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

and Applied Analysis 3 Our main result is as follows. Theorem 2.3. Suppose a x , b x , and F satisfy A , F1 – F3 . Then System 1.2 has infinitely many distinct pairs of solutions zn un, vn , which are critical points of the functional I : X → R, and I zn → ∞ as n → ∞. 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 X L ( H2 0, L ∩H1 0 0, L )2 . 3.1 Then X L is a Hilbert space. The norm is


Introduction
The existence of periodic solutions of fourth-order differential equations has been studied by more and more researchers 1-6 .The application methods contain mainly Clark theorem 2-4 , 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: 1.1 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.

Abstract and Applied Analysis
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 1.2 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.

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 : F 2 There exists β > 2, as u 2 v 2 / 0, we have From condition A , we obtain 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.

Variational Structure and the Proof of Result
In this section, we prove the main result stated in Section 2.

3.1
Then X L is a Hilbert space.The norm is where

3.4
For every z u, v ∈ X L , using Poincaré inequality 7 , we obtain Thus, we can define another norm • 1 in X L .That is, for every z ∈ X L ,

3.6
The inner product in X L as follows: The two different norms 3.2 and 3.6 are equivalent in X L .
In this section we consider System 1.2 .The Fréchet derivative of I is given by the following: where z u, v ∈ X L .
Remark 3.1.In general, the growth of F is limited by the differentiability of functional I, 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 I is Fréchet differentiable.
In fact, for every z u, v ∈ X L , z u, v ∈ X L , we obtain where

3.10
and It is similar to the discussion of 8 , the solutions of System 1.2 corresponds to the critical point of the functional I, so we need to discuss the critical point of functional I.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, I ∈ C 1 X, R , I is a Fréchet continuously differentiable functional in X L .I is said to be satisfying Palais-Smale PS condition if any sequence {u n } ⊂ X for which {I u n } is bounded and {I u n } → 0 as j → ∞, possesses a convergent subsequence.Lemma 3.3 see 8 .Let X be an infinite dimensional Banach space and X n n be a sequence of finite dimensional subspaces of X such that dimX n n,

3.11
Let I ∈ C 1 X, R be an even functional, I 0 0, and I satisfy PS condition.Suppose that A 1 there are constants ρ, α > 0 such that I| ∂ Bρ ≥ α, and Then I possesses infinitely many pairs of critical points with unbounded sequence of critical values.

The Proof of Result
Step 1 Functional I satisfies PS condition .Let {z n } { u n , v n } be a PS sequence in X, that is, {I z n } is bounded and where γ ≥ 4. Letting n → ∞ in 3.13 , we have a contradiction with z n → ∞ as n → ∞.Therefore {z n } is a bounded sequence in X L .Passing if necessary to a subsequence we may assume that {z n } is weakly convergent to a function z ∈ X L , z n z in X L , and From the Lebesgue theorem, z ∈ X L , z n z in X L , and z n → z in C 0, L , letting n → ∞ in 3.9

3.15
From 3.15 and z ∈ X L , z n z in X L , we have z n − z → 0 as n → ∞.
Remark 3.4.γ is the largest sum of the order of u and v.
Step 2 Geometric conditions .Let e 1 1, 0 , e 2 0, 1 , then {e 1 , e 2 } constitutes a pair of standard orthogonal base in R 2 .Let us define X 2m to be the subspace of X L X 2m span sin kπx L e i , i for every m ∈ N. We have dim X 2m 2m, Define mapping H : X 2m → R 2m .For any z ∈ X 2m , we obtain

3.18
It is clear that H is a linear odd mapping.For every z ∈ X 2m , we have

3.19
Abstract and Applied Analysis 7 So

3.20
From 3.20 , we obtain H is an odd homeomorphism from X 2m to R 2m .Then H is an odd homeomorphism from K to S 2m−1 , since H K S 2m−1 .On one hand, from functional 3.8 and using Sobolev's embedding theorem, we obtain

3.23
So A 2 holds.The proof of Theorem 2.3 is completed.
Example 3.5.In System 1.2 , consider the problem: where p i x ≥ 0, but there exists at least one p i x / 0, n is an even and n ≥ 4, i 0, 1, 2, . . ., n.
It is obvious that F x, −u, −v F x, u, v and F x, u, v o u 2 v 2 as u 2 v 2 → 0.
For the superlinear property, we calculate that

3.26
So F satisfies the conditions F 1 -F 3 .We only choose a x > 0, b x > 0, c > −π 2 /L 2 , d > −π 2 /L 2 , then the condition A is satisfied.Therefore, System 1.2 has infinitely many distinct pairs of solutions by using Theorem 2.3.