Open Access

Antiperiodic Boundary Value Problems for Second-Order Impulsive Ordinary Differential Equations

Boundary Value Problems20092008:585378

DOI: 10.1155/2008/585378

Received: 23 July 2008

Accepted: 23 November 2008

Published: 15 January 2009

Abstract

We consider a second-order ordinary differential equation with antiperiodic boundary conditions and impulses. By using Schaefer's fixed-point theorem, some existence results are obtained.

1. Introduction

Impulsive differential equations, which arise in biology, physics, population dynamics, economics, and so forth, are a basic tool to study evolution processes that are subjected to abrupt in their states (see [14]). Many literatures have been published about existence of solutions for first-order and second-order impulsive ordinary differential equations with boundary conditions [519], which are important for complementing the theory of impulsive equations. In recent years, the solvability of the antiperiodic boundary value problems of first-order and second-order differential equations were studied by many authors, for example, we refer to [2032] and the references therein. It should be noted that antiperiodic boundary value problems appear in physics in a variety of situations [33, 34]. Recently, the existence results were extended to antiperiodic boundary value problems for first-order impulsive differential equations [35, 36]. Very recently, Wang and Shen [37] investigated the antiperiodic boundary value problem for a class of second-order differential equations by using Schauder's fixed point theorem and the lower and upper solutions method.

Inspired by [3537], in this paper, we investigate the antiperiodic boundary value problem for second-order impulsive nonlinear differential equations of the form
(1.1)

where , , is continuous on , , exist, ; , ; .

To the best of the authors knowledge, no one has studied the existence of solutions for impulsive antiperiodic boundary value problem (1). The following Schaefer's fixed-point theorem is fundamental in the proof of our main results.

Lemma 1.1 (see [38] (Schaefer)).

Let be a normed linear space with a compact operator. If the set
(1.2)

is bounded, then has at least one fixed point.

The paper is formulated as follows. In Section 2, some definitions and lemmas are given. In Section 3, we obtain two new existence theorems by using Schaefer's fixed point theorem. In Section 4, an illustrative example is given to demonstrate the effectiveness of the obtained results.

2. Preliminaries

In order to define the concept of solution for (1), we introduce the following spaces of functions:

is continuous for any , , exist, and ,

is continuously differentiable for any , , exist, and .

and are Banach space with the norms
(2.1)

A solution to the impulsive BVP (1) is a function that satisfies (1) for each .

Consider the following impulsive BVP with
(2.2)

where .

For convenience, we set .

Lemma 2.1.

is a solution of (2.2) if and only if is a solution of the impulsive integral equation
(2.3)
where
(2.4)

Proof.

If is a solution of (2.2), setting
(2.5)
then, by the first equation of (2.2) we have
(2.6)
Multiplying (2.6) by and integrating on and ( ), respectively, we get
(2.7)
So
(2.8)
In the same way, we can obtain that
(2.9)
where . Integrating (2.5), we have
(2.10)
By (2.9), we get
(2.11)
Substituting (2.11) into (2.10), we obtain
(2.12)
(2.13)
In view of and , we have
(2.14)

Substituting (2.14) into (2.12), by routine calculation, we can get (2.3).

Conversely, if is a solution of (2.3), then direct differentiation of (2.3) gives , . Moreover, we obtain , , and . Hence, is a solution of (2.2).

Remark 2.2.

We call above the Green function for the following homogeneous BVP:
(2.15)
Define a mapping by
(2.16)

In view of Lemma 2.1, we easily see that is a fixed point of operator if and only if is a solution to the impulsive boundary value problem (1).

It is easy to check that
(2.17)

Lemma 2.3.

If and , then
(2.18)

Proof.

Since , we have
(2.19)
Set , we obtain from that
(2.20)
Substituting (2.20) into (2.19), we get
(2.21)

The proof is complete.

3. Main results

In this section, we study the existence of solutions for BVP (1). For this purpose we assume that there exist constants , functions , and nonnegative constants ( ) such that

(H1) , and

(H2) , ,

hold.

Remark 3.1.

means that the nonlinearity growths at most linearly in , implies that the impulses are (at most) linear.

For convenience, let
(3.1)

Theorem 3.2.

Suppose that conditions and are satisfied. Further assume that
(3.2)

holds, where and as in (3.1). Then, BVP (1) has at least one solution.

Proof.

It is easy to check by Arzela-Ascoli theorem that the operator is completely continuous. Assume that is a solution of the equation
(3.3)
Then,
(3.4)
(3.5)
Integrating (3.4) from 0 to , we get that
(3.6)
In view of , we obtain by (3.6) that
(3.7)
Integrating (3.4) from 0 to , we obtain that
(3.8)
From (3.7) and (3.8), we have
(3.9)
that is,
(3.10)
Thus,
(3.11)
where are as in (3.1). Integrating (3.5) from 0 to , we get that
(3.12)
In view of and , we have
(3.13)

Substituting (3.13) into (3.12), we obtain by and (3.11) that

(3.14)
Thus,
(3.15)
where
(3.16)
By Lemma 2.3 and (3.15), we have
(3.17)
It follows from the above inequality and (3.2) that there exists such that . Hence, we get by (3.11) that
(3.18)

Thus, . It follows from Lemma 1.1 that BVP (1) has at least one solution. The proof is complete.

Theorem 3.3.

Assume that holds. Suppose that there exist a continuous and nondecreasing function and a nonnegative function with
(3.19)
Moreover suppose that
(3.20)
holds, where
(3.21)

Then, BVP (1) has at least one solution.

Proof.

From (3.20), there exist and such that
(3.22)
Thus, there exists such that
(3.23)
Assume that is a solution of the equation
(3.24)
Then, we have by (3.19), (2.17), and (3.23) that
(3.25)
Thus, we have
(3.26)
that is,
(3.27)
which implies that there exists such that . By (3.7), (3.8), and (3.23), we get
(3.28)
which implies that
(3.29)

Hence, . It follows from Lemma 1.1 that BVP (1) has at least one solution. The proof is complete.

4. Example

In this section, we give an example to illustrate the effectiveness of our results.

Example 4.1.

Consider the problem
(4.1)
Let , , , , . It is easy to show that
(4.2)
where , , . And
(4.3)
Thus, and hold. Obviously, , , , , and . Let , we have
(4.4)
Therefore,
(4.5)

which implies that (3.2) holds. So, all the conditions of Theorem 3.2 are satisfied. By Theorem 3.2, antiperiod boundary value problem (4.1) has at least one solution.

Declarations

Acknowledgments

The author would like to thank the referees for their valuable suggestions and comments. This project is supported by the National Natural Science Foundation of China (10771212) and the Natural Science Foundation of Jiangsu Education Office (06KJB110010).

Authors’ Affiliations

(1)
Department of Mathematics, Huaiyin Teachers College

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© Chuanzhi Bai. 2008

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