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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 724854, 8 pages
http://dx.doi.org/10.1155/2013/724854
Research Article

An Existence Result for Nonlocal Impulsive Second-Order Cauchy Problems with Finite Delay

School of Mathematics, Yunnan Normal University, Kunming 650092, China

Received 1 October 2012; Revised 7 December 2012; Accepted 9 December 2012

Academic Editor: Toka Diagana

Copyright © 2013 Fang Li and Huiwen Wang. 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.

Abstract

We deal with the existence of mild solutions of a class of nonlocal impulsive second-order functional differential equations with finite delay in a real Banach space . An existence result on the mild solution is obtained by using the theory of the measures of noncompactness. An example is presented.

1. Introduction

The Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (see, e.g., [15]).

The literature concerning second- and higher-order ordinary functional differential equations is very extensive. We only mention the works [1, 615], which are directly related to this work.

On the other hand, the impulsive conditions have advantages over traditional initial value problems because they can be used to model phenomena that cannot be modeled by traditional initial value problems, such as the dynamics of populations subject to abrupt changes (harvesting, diseases, etc.) (see [1627] and references therein). For this reason, the theory of impulsive differential equations has become an important area of investigation in recent years. Partial differential equations of first and second order with impulses have been studied by Rogovchenko [26], Liu [25], Cardinali and Rubbioni [19], Liang et al. [24], Henríquez and Vásquez [1], Hernández et al., [2123], Arthi and Balachandran [17], and so forth.

Moreover, we consider the nonlocal condition , where is a mapping from some space of functions so that it constitutes a nonlocal condition (see [24, 2830] and the references therein), where it is demonstrated that nonlocal conditions have better effects in applications than traditional initial value problems.

In this paper, we pay our attention to the investigation of the existence of mild solutions to the following impulsive second-order functional differential equations with finite delay in a real Banach space : where is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators on . , are given functions to be specified later. , where denotes the space of all continuous functions from to .

The impulsive moments are given such that , are appropriate functions, represents the jump of a function at , which is defined by , where and are, respectively, the right and the left limits of at .

For any continuous function defined on the interval and any , we denote by the element of defined by for .

In this paper, motivated by above works, we study (1)–(4) in and obtain the existence theorem based on theory on measures of noncompactness without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the cosine family of bounded linear operators generated by is compact.

2. Preliminaries

Throughout this paper, we set , a compact interval in . We denote by a Banach space with norm , by the Banach space of all linear and bounded operators on . We abbreviate with , for any .

Let

It is easy to check that is a Banach space with the norm

We let , .

For , we denote by the set .

A family in is called a cosine function on if (i) is the identity operator in ; (ii) for all ; (iii)The map is strongly continuous for each . The associated sine function is the family of operators defined by

One can define the infinitesimal generator of by

In this paper, we assume there exist positive constants and such that The following properties are well known [6, 7, 11, 12]:

For more details on strongly continuous cosine and sine families, we refer the reader to [6, 7, 11, 12].

Next, we recall that the Hausdorff measure of noncompactness on each bounded subset of Banach space is defined by

This measure of noncompactness satisfies some basic properties as follows.

Lemma 1 (see [31]). Let be a real Banach space, and let be bounded. Then (1) if and only if is precompact; (2), where and mean the closure and convex hull of , respectively; (3) if ; (4); (5), where ; (6), for any ; (7)let be a Banach space and Lipschitz continuous with constant . Then for all being bounded.

Proposition 2 (see [32], Page 125). Let be a bounded set for a real Banach space . Then, for every there exists a sequence in such that

In the sequel, we make use of the following formulation of Theorem 4.2.2 of [33] obtained by using Theorem 2 of [34].

Proposition 3. Let be a sequence in such that there exist with the properties:(i); (ii). Then, for every , we have where is from (10).

A continuous map is said to be a -contraction if there exists a positive constant such that for any bounded closed subset .

Theorem 4 (see [31] (Darbo-Sadovskii)). If is bounded closed and convex, the continuous map is a -contraction, then the map has at least one fixed point in .

Definition 5. A function is called a mild solution of system (1)–(4) if , and

Remark 6. A mild solution of (1)–(4) satisfies (2) and (4). However, a mild solution may be not differentiable at zero.

3. Existence Result and Proof

In this section, we study the existence of mild solutions for the system (1)–(4).

Let stand for the space endowed with norm

We will require the following assumptions.

(H1)     satisfies is measurable for all and is continuous for a.e. , and there exists a function such that for almost all ;

   there exists a function such that for any bounded sets , (H2) are compact operators and there exist positive constants such that (H3) is a compact operator and there exists a constant such that where .  (H4) There exists such that .

Theorem 7. Assume that (H1)–(H4) are satisfied, then there exists a mild solution of (1)–(4) on provided that .

Proof. Define the operator in the following way: It is clear that the operator is well defined, and the fixed point of is the mild solution of problems (1)–(4).
The operator can be written in the form , where the operators are defined as follows:
Obviously, under the assumptions of , is continuous. For , we can prove that is continuous.
Indeed, let be a sequence such that in as . Since satisfies (H1)(i), for almost every , we get
Noting that in , we can see that there exists such that for sufficiently large. Therefore, we have It follows from the Lebesgue’s dominated convergence theorem that Moreover, noting that (H2), we obtain that This shows that is continuous. Therefore, is continuous.
Let us introduce in the space the equivalent norm defined as where is a constant chosen so that Noting that for any , we have so, we can take the appropriate to satisfy (29).
Consider the set where is a constant chosen so that where and .
Now, if , , then For , , we have then therefore, It results that
Let , we obtain Hence for some positive number , .
Using the strong continuity of and the compactness condition on the operators , for , there exists such that when . If and such that , then
For , by the hypothesis (H1)(i) and (40), we get As and , the right-hand side of the inequality above tends to zero independent of , so maps bounded sets into equicontinuous sets.
For bounded set , we consider the map where is the Hausdorff measure of noncompactness on the Banach space and is defined in (7).
Furthermore, we define the Hausdorff measure of noncompactness on as follows:
For every bounded subset , by applying Proposition 2, for any there exists a sequence such that noting that the definition of , we have Then, noting the equicontinuity of ,  , we can apply Lemmas 2.1 and 2.2 of [35] to obtain Then from (45) and (46), we have
For every bounded subset , we have moreover, by applying Proposition 2, for any there exists a sequence such that Combining with (48) and (49), we have Using the induction of (45)–(47) above, we can see Thus, from (51), (H2) and Proposition 3 and (3) in Lemma 1, we can see where .
Noting that Thus, by (52), we see
Since is arbitrary, we can obtain
Combining with (H3), we have hence is a -contraction on . According to Theorem 4, the operator has at least one fixed point in . This completes the proof.

Next, we establish a condition that guarantee that a mild solution satisfies (3).

Proposition 8. Assume that the hypotheses of Theorem 7 are fulfilled and that . If is a mild solution of (1)–(4), then condition (3) holds.

Proof. Clearly, as . Moreover, noting that and (11), we have is of class . Therefore, we can see that which shows the assertion.

4. Application

In this section, we consider an application of the theory developed in Section 3 to the study of an impulsive partial differential equation with unbounded delay.

Example 9. , is the map defined by with domain .
We consider the following integrodifferential model: where , , , , and . and satisfy the following assumptions. (1)The function is a continuous function and . (2)The function is measurable and there exists a constant such that . (3)For every , the function , is measurable and there exists a constant such that
To treat the above problem, we define is the infinitesimal generator of a strongly continuous cosine function on . Moreover, has a discrete spectrum, the eigenvalues are , with the corresponding normalized eigenvectors ; the set is an orthonormal basis of and the following properties hold. (a)If , then . (b)For each , and . Consequently, for , and is compact for every .
For and , we set Then the above equation (59) can be reformulated as the abstract (1)–(4).
For , we can see where .
For any ,  ,
Therefore, for any bounded sets , , we have where .
For ,
Suppose further that there exists a constant such that and , then (59) has at least a mild solution by Theorem 7.

Acknowledgments

This work was partly supported by the NSF of China (11201413), the NSF of Yunnan Province (2009ZC054M), the Educational Commission of Yunnan Province (2012Z010) and the Foundation of Key Program of Yunnan Normal University.

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