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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 (MSC2010)
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