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 retardations Δ n ( r ( t − n ) Δ n u ( 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.