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
The authors consider the second-order nonlinear difference equation of the type using critical point theory, and they obtain some new results on the existence of periodic solutions.
We denote by the set of all natural numbers, integers, and real numbers, respectively. For , define when .
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 ).
Let be a real Hilbert space, and assume that satisfies the P-S condition and the following conditions:
(I1) there exist constants and such that for all , where
(I2) and there exists such that
Let be a real Banach space, where and is finite dimensional. Suppose satisfies the P-S condition and
(I3) there exist constants , such that
(I4) there is and a constant such that .
In this section, we are going to establish the corresponding variational framework for (1.1).
for all , and .
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.
Since the above inequality implies that is a bounded sequence in Thus possesses convergent subsequences, and the proof is complete.
Suppose that and following conditions hold:
Then there exist at least two nontrivial -periodic solutions for (1.1).
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:
possesses at least one -periodic solution.
First, we proved the following lemma.
satisfies P-S condition on .
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.
This implies that the assumption of saddle point theorem is satisfied. Thus there exists at least one critical point of on , and the proof is complete. When we have the following result.
Assume that the following conditions are satisfied:
Then (3.21) possesses at least one -periodic solution.
Before proving Theorem 3.5, first, we prove the following result.
Assume that holds, then defined by (3.22) satisfies P-S condition.
For any sequence with being bounded and as there exists a positive constant such that
By , the above inequality implies that is a bounded sequence in Thus possesses convergent subsequences, and the proof is complete.
Proof of Theorem 3.5.
Set then Thus satisfies the assumption of saddle point theorem, that is, there exists at least one critical point of on This completes the 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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