Homoclinic orbits for second order self-adjoint difference equations

Abstract

In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations Δ [ p ( t ) Δ u ( t − 1 ) ] + q ( t ) u ( t ) = f ( t , u ( t ) ) , t ∈ Z , without any periodicity assumptions on p ( t ) , q ( t ) and f, providing that f ( t , x ) grows superlinearly both at origin and at infinity or is an odd function with respect to x ∈ R , and satisfies some additional assumptions.