- Research Article
- Open Access

- Fengjuan Cao
^{1}, - Zhenlai Han
^{1, 2}Email author, - Ping Zhao
^{2}and - Shurong Sun
^{1, 3}

**2010**:235749

https://doi.org/10.1155/2010/235749

© Fengjuan Cao et al. 2010

**Received: **31 December 2009

**Accepted: **23 February 2010

**Published: **28 February 2010

## Abstract

By topological degree theory and some analysis skills, we consider a class of generalized Liénard type -Laplacian equations. Upon some suitable assumptions, the existence and uniqueness of periodic solutions for the generalized Liénard type -Laplacian differential equations are obtained. It is significant that the nonlinear term contains two variables.

## Keywords

- Periodic Solution
- Periodic Problem
- Homotopic Equation
- Rayleigh Equation
- Classical Inequality

## 1. Introduction

As it is well known, the existence of periodic and almost periodic solutions is the most attracting topics in the qualitative theory of differential equations due to their vast applications in physics, mathematical biology, control theory, and others. More general equations and systems involving periodic boundary conditions have also been considered. Especially, the existence of periodic solutions for the Duffing equation, Rayleigh equation, and Liénard type equation, which are derived from many fields, such as fluid mechanics and nonlinear elastic mechanics, has received a lot of attention.

Many experts and scholars, such as Manásevich, Mawhin, Gaines, Cheung, Ren, Ge, Lu, and Yu, have contributed a series of existence results to the periodicity theory of differential equations. Fixed point theory, Mawhin's continuation theorem, upper and lower solutions method, and coincidence degree theory are the common tools to study the periodicity theory of differential equations. Among these approaches, the Mawhin's continuation theorem seems to be a very powerful tool to deal with these problems.

Some contributions on periodic solutions to differential equations have been made in [1–13]. Recently, periodic problems involving the scalar -Laplacian were studied by many authors. We mention the works by Manásevich and Mawhin [3] and Cheung and Ren [4, 8, 10].

where the function
is quite general and satisfies some monotonicity conditions which ensure that
is homeomorphism onto
Applying Leray-Schauder degree theory, the authors brought us the widely used Manásevich-Mawhin continuation theorem. When
is the so-called one-dimensional *p*-Laplacian operator given by

and two results (Theorems 3.1 and 3.2 ) on the existence of periodic solutions were obtained.

Ge and Ren [5] promoted Mawhin's continuation theorem to the case which involved the quasilinear operator successfully; this also prepared conditions for using Mawhin's continuation theorem to solve nonlinear boundary value problem.

by using topological degree theory, and one sufficient condition for the existence and uniqueness of -periodic solutions of this equation was established.

where , is given by for , , , and -periodic in the first variable, where is a given constant, and

The paper is organized as follows. In Section 2, we give the definition of norm in Banach space and the main lemma. In Section 3, combining Lemma 2.1 with some analysis skills, two sufficient conditions about the existence of solutions for (1.4) are obtained. The nonlinear terms and contain two variables in this paper, which is seldom considered in the other papers, and the results are new.

## 2. Preliminary Results

Clearly, is a Banach space endowed with such norm.

where is a continuous function and -periodic in the first variable, we have the following result.

Lemma 2.1 (see [3]).

then the periodic boundary value problem (2.3) has at least one -periodic solution on

Lemma 2.2.

Suppose the following condition holds:

Then (1.4) has at most one -periodic solution.

Proof.

this contradicts the second equation of (2.12). So we have , for all

For every if then it contradicts (2.19), so ; it implies that then , for all

Hence, (1.4) has at most one -periodic solution. The proof of Lemma 2.2 is completed now.

## 3. Main Results

Theorem 3.1.

Let ( ) hold. Suppose that there exists a positive constant d such that

Then (1.4) has one unique -periodic solution if

Proof.

By Lemma 2.2, combining it is easy to see that (1.4) has at least one -periodic solution. For the remainder, we will apply Lemma 2.1 to study (3.1). Firstly, we will verify that all the possible -periodic solutions of (3.1) are bounded.

then we consider the following two cases.

Case 1.

Case 2.

so condition (ii) holds.

So condition (iii) holds. In view of Lemma 2.1, there exists at least one solution with period . This completes the proof.

Theorem 3.2.

Let ( ) hold. Suppose that there exist positive constants and satisfying the following conditions:

Then for (1.4) there exists one unique -periodic solution when

Proof.

From the first equation of (3.26) we have so

By way of contradiction, (3.29) does not hold, then So there exists such that , for ; therefore, , for so , that is, , for This contradicts the definition of so we have

By condition ( ) we have Similarly, we get

Case 1.

Case 2.

If from the fact that is a continuous function in there exists a constant between and such that

So we have that (3.27) holds. Next, in view of there are integer and constant such that hence

We claim that all the periodic solutions of (3.1) are bounded and

hence together with (3.31), we have

This proves the claim and that the rest of the proof of the theorem is identical to that of Theorem 3.1.

## Declarations

### Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), Shandong Postdoctoral Funded Project (200802018), and the Natural Scientific Foundation of Shandong Province (Y2008A28, ZR2009AL003) and is also supported by University of Jinan Research Funds for Doctors (XBS0843).

## Authors’ Affiliations

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