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Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system. (English) Zbl 1129.39008
Consider the nonlinear second order discrete Hamiltonian system $$\Delta^2u(t-1)+\nabla F(t,u(t))=0, \quad \forall t\in Z,$$ where $\Delta u(t)=u(t+1)-u(t)$, $\Delta^2u(t)=\Delta (\Delta u(t))$, $F: Z\times R^N\to R$, $F(t,x)$ is continuously differentiable in $x$ for every $t\in Z$ and $T$-periodic in $t$ for all $x\in R^N$, $T$ is a positive integer, $Z$ is the set of all integers, $\nabla F(t,x)$ denotes the gradient of $F(t,x)$ in $x$. Several criteria on the existence of at least one $T$-periodic solution are presented, which are established by employing minimax methods in critical point theory. The obtained results improve some exisiting ones.

##### MSC:
 39A12 Discrete version of topics in analysis 39A11 Stability of difference equations (MSC2000) 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
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##### References:
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