Existence of infinitely many homoclinic orbits for fourth-order difference systems containing both advance and retardation

By using the symmetric mountain pass theorem, we establish some existence criteria to guarantee the fourth-order difference system Δ4u(n − 2) + q(n)u(n) = f(n, u(n + 1), u(n), u(n − 1)) have infinitely many homoclinic orbits, where n∈Z,u∈RN,q:Z→RN×N and f:Z×R3N→R are no periodic in n.