Existence theorems of periodic solutions for second-order nonlinear difference equations. (English) Zbl 1146.39006

Cai, Xiaochun; Yu, Jianshe

Adv. Difference Equ. 2008, Article ID 247071, 11 p. (2008).

Summary: The authors consider the second-order nonlinear difference equation of the type \(\Delta(p_n(\Delta x_{n-1})^\delta)+q_nx_n^\delta=f(n,x_n)\), \(n\in\mathbb Z\), using critical point theory, and they obtain some new results on the existence of periodic solutions.

MSC:

39A11	Stability of difference equations (MSC2000)
39A10	Additive difference equations

Keywords:

mountain pass lemma; saddle point theorem; second-order nonlinear difference equation; critical point theory; periodic solutions

References:

[1]	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. · Zbl 0860.39001
[2]	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)90191-1 · Zbl 0698.39002
[3]	Peil T, Peterson A:Criteria for [InlineEquation not available: see fulltext.]-disfocality of a selfadjoint vector difference equation. Journal of Mathematical Analysis and Applications 1993, 179(2):512-524. 10.1006/jmaa.1993.1366 · Zbl 0802.39003
[4]	Dannan F, Elaydi S, Liu P: Periodic solutions of difference equations. Journal of Difference Equations and Applications 2000, 6(2):203-232. 10.1080/10236190008808222 · Zbl 0953.39007
[5]	Elaydi S, Zhang S: Stability and periodicity of difference equations with finite delay. Funkcialaj Ekvacioj 1994, 37(3):401-413. · Zbl 0819.39006
[6]	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.
[7]	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.
[8]	Agarwal RP, Wong PJY: Advanced Topics in Difference Equations, Mathematics and Its Applications. Volume 404. Kluwer Academic, Dordrecht, The Netherlands; 1997:viii+507.
[9]	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)00249-8 · Zbl 1007.39006
[10]	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, II · Zbl 0933.39025
[11]	Wong PJY, Agarwal RP: Oscillation and monotone solutions of second order quasilinear difference equations. Funkcialaj Ekvacioj 1996, 39(3):491-517. · Zbl 0871.39005
[12]	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)90165-4 · Zbl 0694.34035
[13]	Marini M: On nonoscillatory solutions of a second-order nonlinear differential equation. Bollettino della Unione Matematica Italiana 1984, 3(1):189-202. · Zbl 0574.34022
[14]	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. · Zbl 0609.58002
[15]	Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277. · Zbl 0676.58017
[16]	Chang KC, Lin YQ: Functional Analysis. Peking University Press, Beijing, China; 1986.