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

- Chuanzhi Bai
^{1}Email author

**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.

## Keywords

## 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 [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 [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.

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)).

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 .

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

Lemma 2.1.

Proof.

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.

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).

Lemma 2.3.

Proof.

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

hold.

Remark 3.1.

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

Theorem 3.2.

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

Proof.

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

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

Theorem 3.3.

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

Proof.

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.

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

## References

- Lakshmikantham V, Bainov DD, Simeonov PS:
*Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics*.*Volume 6*. World Scientific, Singapore; 1989:xii+273.View ArticleGoogle Scholar - Samoilenko AM, Perestyuk NA:
*Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*.*Volume 14*. World Scientific, Singapore; 1995:x+462.Google Scholar - Zavalishchin ST, Sesekin AN:
*Dynamic Impulse Systems: Theory and Application, Mathematics and Its Applications*.*Volume 394*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xii+256.View ArticleMATHGoogle Scholar - Liu X (Ed): Advances in impulsive differential equations In Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2002,9(3):313-462.Google Scholar
- Bai C, Yang D:
**Existence of solutions for second-order nonlinear impulsive differential equations with periodic boundary value conditions.***Boundary Value Problems*2007,**2007:**-13.Google Scholar - Li J, Shen J:
**Periodic boundary value problems for second order differential equations with impulses.***Nonlinear Studies*2005,**12**(4):391-400.MathSciNetMATHGoogle Scholar - Nieto JJ:
**Periodic boundary value problems for first-order impulsive ordinary differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2002,**51**(7):1223-1232. 10.1016/S0362-546X(01)00889-6MathSciNetView ArticleMATHGoogle Scholar - Chen J, Tisdell CC, Yuan R:
**On the solvability of periodic boundary value problems with impulse.***Journal of Mathematical Analysis and Applications*2007,**331**(2):902-912. 10.1016/j.jmaa.2006.09.021MathSciNetView ArticleMATHGoogle Scholar - Benchohra M, Henderson J, Ntouyas S:
*Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications*.*Volume 2*. Hindawi, New York, NY, USA; 2006:xiv+366.View ArticleMATHGoogle Scholar - Jankowski T, Nieto JJ:
**Boundary value problems for first-order impulsive ordinary differential equations with delay arguments.***Indian Journal of Pure and Applied Mathematics*2007,**38**(3):203-211.MathSciNetMATHGoogle Scholar - Zeng G, Wang F, Nieto JJ:
**Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response.***Advances in Complex Systems*2008,**11**(1):77-97. 10.1142/S0219525908001519MathSciNetView ArticleMATHGoogle Scholar - Akhmetov MU, Zafer A, Sejilova RD:
**The control of boundary value problems for quasilinear impulsive integro-differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2002,**48**(2):271-286. 10.1016/S0362-546X(00)00186-3MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, O'Regan D:
**Variational approach to impulsive differential equations.***Nonlinear Analysis: Real World Applications*2009,**10**(2):680-690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleMATHGoogle Scholar - Zhang H, Chen L, Nieto JJ:
**A delayed epidemic model with stage-structure and pulses for pest management strategy.***Nonlinear Analysis: Real World Applications*2008,**9**(4):1714-1726. 10.1016/j.nonrwa.2007.05.004MathSciNetView ArticleMATHGoogle Scholar - Qian D, Li X:
**Periodic solutions for ordinary differential equations with sublinear impulsive effects.***Journal of Mathematical Analysis and Applications*2005,**303**(1):288-303. 10.1016/j.jmaa.2004.08.034MathSciNetView ArticleMATHGoogle Scholar - Yan J, Zhao A, Nieto JJ:
**Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems.***Mathematical and Computer Modelling*2004,**40**(5-6):509-518. 10.1016/j.mcm.2003.12.011MathSciNetView ArticleMATHGoogle Scholar - Mohamad S, Gopalsamy K, Akça H:
**Exponential stability of artificial neural networks with distributed delays and large impulses.***Nonlinear Analysis: Real World Applications*2008,**9**(3):872-888. 10.1016/j.nonrwa.2007.01.011MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Boundary value problems for a class of impulsive functional equations.***Computers & Mathematics with Applications*2008,**55**(12):2715-2731. 10.1016/j.camwa.2007.10.019MathSciNetView ArticleMATHGoogle Scholar - Li J, Nieto JJ, Shen J:
**Impulsive periodic boundary value problems of first-order differential equations.***Journal of Mathematical Analysis and Applications*2007,**325**(1):226-236. 10.1016/j.jmaa.2005.04.005MathSciNetView ArticleMATHGoogle Scholar - Aftabizadeh AR, Aizicovici S, Pavel NH:
**On a class of second-order anti-periodic boundary value problems.***Journal of Mathematical Analysis and Applications*1992,**171**(2):301-320. 10.1016/0022-247X(92)90345-EMathSciNetView ArticleMATHGoogle Scholar - Franco D, Nieto JJ, O'Regan D:
**Anti-periodic boundary value problem for nonlinear first order ordinary differential equations.***Mathematical Inequalities & Applications*2003,**6**(3):477-485.MathSciNetView ArticleMATHGoogle Scholar - Jankowski T:
**Ordinary differential equations with nonlinear boundary conditions of antiperiodic type.***Computers & Mathematics with Applications*2004,**47**(8-9):1419-1428. 10.1016/S0898-1221(04)90134-4MathSciNetView ArticleMATHGoogle Scholar - Yin Y:
**Remarks on first order differential equations with anti-periodic boundary conditions.***Nonlinear Times and Digest*1995,**2**(1):83-94.MathSciNetMATHGoogle Scholar - Yin Y:
**Monotone iterative technique and quasilinearization for some anti-periodic problems.***Nonlinear World*1996,**3**(2):253-266.MathSciNetMATHGoogle Scholar - Ding W, Xing Y, Han M:
**Anti-periodic boundary value problems for first order impulsive functional differential equations.***Applied Mathematics and Computation*2007,**186**(1):45-53. 10.1016/j.amc.2006.07.087MathSciNetView ArticleMATHGoogle Scholar - Ahmad B, Nieto JJ:
**Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(10):3291-3298. 10.1016/j.na.2007.09.018MathSciNetView ArticleMATHGoogle Scholar - Franco D, Nieto JJ, O'Regan D:
**Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions.***Applied Mathematics and Computation*2004,**153**(3):793-802. 10.1016/S0096-3003(03)00678-7MathSciNetView ArticleMATHGoogle Scholar - Chen Y, Nieto JJ, O'Regan D:
**Anti-periodic solutions for fully nonlinear first-order differential equations.***Mathematical and Computer Modelling*2007,**46**(9-10):1183-1190. 10.1016/j.mcm.2006.12.006MathSciNetView ArticleMATHGoogle Scholar - Ou C:
**Anti-periodic solutions for high-order Hopfield neural networks.***Computers & Mathematics with Applications*2008,**56**(7):1838-1844. 10.1016/j.camwa.2008.04.029MathSciNetView ArticleMATHGoogle Scholar - Wu R:
**An anti-periodic LaSalle oscillation theorem.***Applied Mathematics Letters*2008,**21**(9):928-933. 10.1016/j.aml.2007.10.004MathSciNetView ArticleMATHGoogle Scholar - Li Y, Huang L: Anti-periodic solutions for a class of Liénard-type systems with continuously distributed delays. Nonlinear Analysis: Real World Applications. In pressGoogle Scholar
- Wang K:
**A new existence result for nonlinear first-order anti-periodic boundary value problems.***Applied Mathematics Letters*2008,**21**(11):1149-1154. 10.1016/j.aml.2007.12.013MathSciNetView ArticleMATHGoogle Scholar - Ahn C, Rim C:
**Boundary flows in general coset theories.***Journal of Physics A*1999,**32**(13):2509-2525. 10.1088/0305-4470/32/13/004MathSciNetView ArticleMATHGoogle Scholar - Abdurrahman A, Anton F, Bordes J:
**Half-string oscillator approach to string field theory (ghost sector. I).***Nuclear Physics B*1993,**397**(1-2):260-282. 10.1016/0550-3213(93)90344-OMathSciNetView ArticleMATHGoogle Scholar - Franco D, Nieto JJ:
**First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions.***Nonlinear Analysis: Theory, Methods & Applications*2000,**42**(2):163-173. 10.1016/S0362-546X(98)00337-XMathSciNetView ArticleMATHGoogle Scholar - Luo Z, Shen J, Nieto JJ:
**Antiperiodic boundary value problem for first-order impulsive ordinary differential equations.***Computers & Mathematics with Applications*2005,**49**(2-3):253-261. 10.1016/j.camwa.2004.08.010MathSciNetView ArticleMATHGoogle Scholar - Wang W, Shen J:
**Existence of solutions for anti-periodic boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(2):598-605. 10.1016/j.na.2007.12.031MathSciNetView ArticleMATHGoogle Scholar - Lloyd NG:
*Degree Theory*. Cambridge University Press, Cambridge, UK; 1978:vi+172. Cambridge Tracts in Mathematics, no. 7Google Scholar

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