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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 293846, 9 pages
http://dx.doi.org/10.1155/2010/293846
Research Article

An Existence Result for Second-Order Impulsive Differential Equations with Nonlocal Conditions

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

Received 2 May 2010; Accepted 17 June 2010

Academic Editor: Guang Y. Zhang

Copyright © 2010 Meili Li and Haiqiang Liu. 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

The Leray-Schauder alternative is used to investigate the existence of solutions for second-order impulsive differential equations with nonlocal conditions in Banach spaces. The results improve some recent results.

1. Introduction

The theory of impulsive differential equations is emerging as an important area of investigation since it is a lot richer than the corresponding theory of nonimpulsive differential equations. Many evolutionary processes in nature are characterized by the fact that at certain moments in time an abrupt change of state is experienced. That is the reason for the rapid development of the theory of impulsive differential equations; see the monographs [1, 2].

This paper is concerned with the study on existence of second-order impulsive differential equations with nonlocal conditions of the form where the state takes values in Banach space with the norm ,   , and are given functions to be specified later.

The nonlocal condition is a generalization of the classical initial condition. The first results concerning the existence and uniqueness of mild solutions to Cauchy problems with nonlocal conditions were studied by Byszewski [3]. Recently, theorems about existence, uniqueness and continuous dependence of impulsive differential abstract evolution Cauchy problems with nonlocal conditions have been studied by Fu and Cao [4], Anguraj and Karthikeyan [5], Abada et al. [6], Li and Han [7], and in the references therein.

Up to now there have been very few papers in this direction dealing with the existence of solutions for second-order impulsive differential equations with nonlocal conditions. Our purpose here is to extend the results of first-order impulsive differential equations to second-order impulsive differential equations with nonlocal conditions.

Our main results are based on the following lemma [8].

Lemma 1.1 (Leray-Schauder alternative). Let be a convex subset of a normed linear space and assume that . Let be a completely continuous operator, and let

Then either is unbounded or has a fixed point.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Denote We define the following classes of functions:

, and there exist with ,

, and there exist with where and represent the restriction of and to , respectively and

Obviously, is a Banach space with the norm and is also a Banach space with the norm

Definition 2.1. A map is said to be an -Carathéodory if(i) is measurable for every ,(ii) is continuous for almost all ,(iii)for each , there exists such that for almost all

Definition 2.2. A function is said to be a solution of (1.1) if satisfies the equation a.e. on , the conditions , and .

Lemma 2.3. If satisfies then

Proof. Assume that ( here ). Then
Adding these together, we get that is, (2.3) holds.
Similarly, we have
Substitution of (2.3) in (2.7) gives that is, (2.4) holds.

We assume the following hypotheses: is an -Carathéodory map;, and there exist constants such that for every ; is a continuous function and there exists a constant such that there exists a function such that where is a continuous nondecreasing function with where for each bounded and the set is relatively compact in .

3. Main Results

Theorem 3.1. If the hypotheses are satisfied, then the second-order impulsive nonlocal initial value problem (1.1) has at least one solution on .

Proof. Consider the space with norm
We will now show that the operator defined by has a fixed point. This fixed point is then a solution of (1.1).
First we obtain a priori bounds for the following equation:
We have
Denoting by the right-hand side of the above inequality, we have But Thus we have Denoting by the right-hand side of the above inequality, we have Let Then
This implies that
This inequality implies that there is a constant such that Then and hence
Second, we must prove that the operator is a completely continuous operator.
Let for some We first show that maps into an equicontinuous family. Let and Then for we have and similarly
The right-hand sides are independent of and tend to zero as Thus maps into an equicontinuous family of functions. It is easy to see that the family is uniformly bounded. And from , we know that is compact. Then by Arzela-Ascoli theorem, we can conclude that the map is compact.
Next, we show that is continuous. Let with in Then there is an integer such that for all and so and By , for almost all , and since , we have by the dominated convergence theorem that Thus is continuous. This completes the proof that is completely continuous.
Finally, the set is bounded, as we proved in the first step. As a consequence of Lemma 1.1, we deduce that has a fixed point which is a solution of (1.1).

Acknowledgments

This work is supported by NNSF of China (no. 10971139) and Chinese Universities Scientific Fund (no. B08-1).

References

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