Periodic Solutions Of Second-order Discrete Hamiltonian Systems

Consider the following two nonlinear second order discrete Hamiltonian systemswhere Δu(t)=u(t+1)-u(t),Δ~2u(t) =Δ(Δu(t)), F : Z × R~N → R, F(t,x) is continuously differential in x for every t ∈ Z and T-periodic in t for all x ∈ R~N, T is a positive integer, ▽F(t,x) denotes the gradient of F(t, x) in x.In this paper, we first define some corresponding functions, and prove the critical points of these functions are just the T—periodic solutions of systems (DHS±), then, we consider the existence and multiplicity of periodic solutions for systems (DHS±) by critical point theory.The main results are the following theorems.Theorem 1 Let φ± : H_T → R be defined bywhere H_T = {u : Z → R~N |u(t+T)=u(t), t ∈ Z}, with the inner productand the normwhere (·,·) and |·| denote the usual inner product and the usual norm in R~N. If u ∈ H_T is a solution of the corresponding Euler equation φ'±(u) = 0, u is a T—periodic solution of...