Periodic Solutions for a System of Difference Equations
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.
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 , the nonlinear difference equationhas been considered. In , 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 , 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  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 ). Let be a real Hilbert space, and assume that satisfies the P-S condition and the following
()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 .
()there exist constants , , such thatfor any ;()
Thensatisfies 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.
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 ,
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.
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.