# Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with -Laplacian-Like Operators

- Hui-Sheng Ding
^{1}, - Guo-Rong Ye
^{1}and - Wei Long
^{1}Email author

**2010**:197263

**DOI: **10.1155/2010/197263

© Hui-Sheng Ding et al. 2010

**Received: **1 February 2010

**Accepted: **19 March 2010

**Published: **30 March 2010

## Abstract

We investigate the following nonlinear equations with
-Laplacian-like operators
: some criteria to guarantee the existence and uniqueness of periodic solutions of the above equation are given by using Mawhin's continuation theorem. Our results are new and extend some recent results due to Liu (B. Liu, Existence and uniqueness of periodic solutions for a kind of Lienard type
-Laplacian equation, *Nonlinear Analysis TMA*, 69, 724–729, 2008).

## 1. Introduction

In this paper, we deal with the existence and uniqueness of periodic solutions for the following nonlinear equations with -Laplacian-like operators:

where , are continuous functions on , and is a continuous function on with period ; moreover, is a continuous function satisfying the following:

(H1) for any and ;

It is obvious that under these two conditions, is an homeomorphism from onto and is increasing on

Recall that -Laplacian equations have been of great interest for many mathematicians. Especially, there is a large literature (see, e.g., [1–7] and references therein) about the existence of periodic solutions to the following -Laplacian equation:

and its variants, where for and . Obviously, (1.3) is a special case of (1.1).

However, there are seldom results about the existence of periodic solutions to (1.1). The main difficulty lies in the -Laplacian-like operator of (1.1), which is more complicated than in (1.3). Since there is no concrete form for the -Laplacian-like operator of (1.1), it is more difficult to prove the existence of periodic solutions to (1.1).

Therefore, in this paper, we will devote ourselves to investigate the existence of periodic solutions to (1.1). As one will see, our theorem generalizes some recent results even for the case of (see Remark 2.2).

Next, let us recall some notations and basic results. For convenience, we denote

which is a Banach space endowed with the norm , where

In the proof of our main results, we will need the following classical Mawhin's continuation theorem.

Lemma 1.1 ([8]).

Let (H1), (H2) hold and is Carathéodory. Assume that is an open bounded set in such that the following conditions hold.

has no solution on .

has no solution on

has at least one -periodic solution on .

## 2. Main Results

In this section, we prove an existence and uniqueness theorem for (1.1).

Theorem 2.1.

Suppose the following assumptions hold:

(A1) and for all ;

(A2) there exist a constant and a function such that for all and ,

Then (1.1) has a unique -periodic solution.

Proof

*Existence*. For the proof of existence, we use Lemma 1.1. First, let us consider the homotopic equation of (1.1):

Let be an arbitrary solution of (2.2). By integrating the two sides of (2.2) over , and noticing that and , we have

Since , there is a constant such that

On the other hand, it follows from

that for . In addition, since , there exists such that . Thus .

Then, for all , we have

which yields that .

Now, we have proved that any solution of (2.2) satisfies

that is, the assumption (S3) of Lemma 1.1 holds. By applying Lemma 1.1, there exists at least one solution with period to (1.1).

*Uniqueness*. Let

Setting

If this is not true, we consider the following two cases.

Case 1.

which contradicts with .

Case 2.

Also, we have and . Then, similar to the proof of Case 1, one can get a contradiction.

Now, we have proved that

Hence, (1.1) has a unique -periodic solution. The proof of Theorem 2.1 is now completed.

Remark 2.2.

In Theorem 2.1, setting , then (A2) becomes as follows:

In the case , Liu [7, Theorem ] proved that (1.1) has a unique -periodic solution under the assumptions (A1) and ( ). Thus, even for the case of , Theorem 2.1 is a generalization of [7, Theorem 1].

In addition, we have the following interesting corollary.

Corollary 2.3.

Suppose (A1) and

(A2″) there exist a constant such that

hold. Then (1.1) has a unique -periodic solution.

Proof.

we know that (A2) holds with . This completes the proof.

At last, we give two examples to illustrate our results.

Example 2.4.

it is easy to verify that (A2) holds. By Theorem 2.1, (2.49) has a unique -periodic solution.

Example 2.5.

So ( ) holds. Then, by Corollary 2.3, (2.51) has a unique -periodic solution.

Remark 2.6.

Thus, ( ) does not hold. So [7, Theorem 1] cannot be applied to Example 2.5. This means that our results generalize [7, Theorem 1] in essence even for the case of .

## Declarations

### Acknowledgments

The authors are grateful to the referee for valuable suggestions and comments, which improved the quality of this paper. The work was supported by the NSF of China, the NSF of Jiangxi Province of China (2008GQS0057), the Youth Foundation of Jiangxi Provincial Education Department (GJJ09456), and the Youth Foundation of Jiangxi Normal University.

## Authors’ Affiliations

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