Existence of Periodic Solutions for a Class of Discrete Hamiltonian Systems

By applying minimax methods in critical point theory, we prove the existence of periodic solutions for the following discrete Hamiltonian systems Δ2u(t-1)


Introduction
Consider the following discrete Hamiltonian system: where Δ is the forward difference operator defined by Δu t u t 1 − u t , Δ 2 u t Δ Δu t , t ∈ Z, u ∈ R N , F : Z × R N → R, and F t, x is continuously differentiable in x for every t ∈ Z and is T -periodic in t; T is a positive integer.
Difference equations usually describe evolution of certain phenomena over the course of time.For example, if a certain population has discrete generations, the size of the t 1 th generation x t 1 is a function of the tth generation x t .In fact, difference equations provide a natural description of many discrete models in real world.Since discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, optimal control, and so on, it is of practical importance to investigate the solutions of difference equations.For more details about difference equations, we refer the readers to the books 1-3 .
In some recent papers 4-15 , the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory.These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations.In 2007, Xue and Tang 11 investigated the existence of periodic solutions for 1.1 and obtained the main result.
Theorem A see 11 .Suppose that F satisfies the following conditions: F1 there exists a positive constant T such that F t T, x F t, x for all t, x ∈ Z × R N ; F2 there are constants L 1 > 0, L 2 > 0, and 0 ≤ α < 1 such that where ε > 0 and a ε is a positive constant and is dependent on ε.The above inequality shows that there are functions not satisfying condition F2 .If we let T t 1 f t 0, α 3/4, T 2, then we have But the above equality does not satisfy F3 .This example shows that it is valuable to further improve conditions F2 and F3 .Before stating our main results, we first introduce some preliminaries.

2.4
Obviously, u u 2 and u is equivalent to u r .Hence, there exist two positive constants C 1 , C 2 , which are independent on r, such that 2.5 If we define u ∞ sup t∈Z 1,T |u t |, it is easy to see that for any r > 1, u ∞ ≤ u r , ∀u ∈ H T .

2.6
For any u ∈ H T , let We can compute the Fréchet derivative of 2.7 as Hence, u is a critical point of ϕ on H T if and only if So, the critical points of ϕ are classical solutions of 1.1 .The following lemmas are useful in our proof.

Lemma 2.1 see 11 . As a subspace of H T , N k is defined by
where • and denote the Gauss Function.Then there hold

Main Results and Proofs
Theorem 3.1.Suppose that F satisfies (F1) and the following conditions Then problem 1.1 has at least one periodic solution with period T .
Theorem 3.2.Suppose that F satisfies (F1) and (F2) with α 1.Moreover, assume the following conditions hold: Then problem 1.1 has at least one periodic solution with period T .
Remark 3.3.It is easy to see that F2 is more general than F2 and F3 is weaker than F3 .Theorem 3.2 is a new result, which completes Theorem A when α 1.
For the sake of convenience, we denote Proof of Theorem 3.1.First we prove that ϕ satisfies the PS condition.Suppose that a consequence {u n } ⊂ H T is such that −C 3 ≤ ϕ u n ≤ C 3 , where C 3 > 0 and ϕ u n → 0 as n → ∞.Then for sufficiently large n, From Lemma 2.1, we can write u as u u u ∈ H 0 H ⊥ 0 , where H 0 N 0 , and From F2 , 2.6 , Hölder inequality, and Young inequality, we have

3.6
In a similar way, we have Let u n u n u n ∈ H 0 H ⊥ 0 .From 2.12 and 3.7 , we have 3.9 It follows from 3.8 and 3.9 that where and so where C 5 > 0. It follows from the boundedness of ϕ u n , 2.11 , 3.6 , 3.11 , and 3.12 that F t, 0 .

3.13
Inequalities 3.5 and 3.13 imply that {u n } is bounded.Hence, { u n } is bounded by 3.12 , and then {u n } is bounded.Since H T is finite dimensional, there exists a subsequence of {u n } convergent in H T .Thus, we conclude that PS condition is satisfied.
In order to use the saddle point theorem 16, Theorem 4.6 , we only need to verify the following conditions: 3.14 For any x ∈ H 0 , since T t 1 |Δx| 2 0, we have It follows from 3.14 and the above inequality that Thus I1 is easy to verify.
Proof of Theorem 3.2.By 3.2 and F4 , we can choose an a 2 ∈ R such that

3.21
In a similar way, we have

3.22
From 3.8 and 3.22 , we have

3.23
It follow from 3.23 that where and so where C 7 > 0. It follows from the boundedness of ϕ u n , 2.11 , 3.21 , 3.25 , and 3.26 that F t, 0 .

3.27
Inequalities 3.20 and 3.27 imply that {u n } is bounded.Hence, { u n } is bounded by 3.26 , and then {u n } is bounded.Since H T is finite dimensional, there exists a subsequence of {u n } convergent in H T .Thus, we conclude that PS condition is satisfied.
In the following, we prove that ϕ satisfies I1 and I2 .In fact, from F4 , we have

3.28
It follows from 3.27 and T t 1 |Δx| 2 0 that Thus I1 is easy to verify.
Next, for all u ∈ H ⊥ 0 , from F2 with α 1 and 2.6 , we have

3.31
Since The proof of Theorem 3.2 is complete.

Examples
In this section, we give two examples to illustrate our results.
Then from Theorem 3.2, problem 1.1 has at least one periodic solution with period T .
where h : Z 1, T → R N and h t T h t .It is easy to see that F t, x satisfies F1 andε |x| b ε |h t |, ∀ t, x ∈ Z 1, T × R N , 4.7where ε > 0 and b ε is a positive constant and is dependent on ε.The above shows that F2 holds with α 1 and