- Research Article
- Open Access

# Antiperiodic Solutions for Liénard-Type Differential Equation with -Laplacian Operator

- Taiyong Chen
^{1}Email author, - Wenbin Liu
^{1}and - Cheng Yang
^{1}

**Received:**2 March 2010**Accepted:**19 August 2010**Published:**25 August 2010

## Abstract

The existence of antiperiodic solutions for Liénard-type and Duffing-type differential equations with -Laplacian operator has been studied by using degree theory. The results obtained improve and enrich some known works to some extent.

## Keywords

- Fourier Series
- Laplacian Equation
- Linear Normal Space
- Impulsive Differential Equation
- Fourier Series Expansion

## 1. Introduction

Antiperiodic problems arise naturally from the mathematical models of various of physical processes (see [1, 2]), and also appear in the study of partial differential equations and abstract differential equations (see [3–5]). For instance, electron beam focusing system in travelling-wave tube's theories is an antiperiodic problem (see [6]).

During the past twenty years, antiperiodic problems have been studied extensively by numerous scholars. For example, for first-order ordinary differential equations, a Massera's type criterion was presented in [7] and the validity of the monotone iterative technique was shown in [8]. Moreover, for higher-order ordinary differential equations, the existence of antiperiodic solutions was considered in [9–12]. Recently, existence results were extended to antiperiodic boundary value problems for impulsive differential equations (see [13]), and antiperiodic wavelets were discussed in [14].

using of the main assumption as follows:

where , . Obviously, the inverse operator of is , where is a constant such that .

Notice that, when , the nonlinear operator reduces to the linear operator .

In the past few decades, many important results relative to (1.3) with certain boundary conditions have been obtained. We refer the reader to [17–20] and the references cited therein. However, to the best of our knowledge, there exist relatively few results for the existence of antiperiodic solutions of (1.3). Moreover, it is well known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [21]). Thus, it is worthwhile to continue to investigate the existence of antiperiodic solutions for (1.3).

Note that, is also a -periodic solution of (1.4) or (1.5) if is a -antiperiodic solution of (1.4) or (1.5). Hence, from the arguments in this paper, we can also obtain the existence results of periodic solutions for above equations.

The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries. In Section 3, we establish some sufficient conditions for the existence of antiperiodic solutions of (1.4), basing on Leray-Schauder principle. Then, in Section 4, we obtain two existence results of antiperiodic solutions with symmetry for (1.5). Finally, in Section 5, some explicit examples are given to illustrate the main results. Our results are different from those of bibliographies listed above.

## 2. Preliminaries

and denotes norm in .

It is easy to prove that the mappings are completely continuous by using Arzelà-Ascoli theorem.

Next, we introduce a Wirtinger inequality (see [22]) and a continuation theorem (see [23, 24]) as follows.

Lemma 2.1 (Wirtinger inequality).

Lemma 2.2 (Continuation theorem).

Then equation has at least one solution in .

## 3. Antiperiodic Solutions for (1.4)

In this section, an existence result of antiperiodic solutions for (1.4) will be given.

Theorem 3.1.

Assume that

Then (1.4) has at least one antiperiodic solution.

Remark 3.2.

When , is equal to 1. It is easy to see that condition ( ) in [15] is stronger than condition ( ) of Theorem 3.1.

We begin with some lemmas below.

Lemma 3.3.

where is a positive constant only dependent of and .

Proof.

where .

The proof is complete.

Lemma 3.4.

where is a positive constant independent of .

Proof.

where . The proof is complete.

Now we give the proof of Theorem 3.1.

Proof of Theorem 3.1..

Obviously, the set is an open-bounded set in and zero element .

Obviously, the operator is completely continuous in and the fixed points of operator are the antiperiodic solutions of (1.4).

Consequently, the operator has at least one fixed point in by using Lemma 2.2. Namely, (1.4) has at least one antiperiodic solution. The proof is complete.

## 4. Antiperiodic Solutions with Symmetry for (1.5)

In this section, we will prove the existence of even antiperiodic solutions or odd antiperiodic solutions for (1.5).

Theorem 4.1.

Assume that

Proof.

Obviously, the operator is continuous.

*a prior bounds*in , that is, satisfies

where is a positive constant independent of . So that, our problem is reduced to construct one completely continuous operator in which sends into , such that the fixed points of operator in some open-bounded set are the even antiperiodic solutions of (1.5).

Obviously, the set is an open-bounded set in and zero element .

From the similar arguments in the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator in . Thus, (1.5) has at least one even antiperiodic solution. The proof is complete.

Theorem 4.2.

Assume that

Proof.

Obviously, the operator is continuous.

Based on the proof of Theorem 3.1, for the possible odd antiperiodic solutions of (4.3), there exists *a prior bounds* in
. Hence, our problem is reduced to construct one completely continuous operator
in
which sends
into
, such that the fixed points of operator
in some open-bounded set are the odd antiperiodic solutions of (1.5).

Obviously, the set is an open-bounded set in and zero element .

By a similar way as the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator in . So that, (1.5) has at least one odd antiperiodic solution. The proof is complete.

## 5. Examples

In this section, we will give some examples to illustrate our main results.

Example 5.1.

For , by direct calculation, we can get . Choosing , then (5.1) satisfies the condition of Theorem 3.1. So it has at least one antiperiodic solution.

Moreover, the conditions of Theorem 4.1 are also satisfied. Thus (5.1) has at least one even antiperiodic solution.

Example 5.2.

We choose . Obviously, (5.1) satisfies all the conditions of Theorem 4.2. Hence it has at least one odd antiperiodic solution.

## Declarations

### Acknowledgments

The authors would like to thank the referees very much for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China (10771212), the Fundamental Research Funds for the Central Universities, and the Science Foundation of China University of Mining and Technology (2008A037).

## Authors’ Affiliations

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