Existence of periodic solutions for a \(2n\)th-order nonlinear difference equation. (English) Zbl 1153.39302

Cai, Xiaochun; Yu, Jianshe

J. Math. Anal. Appl. 329, No. 2, 870-878 (2007).

Summary: The authors consider the \(2n\)th-order difference equation \[ \Delta^n(r_{t-n}\Delta^n x_{t-n})+f(t,x_t)=0,\quad n \in \mathbb{Z}(3), t\in \mathbb{Z} \] where \(f: \mathbb{Z} \times \mathbb{R} \to \mathbb{R}\) is a continuous function in the second variable, \(f(t+T,z)=f(t,z)\) for all \((t,z)\in \mathbb{Z}\times \mathbb{R}, r_{t+T}=r_t\) for all \(t \in \mathbb{Z}\), and \(T\) a given positive integer. By the Linking Theorem, some new criteria are obtained for the existence and multiplicity of periodic solutions of the above equation.

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis
39B22	Functional equations for real functions

Keywords:

nonlinear difference equations; periodic solutions; critical points

References:

[1]	Ahlbrandt, C. D.; Peterson, A. C., The \((n, n)\)-disconjugacy of a \(2n\) th-order linear difference equation, Comput. Math. Appl., 28, 1-9 (1994) · Zbl 0815.39003
[2]	Anderson, Doug, A \(2n\) th-order linear difference equation, Comm. Appl. Anal., 2, 4, 521-529 (1998) · Zbl 0903.39001
[3]	Chang, K. C.; Lin, Y. Q., Functional Analysis (1986), Peking University Press: Peking University Press Beijing, China, (in Chinese)
[4]	Guo, Z. M.; Yu, J. S., The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc. (2), 68, 419-430 (2003) · Zbl 1046.39005
[5]	Guo, Z. M.; Yu, J. S., The existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 3, 226-235 (2003)
[6]	Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993), Kluwer: Kluwer Dordrecht · Zbl 0787.39001
[7]	Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0676.58017
[8]	Migda, M., Existence of nonoscillatory solutions of some higher order difference equations, Appl. Math. E-Notes, 4, 33-39 (2004) · Zbl 1069.39010
[9]	Peil, T.; Peterson, A., Asymptotic behavior of solutions of a two-term difference equation, Rocky Mountain J. Math., 24, 233-251 (1994) · Zbl 0809.39006
[10]	Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65 (1986) · Zbl 0609.58002