The existence of periodic and subharmonic solutions of subquadratic second order difference equations. (English) Zbl 1046.39005

Guo, Zhiming; Yu, Jianshe

J. Lond. Math. Soc., II. Ser. 68, No. 2, 419-430 (2003).

Of concern is the nonlinear second order difference equation \[ x_{n+1}-2x_{n}+x_{n-1}+f(n,x_{n})=0,\;n\in \mathbb{Z}, \] where \(f=(f_1,\dots,f_{m})^T\in C(\mathbb{R}\times\mathbb{R}^m,\mathbb{R}^{m})\) and \(f(t+M,z)=f(t,z)\) for some positive integer \(M\) and for all \((t,z)\in\mathbb{R}\times \mathbb{R}^{m}\). One supposes there exists a function \(F(t,z)\in C^{1}(\mathbb{R}\times \mathbb{R}^{m},\mathbb{R} ^{m})\) such that the gradient of \(F(t,z)\) in \(z\) coincides with \(f(t,z)\). Let \(p\) be a given positive integer. In this paper, the existence of \(pM\)-periodic solutions of the above difference equation is studied, under different hypotheses on \(f\) and \(F\). The method used here is from the critical point theory. These results are the discrete analogues of some theorems obtained in the continuous case for the second order differential equation \(x^{\prime\prime }+f(t,x)=0\), \(t\in\mathbb{R}\).

Reviewer: N. C. Apreutesei (Iaşi)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis

Keywords:

\(pM\)-periodic solution; subharmonic solution; critical point theory; subquadratic difference equation; nonlinear second order difference equation