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

*Advances in Difference Equations*
**volume 2010**, Article number: 197263 (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 ;

(H2) there exists a function such that 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.

(S1) For each , the problem

has no solution on .

(S2) The equation

has no solution on

(S3) The Brouwer degree

Then the periodic boundary value problem

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

that is,

Since is continuous, there exists such that

In view of (A1), we obtain

where . Then, for each , we have

which gives that

Thus,

Since , there is a constant such that

Set

In view of (A2) and

we get

where

By (2.9), we have

Noticing that

there exists 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

For the above , it follows from that there exists such that

Combining this with , we get

which yields that .

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

Since , we have

In view of being strictly decreasing, we get

Set

Then, we know that (2.2) has no solution on for each , that is, the assumption (S1) of Lemma 1.1 holds. In addition, it follows from (2.24) that

So the assumption (S2) of Lemma 1.1 holds. Let

For and , by (2.24), we have

Thus, is a homotopic transformation. So

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

Then (1.1) is transformed into

Let and being two -periodic solutions of (1.1); and

Then we obtain

Setting

it follows from (2.33) that

Now, we claim that

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

Case 1.

There exists such that

which implies that

By (A1), . So it follows from that . Thus, in view of

and (H1), we obtain

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

Analogously, one can show that

So we have . Then, it follows from (2.35) that

which implies 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:

(A2′) there exists a constant such that for all and ,

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.

Let . Noticing that

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

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

Example 2.4.

Consider the following nonlinear equation:

where , , , and . One can easily check that satisfy (H1) and (H2). Obviously, (A1) holds. Moreover, since

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

Example 2.5.

Consider the following -Laplacian equation:

where , and . Obviously, (A1) holds. Moreover, we have

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

Remark 2.6.

In Example 2.5, , we have

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 .

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## 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.

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Ding, HS., Ye, GR. & Long, W. Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with -Laplacian-Like Operators.
*Adv Differ Equ* **2010, **197263 (2010). https://doi.org/10.1155/2010/197263

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### Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Periodic Solution