Boundary value problems for 2n2n-order ϕcϕc-Laplacian difference equations containing both advance and retardation

In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of solutions of the boundary value problem for a 2n2n-order ϕcϕc-Laplacian difference equation containing both advance and retardation. To the best of our knowledge, this is the first time to discuss the boundary value problem for a special ϕϕ-Laplacian difference equation.