any bounded subset D ⊂ PC ( [ 0 , a ] , X ) .

Here we let k ^ ( s ) = ^{∫ 0 s} ... k ( s , τ ) d τ and ( 1 + 2 _{k 1} ^ ( s ) + 2 _{k 2} ^ ( s ) ) ...4; Q .

Theorem 8.

Assume that the hypotheses ( H _{g 1} ) , ( H _{g 2} ) , I, ( H _{f 1} ) , and ( H _{f 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); the map K : PC ( [ 0 , a ] ; X ) [arrow right] PC ( [ 0 , a ] ; X ) is defined by [figure omitted; refer to PDF] with [figure omitted; refer to PDF] for all t ∈ [ 0 , a ] .

It is easy to see that the fixed point of K is the mild solution of nonlocal impulsive problem (1).

From our hypotheses, the continuity of K is proved as follows.

For this purpose, we assume that _{u n} [arrow right] u in PC ( [ 0 , a ] ; X ). It comes from the continuity of k and h that k ( s , τ , _{u n} ( τ ) ) [arrow right] k ( s , τ , u ( τ ) ) and h ( s , τ , _{u n} ( τ ) ) [arrow right] h ( s , τ , u ( τ ) ) , respectively.

By Lebesgue convergence theorem, [figure omitted; refer to PDF] Similarly we have [figure omitted; refer to PDF] for all s ∈ [ 0 , a ] . Consider [figure omitted; refer to PDF] as n [arrow right] ∞ . So K _{u n} [arrow right] K u in PC ( [ 0 , a ] ; X ) . That is, K is continuous.

We denote _{W 0} = { u ∈ PC ( [ 0 , a ] ) ; X ) , || u ( t ) || ...4; m ( t ) for all t ∈ [ 0 , a ] } ; then W ⊆ PC ( [ o , a ] ; X ) is bounded and convex.

Define _{W 1} = conv ¯ K ( _{W 0} ) , where conv ¯ means that the closure of the convex hull in PC ( [ 0 , a ] ; X ) .

For any u ∈ K ( _{W 0} ) , we know that [figure omitted; refer to PDF] and by ( H _{f 2} ) , _{W 1} ⊂ _{W 0} .

From the Arzela-Ascoli theorem, to prove the compactness of K , we can prove that _{K 1} u : u ∈ _{W 0} is equicontinuous and _{K 1} u ( t ) ⊂ X is precompact for t ∈ [ 0 , a ] : [figure omitted; refer to PDF] Since f is compact, M N || T ( σ ) - I || and || [ T ( σ ) - I ] f ( s , u ( s ) , ^{∫ 0 s} ... k ( s , τ , u ( τ ) ) d τ , ^{∫ 0 a} ... h ( s , τ , u ( τ ) ) d τ ) || [arrow right] 0 as σ [arrow right] 0 uniformly for s ∈ [ 0 , a ] and u ∈ PC ( [ 0 , a ] ; X ) . This implies that for any _{[varepsilon] 1} > 0 and _{[varepsilon] 2} > 0 , there exist a δ > 0 such that [figure omitted; refer to PDF] for 0 ...4; σ < δ and all u ∈ PC ( [ 0 , a ] ; X ) . Therefore [figure omitted; refer to PDF]

We know that [figure omitted; refer to PDF] for 0 ...4; σ < δ and all u ∈ PC ( [ 0 , a ] ; X ) . So { _{K 1} u : u ∈ _{W 0} } is equicontinuous.

The set { T ( t - s ) f ( s , u ( s ) , ^{∫ 0 s} ... k ( s , τ , u ( τ ) ) d τ , ^{∫ 0 a} ... h ( s , τ , u ( τ ) ) d τ ) ; t , s ∈ [ 0 , a ] , u ∈ PC ( [ 0 , a ] ; X ) } is precompact as f is compact and T ( · ) is a _{C 0} -semigroup.

So _{K 1} u ( t ) ⊂ X is precompact as _{K 1} u ( t ) ⊂ t conv ¯ { T ( t - s ) f ( s , u ( s ) , ^{∫ 0 s} ... k ( s , τ , u ( τ ) ) d τ , ^{∫ 0 a} ... h ( s , τ , u ( τ ) ) d τ ) ; s ∈ [ 0 , t ] , u ∈ PC ( [ 0 , a ] ; X ) } for all t ∈ [ 0 , a ] . _{W 1} is equicontinuous on each interval _{J i} of [ 0 , a ] . For _{t i} ...4; t < t + σ ...4; _{t i + 1} , i = 1,2 , ... , p , we have, using the semigroup properties [figure omitted; refer to PDF] which follows that { _{K 2} u : u ∈ _{W 0} } is equicontinuous on each _{J i} due to the equicontinuous of T ( t ) and hypotheses (I ). Therefore, _{W 1} ⊂ PC ( [ 0 , a ] ; X ) is bounded closed convex nonempty and equicontinuous on each interval _{J i} , i = 0,1 , 2 , ... , p .

We define _{W n + 1} = conv ¯ K ( _{W n} ) , for n = 1,2 , ... , p . From above we know that ^{{ _{W n} } n = 1 ∞} is a decreasing sequence of bounded, closed, convex nonempty subsets in PC ( [ 0 , a ] ; X ) and equicontinuous on each _{J i} , i = 1,2 , ... , p .

Now for n ...5; 1 and t ∈ [ 0 , a ] , _{W n} ( t ) and K ( _{W n} ( t ) ) are bounded subsets of X . Hence for any [varepsilon] > 0 , there is a sequence ^{{ _{u k} } k = 1 ∞} ⊂ _{W n} such that (see, e.g., [2, page 125]) [figure omitted; refer to PDF] for t ∈ [ 0 , a ] .

From the compactness of g and _{I i} , by Lemmas 2 and 5 and ( H _{f 3} ) , we have [figure omitted; refer to PDF] Since [varepsilon] > 0 is arbitrary, it follows from the above inequality that [figure omitted; refer to PDF] for all t ∈ [ 0 , a ] . Since _{W n} is decreasing for n , we define _{f n} ( t ) = _{lim ... n [arrow right] ∞} β ( _{W n} ( t ) ) for all t ∈ [ 0 , a ] . From (20), we have [figure omitted; refer to PDF] for t ∈ [ 0 , a ] , which implies that _{f n} ( t ) = 0 for all t ∈ [ 0 , a ] . By Lemma 4, we know that [figure omitted; refer to PDF] Using Lemma 2, we also know that [figure omitted; refer to PDF] is convex, compact, and nonempty in PC ( [ 0 , a ] ; X ) and K ( W ) ⊂ W .

By the famous Schauder's fixed point theorem, there exists at least one mild solution u of the problem (1), where u ∈ W is a fixed point of the continuous map K .

4. g Is Lipschitz

In this section, we discuss the problem (1) when g is Lipschitz continuous and _{I i} , i = 1,2 , ... , p is not compact. We replace hypotheses ( H _{f 2} ) , ( I ) by

^{( H _{f 2} ) [variant prime]} There is a constant L ∈ ( 0 , 1 / M ) such that || g ( u ) - g ( v ) || ...4; L _{|| u - v || PC} for all u , v ∈ PC ( [ 0 , a ] ; X ) .

(I [variant prime]) There exists _{L i} > 0 , i = 1,2 , ... , p , such that || _{I i} ( u ) - _{I i} ( v ) || ...4; _{L i} || u - v || , for all u , v ∈ X .

Theorem 9.

Assume that the hypotheses ( H _{g 1} ) , ^{( H _{f 2} ) [variant prime]} , (I[variant prime]), ( H _{f 1} ) - ( H _{f 3} ) are satisfied. Then the nonlocal impulsive problem (1) has at least one mild solution on [0,a], provided that [figure omitted; refer to PDF]

Proof.

Define the operator K : PC ( [ 0 , a ] ; X ) [arrow right] PC ( [ 0 , a ] ; X ) by [figure omitted; refer to PDF] With [figure omitted; refer to PDF] for all u ∈ PC ( [ 0 , a ] ; X ) .

Define _{W 0} = { u ∈ PC ( [ 0 , a ] ; X ) } : || u ( t ) || ...4; m ( t ) for all t ∈ [ 0 , a ] , and let W = K _{W 0} ¯ .

Then from the proof of Theorem 8, we know that W is a bounded closed convex and equicontinuous subset of PC ( [ 0 , a ] ; X ) and K W ⊂ W . We will prove that K is _{β PC} -contraction on W . Then Darbo-Sadovskii fixed point theorem can be used to get a fixed point of K in W , which is a mild solution of (1).

We first show that _{K 1} is Lipschitz on PC ( [ 0 , a ] ; X ) .

In fact, take u , v ∈ PC ( [ 0 , a ] ; X ) arbitrary. Then by ^{( H _{f 2} ) [variant prime]} , we have [figure omitted; refer to PDF]

It follows that || _{K 1} u - _{K 1} v || ...4; M L _{|| u - v || PC} for all u , v ∈ PC ( [ 0 , a ] ; X ) . That is, _{K 1} is Lipschitz with Lipschitz constant M L .

Next, for every bounded subset B ⊂ W , for any [varepsilon] > 0 , there is a sequence ^{{ _{u k} } k = 1 ∞} ⊂ B such that β ( _{K 2} B ( t ) ) ...4; 2 β ( ^{{ _{K 2 u k} ( t ) } k = 1 ∞} ) + [varepsilon] , for t ∈ [ 0 , a ] . Since B and _{K 2} B are equicontinuous, we get from Lemmas 2 and 5 and ( H _{f 3} ) that [figure omitted; refer to PDF] for t ∈ [ 0 , a ] . Since [varepsilon] > 0 arbitrary, we have [figure omitted; refer to PDF] for any bounded subset B ⊂ W . [figure omitted; refer to PDF] for any subset B ⊂ W ; due to Lemma 2, (29), and (30), we have [figure omitted; refer to PDF] From (24), we know that K is _{β PC} -contraction on W . By Lemma 3, there is a fixed point u of K in W , which is a mild solution of problem (1).

5. Classical Solutions

To study the classical solutions, let us recall the following result.

Lemma 10.

Assume that _{u 0} ∈ D ( A ) , _{q i} ∈ D ( A ) , i = 1,2 , ... , p , and that f ∈ ^{C 1} ( [ 0 , a ] × X , X ) .

Then the impulsive differential equation [figure omitted; refer to PDF] has a unique classical solution u ( · ) which satisfies, for t ∈ [ 0 , a ] , [figure omitted; refer to PDF] Now we make the following assumption.

( ^{H 1} ) There exists a function η ∈ ^{L 1} ( 0 , a ; ^{R +} ) such that for any bounded D ⊂ X , [figure omitted; refer to PDF] or a.e. t ∈ [ 0 , a ] and for any bounded subset D ⊂ P C ( [ 0 , a ] , X ) .

Theorem 11.

Let ( ^{H 1} ) be satisfied and u ( · ) a mild solution of the problem (1). Assume that u ( 0 ) ∈ D ( A ) , _{I i} ( u ( _{t i} ) ) ∈ D ( A ) , i = 1,2 , ... , p and that f ∈ ^{C 1} ( [ 0 , a ] × X , X ) . Then u(0) gives rise to a classical solution of the problem (1).

If u ( · ) is a uniquely determined mild solution, then it gives rise to a unique classical solution.

Proof.

We can define _{q i} = _{I i} ( u ( _{t i} ) ) , i = 1,2 , ... , p .

From Lemma 10, [figure omitted; refer to PDF] has a unique classical solution v ( · ) which satisfies, for t ∈ [ 0 , a ] , [figure omitted; refer to PDF] Now, u ( · ) is a mild solution of the problem (1), so that we get for t ∈ [ 0 , a ] , [figure omitted; refer to PDF] Thus we get [figure omitted; refer to PDF] which gives, by ( ^{H 1} ) and an application of Gronwall's inequality, _{|| u - v || PC} = 0 .

This implies that u ( · ) gives rise to a classical solution and completes the proof.

6. Example

Let Ω be a bounded domain in _{R n} with smooth boundary ∂ Ω , and X = ^{L 2} ( Ω ) . Consider the following nonlinear integrodifferential equation in X : [figure omitted; refer to PDF] with nonlocal conditions [figure omitted; refer to PDF] or [figure omitted; refer to PDF] where _{γ 1} , _{γ 2} , _{γ 3} , _{γ 4} , ∈ R , _{h 1} ( t , y ) ∈ PC ( [ 0 , a ] ; Ω ¯ ) . Set A = Δ , D ( A ) = ^{W 2,2} ( Ω ) _... ... ^{W 0 1,2} ( Ω ) , [figure omitted; refer to PDF] Define nonlocal conditions [figure omitted; refer to PDF] or [figure omitted; refer to PDF] It is easy to see that A generates a compact _{C 0} -semigroup in X , and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and || k ( t , s , u ( s ) ) || ...4; | _{γ 2} | ( 1 + || u || ) , || h ( t , s , u ( s ) ) || ...4; | _{γ 3} | ( 1 + || u || ) , || _{I i} ( u ) || ...4; _{l i} ( || u || ) , i = 1,2 , ... , p .

For nonlocal conditions (44), || g ( u ) || ...4; a ( mes ( Ω ) ) _{max ... t ∈ [ 0 , a ] , y ∈ Ω} | _{h 1} ( t , y ) | [ || u || + ^{( mes ( Ω ) ) 1 / 2} ] , u ∈ PC ( [ 0 , a ] ; X ) , and g is compact example of [18].

For nonlocal conditions (45), [figure omitted; refer to PDF] Hence, g is Lipschitz. Furthermore, _{γ 1} , _{γ 2} , _{γ 3} , _{γ 4} and a can be chosen such that (24) is also satisfied. Obviously, it satisfies all the assumptions given in our Theorem 9; the problem has at least one mild solution in PC ( [ 0 , a ] ; ^{L 2} ( Ω ) ) .

Acknowledgment

The authors are thankful for Project MTM2010-16499 from MEC of Spain.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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