Existence of homoclinic orbits in discrete Hamiltonian systems without Palais–Smale condition

Abstract

Some existing results for the second-order self-adjoint discrete Hamiltonian system whose associated functionals do not satisfy a Palais–Smale condition are proved. By using Mountain Pass Theorem, a weak convergence argument and a discrete version of Liebs lemma, we establish that some existing criteria to guarantee the second-order self-adjoint discrete Hamiltonian system have at least one non-trivial homoclinic orbit.