Existence of homoclinic orbits for 2nth-order nonlinear difference equations containing both many advances and retardations

By establishing a proper variational framework and using the critical point theory, we establish some new existence criteria to guarantee the 2nth-order nonlinear difference equation containing both many advances and retardationsΔn(r(t−n)Δnu(t−n))+q(t)u(t)=f(t,u(t+n),…,u(t),…,u(t−n)),n∈Z(3),t∈Z, has at least one or infinitely many homoclinic orbits, where r, q, f are nonperiodic in t. Our conditions on the potential are rather relaxed and some existing results in the literature are improved.