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.


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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