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 equationsΔ[p(t)Δu(t−1)]+q(t)u(t)=f(t,u(t)),t∈Z, without any periodicity assumptions on p(t)p(t), q(t)q(t) and f  , providing that f(t,x)f(t,x) grows superlinearly both at origin and at infinity or is an odd function with respect to x∈Rx∈R, and satisfies some additional assumptions.