# On the Solvability of Second-Order Impulsive Differential Equations with Antiperiodic Boundary Value Conditions

- Yepeng Xing
^{1}Email author and - Valery Romanovski
^{2}

**2008**:864297

**DOI: **10.1155/2008/864297

© Y. Xing and V. Romanovski. 2008

**Received: **3 July 2008

**Accepted: **10 November 2008

**Published: **26 November 2008

## Abstract

We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.

## 1. Introduction and Preliminaries

Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known that many biological phenomena involving threshold, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems do exhibit impulse effects. The branch of modern, applied analysis known as "impulsive" differential equations provides a natural framework to mathematically describe the aforementioned jumping processes. The reader is referred to monographs [1–4] and references therein for some nice examples and applications to the above areas.

where and is continuous on , are continuous functions.

In [4–12], the authors studied the existence of antiperiodic solutions for first-order, second-order, or high-order differential equations without impulses, and in [3, 13–16] the authors were concerned with the antiperiodic solutions of first-order impulsive differential equations. Also we should mention the work by Cabada et al. in [17] which is concerned with a certain th order linear differential equation with constant impulses at fixed times and nonhomogeneous periodic boundary conditions. So far, to the best of our knowledge, this is the first work to deal with the antiperiodic solutions to second-order differential equations with nonconstant impulses. Our method to prove the existence of antiperiodic solutions is based on the works in [13, 18, 19]. We should point out that it is Christopher C. Tisdell who started with this method.

The article is organized as follows. In Section 2, we present the expression of Green's functions of related linear operator in the space of piecewise continuous functions. Section 3 contains the main results of the paper and is devoted to the existence of solutions to (1.1). There, differential inequalities are developed and applied to prove the existence of at least one solution to (1.1). In Section 4, a couple of examples are given to illustrate how the main results work.

with the norm where is the usual Euclidean norm and will be the Euclidean inner product.

with the norm

The following fixed point theorem is our main tool to prove the existence of at least one solution to (1.1).

Schaefer's Fixed Point Theorem (19)

- (i)
the operator equation has a solution for , or

- (ii)
the set is unbounded.

## 2. Expression of Green's Function

In this part, we present the expression of Green's functions for second order impulsive equations with antiperiodic conditions.

Lemma 2.1.

Proof.

Now we are in position to show the expression of for To do that, we need to compute in (2.10). In what follows we present the expression of for step by step and then obtain the general form of for .

we see that is the solution of (2.2).

Now, we prove is a solution of (2.1). Then the proof is completed.

Corollary 2.2.

We now give Green's function of (2.1) for .

Lemma 2.3.

Recall that a mapping between Banach spaces is compact if it is continuous and carries bounded sets into relatively compact sets.

Lemma 2.4.

where and are as given in Lemma 2.1. Then is a compact map.

Proof.

Noting the continuity of and , this follows in a standard step-by-step process and so it is omitted.

## 3. Main Results

In this section, we prove the existence results for (1.1) in presence of Schaefer's fixed-point theorem.

Theorem 3.1.

where is the Euclidean inner product, . Then (1.1) has at least one solution.

Proof.

*G*) and (

*H*) in Lemma 2.1. By Lemma 2.4, is a compact mapping. Also, it follows from Lemma 2.1 that is a fixed point of if and only if satisfies

It is equivalent to say that satisfies

Now we have shown that any possible solution of (3.6) is bounded by which is independent of . By Scheafer's fixed theorem we know that has at least one fixed point. Therefore, the proof is completed.

Suppose both and in Theorem 3.1. We obtain the following theorem.

Theorem 3.2.

where is the Euclidean inner product, , then (1.1) has at least one solution.

Proof.

where .

Then the proof is completed.

Similarly, we can prove the following existence result for in Theorem 3.2.

Theorem 3.3.

where is the Euclidean inner product, , then (1.1) has at least one solution.

## 4. Examples

In this part, we show how our main theorems work by a couple of examples.

Example 4.1.

where have at least one solution.

Proof.

Moreover, , . Then the conclusion follows from Theorem 3.1.

Example 4.2.

We claim that (4.5) has at least one solution.

Proof.

Moreover, , . Then the conclusion follows from Theorem 3.2.

## Declarations

### Acknowledgments

This research is supported by Ad Futura Scientific and Educational Foundation of the Republic of Slovenia, the Ministry of Higher Education, Science and Technology of the Republic of Slovenia; the Nova Kreditna Banka Maribor; TELEKOM Slovenije; National Natural Science Foundation of China (10671127); National Natural Science Foundation of Shanghai (08ZR1416000); and Foundation of Science and Technology Commission of Shanghai Municipality (06XD14034).

## Authors’ Affiliations

## References

- Bainov D, Simeonov PS:
*Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics*.*Volume 66*. Longman Scientific & Technical, Harlow, UK; 1993:x+228.MATHGoogle 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 - 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 - Park JY, Ha TG:
**Existence of antiperiodic solutions for hemivariational inequalities.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(4):747-767. 10.1016/j.na.2006.11.032MathSciNetView 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 - Aftabizadeh AR, Aizicovici S, Pavel NH:
**Anti-periodic boundary value problems for higher order differential equations in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*1992,**18**(3):253-267. 10.1016/0362-546X(92)90063-KMathSciNetView ArticleMATHGoogle Scholar - Aizicovici S, McKibben M, Reich S:
**Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities.***Nonlinear Analysis: Theory, Methods & Applications*2001,**43**(2):233-251. 10.1016/S0362-546X(99)00192-3MathSciNetView ArticleMATHGoogle Scholar - Aizicovici S, Pavel NH:
**Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space.***Journal of Functional Analysis*1991,**99**(2):387-408. 10.1016/0022-1236(91)90046-8MathSciNetView ArticleMATHGoogle Scholar - Cabada A, Vivero DR:
**Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations.***Advances in Difference Equations*2004,**2004**(4):291-310. 10.1155/S1687183904310022MathSciNetView ArticleMATHGoogle 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 - Wang K, Li Y:
**A note on existence of (anti-)periodic and heteroclinic solutions for a class of second-order odes.***Nonlinear Analysis: Theory, Methods & Applications.*2009,**70**(4):1711-1724. 10.1016/j.na.2008.02.054MathSciNetView 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 - Nieto JJ, Tisdell CC:
**Existence and uniqueness of solutions to first-order systems of nonlinear impulsive boundary-value problems with sub-, super-linear or linear growth.***Electronic Journal of Differential Equations*2007,**2007**(105):1-14.MathSciNetMATHGoogle 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 - 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 - 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 - Cabada A, Liz E, Lois S:
**Green's function and maximum principle for higher order ordinary differential equations with impulses.***The Rocky Mountain Journal of Mathematics*2000,**30**(2):435-446. 10.1216/rmjm/1022009274MathSciNetView 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 - Rudd M, Tisdell CC:
**On the solvability of two-point, second-order boundary value problems.***Applied Mathematics Letters*2007,**20**(7):824-828. 10.1016/j.aml.2006.08.028MathSciNetView ArticleMATHGoogle Scholar

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