Homoclinic orbits for second order self-adjoint difference equations

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)Δu(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. © 2005 Elsevier Inc. All rights reserved