#### Abstract

This paper deals with the second-order nonlinear systems of difference equations, we obtain the existence theorems of periodic solutions. The theorems are proved by using critical point theory.

#### 1. Introduction

Let be the set of all natural numbers, integers, and real numbers, respectively. For , note that , where .

In this paper, we consider the existence of periodic solutions for the system of difference equations of the formwhich can be recorded aswhere is a positive integer,and (i.e., ), , , , for any , is a positive integer, , is the ratio of odd positive integers, . For , define . , A sequence is a -periodic solution of (1.2) if substitution of it into (1.2) yields an identity for all .

In [1, 2], the qualitative behavior of linear difference equationshas been investigated. In [3], the nonlinear difference equationhas been considered. In [4], by critical point method, the existence of periodic and subharmonic solutions of equationhas been studied. Other interesting results can been found in [5–8]. In [9], the authors consider the existence of periodic solutions for second-order nonlinear difference equationusing critical point theory, obtaining some new results. It is a discrete analogues of differential equationThey do have physical applications in the study of nuclear physics, gas aerodynamics, and so on (see [10, 11]). In this paper, we obtain some new results of existence of periodic solution for the second-order nonlinear system of difference equations by using critical point theory. We remark, however, the result in [9] is only good for (1.7) which is much less general than our results in what follows.

#### 2. Some Basic Lemmas

Let be a real Hilbert space, mean that is continuously Fréchet differentiable functional defined on . is said to be satisfying Palais-Smale condition (P-S condition) if any bounded sequence and possess a convergent subsequence in . Let be the open ball in with radius and centered at , and let denote its boundary, is null element of .

Lemma 2.1 (see [12]). *Let be a real Hilbert space, and assume that satisfies the P-S condition and the following
conditions:*

()*there exist constants and such that for all ,
where ;*()* and there exists such that .*

Then is a positive critical value of , where

Let be the set of sequenceswhere , that is,For any , , is defined bythen is a vector space. For given positive integer , is defined as a subspace of byObviously, is isomorphic to , for any , defined inner productby which the norm can be induced bywhere It is obvious that with the inner product defined by (2.6) is a finite-dimensional Hilbert space and linearly homeomorphic to . Define the functional on as follows:where such that , that is,for any , , . Clearly , and for any , by and , we haveThus is a critical point of on () if and only ifThat is,By the periodicity of and in the first variable , we know that if is a critical point of the real functional defined by (2.8), then it is a periodic solution of (1.2).

For , , , denoteClearly, , . Because of and being equivalent when , so there exist constants and such that , , , and ,for all , , and .

Lemma 2.2. *
Suppose
that
*()*there exist constants , , such thatfor any ;*()*Then**satisfies P-S condition.*

*Proof. *For
any sequence , is bounded and Then there exists a positive constant ,
such that .
From (), we have SetThen and Because of ,
and ,
in view of Hölder inequality, we haveThusThen we have Thus, for any ,Because of ,
it is easily seen that the inequality (2.24) implies that is a bounded sequence in .
Thus possesses convergent subsequences. The proof
is complete.

#### 3. Main Result

Theorem 3.1. *Suppose that condition () holds, and*

()*for each *()*for any ,*()*Then (1.2) has at least two nontrivial -periodic solutions.
*

*Proof. *By
Lemma 2.2, satisfies P-S condition. Next, we will verify
the conditions () and () of Lemma 2.1. By (), there exists ,
such thatfor any and ,
where .
Thus for any with .
We choose ,
then we havethat is, the condition () of Lemma 2.1 holds.

Obviously, .
For any given with and constant , Thus we can choose a sufficiently
large such that ,
and , .
According to Lemma 2.1, there exists at least one critical value .
We suppose that is a critical point corresponding to ,
then and .

By similar argument of Lemma 2.2, we know that is bounded from above, so there exists such that for any .
Obviously, .
If ,
then the proof is complete. Otherwise, , .
In view of Lemma 2.1,where .
Then for any holds. In view of the continuity of in , , and ,
we know that there exists some such that .
If we choose such thatthen there exist such that .
Then possesses two different critical points and
Ž in ,
hence, we obtain at least two nontrivial critical points which correspond to
the critical value .
Thus (1.2) possesses at least two nontrivial -periodic solutions. The proof is complete.

#### Acknowledgments

This work is supported by Natural Science Foundation of Shanxi Province (2008011002-1) and Shanxi Datong University and by the Development Foundation of Higher Education Department of Shanxi Province.