Skip to main content
  • Research Article
  • Open access
  • Published:

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

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.

References

  1. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Singapore; 1989:xii+273.

    Book  Google Scholar 

  2. 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 

  3. 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.

    Book  MATH  Google Scholar 

  4. Liu X (Ed): Advances in impulsive differential equations In Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2002,9(3):313-462.

  5. 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 

  6. Li J, Shen J: Periodic boundary value problems for second order differential equations with impulses. Nonlinear Studies 2005,12(4):391-400.

    MathSciNet  MATH  Google Scholar 

  7. 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)00889-6

    Article  MathSciNet  MATH  Google Scholar 

  8. 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.2006.09.021

    Article  MathSciNet  MATH  Google Scholar 

  9. 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.

    Book  MATH  Google Scholar 

  10. 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.

    MathSciNet  MATH  Google Scholar 

  11. 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/S0219525908001519

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)00186-3

    Article  MathSciNet  MATH  Google Scholar 

  13. 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.2007.10.022

    Article  MathSciNet  MATH  Google Scholar 

  14. 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.2007.05.004

    Article  MathSciNet  MATH  Google Scholar 

  15. 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.2004.08.034

    Article  MathSciNet  MATH  Google Scholar 

  16. 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.2003.12.011

    Article  MathSciNet  MATH  Google Scholar 

  17. 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.2007.01.011

    Article  MathSciNet  MATH  Google Scholar 

  18. 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.2007.10.019

    Article  MathSciNet  MATH  Google Scholar 

  19. 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.2005.04.005

    Article  MathSciNet  MATH  Google Scholar 

  20. 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-E

    Article  MathSciNet  MATH  Google Scholar 

  21. 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.

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)90134-4

    Article  MathSciNet  MATH  Google Scholar 

  23. Yin Y: Remarks on first order differential equations with anti-periodic boundary conditions. Nonlinear Times and Digest 1995,2(1):83-94.

    MathSciNet  MATH  Google Scholar 

  24. Yin Y: Monotone iterative technique and quasilinearization for some anti-periodic problems. Nonlinear World 1996,3(2):253-266.

    MathSciNet  MATH  Google Scholar 

  25. 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.2006.07.087

    Article  MathSciNet  MATH  Google Scholar 

  26. 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.2007.09.018

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)00678-7

    Article  MathSciNet  MATH  Google Scholar 

  28. 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.2006.12.006

    Article  MathSciNet  MATH  Google Scholar 

  29. Ou C: Anti-periodic solutions for high-order Hopfield neural networks. Computers & Mathematics with Applications 2008,56(7):1838-1844. 10.1016/j.camwa.2008.04.029

    Article  MathSciNet  MATH  Google Scholar 

  30. Wu R: An anti-periodic LaSalle oscillation theorem. Applied Mathematics Letters 2008,21(9):928-933. 10.1016/j.aml.2007.10.004

    Article  MathSciNet  MATH  Google Scholar 

  31. Li Y, Huang L: Anti-periodic solutions for a class of Liénard-type systems with continuously distributed delays. Nonlinear Analysis: Real World Applications. In press

  32. 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.2007.12.013

    Article  MathSciNet  MATH  Google Scholar 

  33. Ahn C, Rim C: Boundary flows in general coset theories. Journal of Physics A 1999,32(13):2509-2525. 10.1088/0305-4470/32/13/004

    Article  MathSciNet  MATH  Google Scholar 

  34. 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-O

    Article  MathSciNet  MATH  Google Scholar 

  35. 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-X

    Article  MathSciNet  MATH  Google Scholar 

  36. 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.2004.08.010

    Article  MathSciNet  MATH  Google Scholar 

  37. 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.2007.12.031

    Article  MathSciNet  MATH  Google Scholar 

  38. Lloyd NG: Degree Theory. Cambridge University Press, Cambridge, UK; 1978:vi+172. Cambridge Tracts in Mathematics, no. 7

    Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chuanzhi Bai.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Bai, C. Antiperiodic Boundary Value Problems for Second-Order Impulsive Ordinary Differential Equations. Bound Value Probl 2008, 585378 (2009). https://doi.org/10.1155/2008/585378

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2008/585378

Keywords