Homoclinic orbits for second order \(p\)-Laplacian difference equations containing both advance and retardation (Q259746)

The difference equation \[ \Delta(\varphi_p(\Delta u_{n-1}))-q_n\varphi_p(u_n)+f(n,u_{n+M},u_n,u_{n-M})=0 \] is considered, with the forward difference operator \(\Delta\), a real sequence \(\{q_n\}_n\), the \(p\)-Laplacian operator \(\varphi_p(s)=|s|^{p-2}s\), some fixed given integer \(M>0\), and \(f(\cdot,v_1,v_2,v_3)\) are \(T\)-periodic. Based on the mountain pass lemma, relaxed sufficient conditions for the existence of a non-trivial homoclinic orbit are given.