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.

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: where is the infinitesimal generator of a -semigroup in a Banach space and , , and are impulsive functions and , and is a real Banach space with norm . , denote the right and left limits of at , 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 and the nonlinearity by several authors [13, 16–18].

Byszwski and Lakshmikantham [19] prove the existence and uniqueness of mild solutions and classical solutions when and satisfy Lipschitz-type conditions. Ntouyas and Tsamotas [20, 21] study the case of compactness conditions of and . 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 . Without loss of generality, we let .

2. Preliminaries

Let () be a real Banach space. We denote by the space of X-valued continuous functions on with the norm and by the space of X-valued Bochner integrable functions on with the norm .

We put , . In order to define the mild solution of problem (1), we introduce the following set.

is continuous onand the right limit .

Definition 1. A function is a mild solution of (1) if The Hausdorff measure of noncompactness is defined by can be covered by finite number of balls with radii for bounded set in a Banach space .

Lemma 2 (see [30]). Let be a real Banach space and be bounded, with the following properties:(1)is precompact if and only if ;(2), where and mean the closure and convex hull of , respectively;(3), where ;(4), where ;(5);(6) for any ;(7)if the map is Lipschitz continuous with constant , then for any bounded subset , where be a Banach space;(8), where means the nonsymmetric (or symmetric) Hausdorff distance between and in ;(9)if is decreasing sequence of bounded closed nonempty subsets of and , then is nonempty and compact in .

The map is said to be a -contraction if there exists a positive constant such that for any bounded closed subset , where is a Banach space.

Lemma 3 (Darbo-Sadovskii [30]). If is bounded closed and convex, the continuous map is a -contraction, then the map has at least one fixed point in .

Lemma 4 (see [2]). If is bounded, then for all , where . Furthermore if is equicontinuous on each interval of [0,a], then is continuous on [0,a], and .

Lemma 5 (see [3]). If is uniformly integrable, then is measurable and

Lemma 6 (see [31]). If the semigroup is equicontinuous and , then the set is equicontinuous on [0,a].

Lemma 7 (see [23]). If is bounded, then for each , there is a sequence , such that .

3. Is Compact

In this section, we give the existence results of nonlocal integrodifferential equation (39). Here we list the following hypotheses.)The semigroup , , generated by is equicontinuous.()(i) is continuous and compact.(ii)There exists such that , for all and .

(I) Let be continuous, compact map and there are nondecreasing functions , satisfying , .()There exists a continuous function and a nondecreasing continuous function such that for all a.e. . And there exists at least one mild solution to the following scalar equation: ()(i) is measurable for is continuous for a.e. .(ii)There exist a function and an increasing continuous function such that for all and a.e. .(iii) is compact.()There exists a function such that for any bounded , for a.e. and for any bounded subset .Here we let and .

Theorem 8. Assume that the hypotheses , , I, , and 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 is defined by with for all .
It is easy to see that the fixed point of is the mild solution of nonlocal impulsive problem (1).
From our hypotheses, the continuity of is proved as follows.
For this purpose, we assume that in ). It comes from the continuity of and that and , respectively.
By Lebesgue convergence theorem, Similarly we have for all . Consider as . So in . That is, is continuous.
We denote , for all ; then is bounded and convex.
Define , where means that the closure of the convex hull in .
For any , we know that and by , .
From the Arzela-Ascoli theorem, to prove the compactness of , we can prove that is equicontinuous and is precompact for : Since is compact, and as uniformly for and . This implies that for any and , there exist a such that for and all . Therefore
We know that for and all . So is equicontinuous.
The set , ,; , is precompact as f is compact and is a -semigroup.
So is precompact as for all . is equicontinuous on each interval of . For , , we have, using the semigroup properties which follows that is equicontinuous on each due to the equicontinuous of and hypotheses (I). Therefore, is bounded closed convex nonempty and equicontinuous on each interval , .
We define , for . From above we know that is a decreasing sequence of bounded, closed, convex nonempty subsets in and equicontinuous on each , .
Now for and , and are bounded subsets of . Hence for any , there is a sequence such that (see, e.g., [2, page 125]) for .
From the compactness of and , by Lemmas 2 and 5 and , we have Since is arbitrary, it follows from the above inequality that for all . Since is decreasing for , we define for all . From (20), we have for , which implies that for all . By Lemma 4, we know that Using Lemma 2, we also know that is convex, compact, and nonempty in and .
By the famous Schauder’s fixed point theorem, there exists at least one mild solution of the problem (1), where is a fixed point of the continuous map .

4. Is Lipschitz

In this section, we discuss the problem (1) when is Lipschitz continuous and , is not compact. We replace hypotheses , by

There is a constant such that for all .

(I′) There exists , , such that , for all .

Theorem 9. Assume that the hypotheses , , (I′), are satisfied. Then the nonlocal impulsive problem (1) has at least one mild solution on [0,a], provided that

Proof. Define the operator by With for all .
Define for all , and let .
Then from the proof of Theorem 8, we know that is a bounded closed convex and equicontinuous subset of and . We will prove that is -contraction on . Then Darbo-Sadovskii fixed point theorem can be used to get a fixed point of in , which is a mild solution of (1).
We first show that is Lipschitz on .
In fact, take arbitrary. Then by , we have
It follows that for all . That is, is Lipschitz with Lipschitz constant .
Next, for every bounded subset , for any , there is a sequence such that , for . Since and are equicontinuous, we get from Lemmas 2 and 5 and that for . Since arbitrary, we have for any bounded subset . for any subset ; due to Lemma 2, (29), and (30), we have From (24), we know that is -contraction on . By Lemma 3, there is a fixed point of in , 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 , , and that .
Then the impulsive differential equation has a unique classical solution u which satisfies, for , Now we make the following assumption.
() There exists a function such that for any bounded , or a.e. and for any bounded subset .

Theorem 11. Let be satisfied and a mild solution of the problem (1). Assume that , , and that . Then u(0) gives rise to a classical solution of the problem (1).
If is a uniquely determined mild solution, then it gives rise to a unique classical solution.

Proof. We can define .
From Lemma 10, has a unique classical solution which satisfies, for , Now, is a mild solution of the problem (1), so that we get for , Thus we get which gives, by and an application of Gronwall’s inequality, .
This implies that gives rise to a classical solution and completes the proof.

6. Example

Let be a bounded domain in with smooth boundary , and . Consider the following nonlinear integrodifferential equation in : with nonlocal conditions or where . Set , Define nonlocal conditions or It is easy to see that generates a compact -semigroup in , and where and , , , .

For nonlocal conditions (44), , and is compact example of [18].

For nonlocal conditions (45), Hence, is Lipschitz. Furthermore, and 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 .

Conflict of Interests

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

Acknowledgment

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