## 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 (MSC2010)
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### References:

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