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

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 $$x^{\prime\prime }+f(t,x)=0$$, $$t\in\mathbb{R}$$.

### MSC:

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