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Zbl 1046.39005
Guo, Zhiming; Yu, Jianshe
The existence of periodic and subharmonic solutions of subquadratic second order difference equations.
(English)
[J] J. Lond. Math. Soc., II. Ser. 68, No. 2, 419-430 (2003). ISSN 0024-6107; ISSN 1469-7750/e

Of concern is the nonlinear second order difference equation $$x_{n+1}-2x_{n}+x_{n-1}+f(n,x_{n})=0,\ n\in \Bbb{Z},$$ where $f=(f_1,\dots,f_{m})^T\in C(\Bbb{R}\times\Bbb{R}^m,\Bbb{R}^{m})$ and $f(t+M,z)=f(t,z)$ for some positive integer $M$ and for all $(t,z)\in\Bbb{R}\times \Bbb{R}^{m}$. One supposes there exists a function $F(t,z)\in C^{1}(\Bbb{R}\times \Bbb{R}^{m},\Bbb{R} ^{m})$ 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 $x^{\prime\prime }+f(t,x)=0$, $t\in\Bbb{R}$.
[N. C. Apreutesei (Iaşi)]
MSC 2000:
*39A11 Stability of difference equations
39A12 Discrete version of topics in analysis

Keywords: $pM$-periodic solution; subharmonic solution; critical point theory; subquadratic difference equation; nonlinear second order difference equation

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