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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 469863, 8 pages
http://dx.doi.org/10.1155/2012/469863
Research Article

Periodic Solutions of a Class of Fourth-Order Superlinear Differential Equations

1School of Mathematics and Information Sciences, Guangzhou University, Guangdong 510006, China
2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong 510006, China

Received 17 July 2012; Accepted 10 September 2012

Academic Editor: Juntao Sun

Copyright © 2012 Yanyan Li and Yuhua Long. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the periodic solutions of a class of fourth-order superlinear differential equations. By using the classical variational techniques and symmetric mountain pass lemma, the periodic solutions of a single equation in literature are extended to that of equations, and also, the cubic growth of nonlinear term is extended to a general form of superlinear growth.

1. Introduction

The existence of periodic solutions of fourth-order differential equations has been studied by more and more researchers [16]. The application methods contain mainly Clark theorem [24], Cone theory [6], and so on.

For a single equation, Tersian and Chaparova [2] study the existence of infinitely many unbounded solutions, using symmetric mountain pass lemma:

It is a natural problem to wonder whether symmetric mountain pass lemma method may be applied not only to single equations but also to systems of differential equations. In this paper we study the existence of periodic solutions of the fourth-order equations, by making use of the classical variational techniques and symmetric mountain pass lemma Through studying System (1.2), (1.1) of the corresponding conclusions are extended.

The paper is organized as follows. In Section 2, we consider the result of System (1.2) under certain conditions. In Section 3, we prove the main result of this paper and give an example.

2. Main Result

In this paper, we state our main result. First we give the following list of assumptions on the parameters in System (1.2):. is an even functional about . That is, for every .There exists , as , we have with respect to consistently, as .

Denote , , , .

From condition , we obtain , when .

Remark 2.1. Let , then condition is transformed to where represents the usual inner product in .

Remark 2.2. From (), we obtain , where represents normal norm in . Besides, from the continuity of , we obtain .

Our main result is as follows.

Theorem 2.3. Suppose , and satisfy , . Then System (1.2) has infinitely many distinct pairs of solutions , which are critical points of the functional , and as .

In this paper, the existence of periodic solutions of a single equation in System (1.1) are extended to the case of equations, and also the cubic growth of nonlinear term is extended to a general form of superlinear growth.

3. Variational Structure and the Proof of Result

In this section, we prove the main result stated in Section 2.

3.1. Variational Structure

Denote Then is a Hilbert space. The norm is where . The corresponding inner product are For every , using Poincaré inequality [7], we obtain Thus, we can define another norm in . That is, for every , The inner product in as follows: The two different norms (3.2) and (3.6) are equivalent in .

In this section we consider System (1.2). The Fréchet derivative of is given by the following: where .

Remark 3.1. In general, the growth of is limited by the differentiability of functional , but we apply truncation techniques in [8]. First, introduce auxiliary functional and the auxiliary functional is Fréchet differentiable. Second, we use critical point theory to prove the existence of critical point of auxiliary functional, then prove the existence of the original equation. However, in order to avoid technical complexity, we assume directly functional is Fréchet differentiable.

In fact, for every , we obtain where and , .

It is similar to the discussion of [8], the solutions of System (1.2) corresponds to the critical point of the functional , so we need to discuss the critical point of functional . In order to prove Theorem 2.3, we introduce below definition and lemma.

Definition 3.2 (see [9]). Let X be a real Banach space, , is a Fréchet continuously differentiable functional in . is said to be satisfying Palais-Smale (PS) condition if any sequence for which is bounded and as , possesses a convergent subsequence.

Lemma 3.3 (see [8]). Let be an infinite dimensional Banach space and be a sequence of finite dimensional subspaces of such that dim, Let be an even functional, , and satisfy condition. Suppose that there are constants such that , and for every there is an such that on .
Then possesses infinitely many pairs of critical points with unbounded sequence of critical values.

3.2. The Proof of Result

Step 1 (Functional satisfies (PS) condition). Let be a (PS) sequence in , that is, is bounded and , as . Suppose that is unbounded in , that is, as . Since it follows that where . Letting in (3.13), we have a contradiction with as .
Therefore is a bounded sequence in . Passing if necessary to a subsequence we may assume that is weakly convergent to a function , in , and in .
From the Lebesgue theorem, , in , and in , letting in (3.9) we obtain From (3.15) and , in , we have as .

Remark 3.4. is the largest sum of the order of and .

Step 2 (Geometric conditions). Let , then constitutes a pair of standard orthogonal base in . Let us define to be the subspace of for every . We have , .
For a given constant , define a bounded closed set

Define mapping . For any , we obtain It is clear that is a linear odd mapping. For every , we have

So From (3.20), we obtain is an odd homeomorphism from to . Then is an odd homeomorphism from to , since .

On one hand, from functional (3.8) and using Sobolev’s embedding theorem, we obtain Thus condition is fulfilled if , .

On the other hand, as , then there exists , such that .

Denote . From functional (3.8), we obtain where . Here choosing , we obtain So holds. The proof of Theorem 2.3 is completed.

Example 3.5. In System (1.2), consider the problem: where , but there exists at least one , is an even and , .

It is obvious that and as .

For the superlinear property, we calculate that Therefore, there exists , as , we have

So satisfies the conditions . We only choose , , then the condition is satisfied. Therefore, System (1.2) has infinitely many distinct pairs of solutions by using Theorem 2.3.

Acknowledgment

This work is supported by the National Natural Science Foundation of China no. 11126063 and no. 111010981.

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