Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equations
© X. Cai and J. Yu. 2008
Received: 14 August 2007
Accepted: 14 November 2007
Published: 29 November 2007
In (1.1), the given real sequences satisfy for any , is continuous in the second variable, and for a given positive integer and for all . and is the ratio of odd positive integers. By a solution of (1.1), we mean a real sequence , satisfying (1.1).
The main purpose here is to develop a new approach to the above problem by using critical point method and to obtain some sufficient conditions for the existence of periodic solutions of (1.1).
Let be a real Hilbert space, , , which implies that is continuously Fréchet differentiable functional defined on . is said to be satisfying Palais-Smale condition (P-S condition) if any sequence is bounded, and as possesses a convergent subsequence in . Let be the open ball in with radius and centered at , and let denote its boundary.
Lemma 1.1 (mountain pass lemma, see ).
In this section, we are going to establish the corresponding variational framework for (1.1).
3. Main Results
In this section, we will prove our main results by using critical point theorem. First, we prove two lemmas which are useful in the proof of theorems.
Assume that the following conditions are satisfied:
satisfies P-S condition.
where Then for any By the continuity of in , and show that there exists some such that If we choose such that the intersection is empty, then there exist such that Thus we obtain two different critical points , of in . In this case, in fact, we may obtain at least two nontrivial critical points which correspond to the critical value The proof of Theorem 3.2 is complete. When , we have the following results.
Assume that the following conditions hold:
First, we proved the following lemma.
By , the above inequality implies that is a bounded sequence in . Thus possesses a convergent subsequence, and the proof of Lemma 3.4 is complete. Now we prove Theorem 3.3 by the saddle point theorem.
Proof of Theorem 3.3.
Assume that the following conditions are satisfied:
Before proving Theorem 3.5, first, we prove the following result.
Proof of Theorem 3.5.
This project is supported by specialized research fund for the doctoral program of higher education, Grant no. 20020532014.
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