Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities

By using critical point theory and periodic approximations, new sufficient conditions are obtained on the existence and nonexistence of homoclinic solutions for a class of discrete nonlinear periodic equations with asymptotically linear nonlinearities. These results partially answer an open problem proposed by Pankov (2006) [2] under rather weaker conditions and greatly improve the related results before.