Antiperiodic Boundary Value Problems for Second-Order Impulsive Ordinary Differential Equations
© Chuanzhi Bai. 2008
Received: 23 July 2008
Accepted: 23 November 2008
Published: 15 January 2009
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.
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 [1–4]). Many literatures have been published about existence of solutions for first-order and second-order impulsive ordinary differential equations with boundary conditions [5–19], 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 [20–32] 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  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.
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  (Schaefer)).
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.
In order to define the concept of solution for (1), we introduce the following spaces of functions:
Substituting (2.14) into (2.12), by routine calculation, we can get (2.3).
The proof is complete.
3. Main results
Then, BVP (1) has at least one solution.
In this section, we give an example to illustrate the effectiveness of our results.
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.
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).
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