- Research Article
- Open Access
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.
- Ahlbrandt CD, Peterson AC: Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts in the Mathematical Sciences. Volume 16. Kluwer Academic, Dordrecht, The Netherlands; 1996:xiv+374.Google Scholar
- Cheng SS, Li HJ, Patula WT: Bounded and zero convergent solutions of second-order difference equations. Journal of Mathematical Analysis and Applications 1989, 141(2):463-483. 10.1016/0022-247X(89)90191-1MATHMathSciNetView ArticleGoogle Scholar
- Peil T, Peterson A:Criteria for -disfocality of a selfadjoint vector difference equation. Journal of Mathematical Analysis and Applications 1993, 179(2):512-524. 10.1006/jmaa.1993.1366MATHMathSciNetView ArticleGoogle Scholar
- Dannan F, Elaydi S, Liu P: Periodic solutions of difference equations. Journal of Difference Equations and Applications 2000, 6(2):203-232. 10.1080/10236190008808222MATHMathSciNetView ArticleGoogle Scholar
- Elaydi S, Zhang S: Stability and periodicity of difference equations with finite delay. Funkcialaj Ekvacioj 1994, 37(3):401-413.MATHMathSciNetGoogle Scholar
- Guo ZM, Yu JS: The existence of periodic and subharmonic solutions for second-order superlinear difference equations. Science in China Series A 2003, 3: 226-235.Google Scholar
- Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar
- Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic, Dordrecht, The Netherlands; 1997:viii+507.View ArticleGoogle Scholar
- Cecchi M, Došlá Z, Marini M: Positive decreasing solutions of quasi-linear difference equations. Computers & Mathematics with Applications 2001, 42(10-11):1401-1410. 10.1016/S0898-1221(01)00249-8MATHMathSciNetView ArticleGoogle Scholar
- Wong PJY, Agarwal RP: Oscillations and nonoscillations of half-linear difference equations generated by deviating arguments. Computers & Mathematics with Applications 1998, 36(10–12):11-26. Advances in difference equations, IIMATHMathSciNetView ArticleGoogle Scholar
- Wong PJY, Agarwal RP: Oscillation and monotone solutions of second order quasilinear difference equations. Funkcialaj Ekvacioj 1996, 39(3):491-517.MATHMathSciNetGoogle Scholar
- Cecchi M, Marini M, Villari G: On the monotonicity property for a certain class of second order differential equations. Journal of Differential Equations 1989, 82(1):15-27. 10.1016/0022-0396(89)90165-4MATHMathSciNetView ArticleGoogle Scholar
- Marini M: On nonoscillatory solutions of a second-order nonlinear differential equation. Bollettino della Unione Matematica Italiana 1984, 3(1):189-202.MATHMathSciNetGoogle Scholar
- Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics. Volume 65. American Mathematical Society, Providence, RI, USA; 1986:viii+100.Google Scholar
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.View ArticleGoogle Scholar
- Chang KC, Lin YQ: Functional Analysis. Peking University Press, Beijing, China; 1986.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.