Homoclinic Orbits for Second Order Self-Adjoint Difference System with a Change of Sign in the Potential

The authors we study the existence of nontrivial homoclinic orbits of the second self-adjoint difference systemΔ2 u(t-1)-q(t)u(t)+▽V(t,u(t))=0 t∈Z,u∈Rm.Under superquadratic conditions,by using the Mountain Pass Theorem and the Symmetric Mountain Pass Theorem,some existence theorems are established for the existence of homoclinic orbits for the above nonautonomous Hamiltonian systems with a change of sign in the potential.