Existence of homoclinic orbits for a higher order difference system

Abstract

By using critical point theory, some new criteria are obtained for the existence of a nontrivial homoclinic orbit to a higher order difference system containing both many advances and retardations. The proof is based on the mountain pass lemma in combination with periodic approximations. Related results in the literature are generalized and improved.

Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

“…Equations similar in structure to (3) have been studied by many researchers using a variety of methods; see, for example, [2][3][4][5][6][7][8][9][10][11][12]. It is well-known that the study of nonlinear difference equations [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] has long been an important one as a result of the fact that they arise in numerical solutions of both ordinary and partial differential equations as well as in applications to different areas of applied mathematics and physics.…”