On variational and topological methods in nonlinear difference equations

Abstract

In this paper, first we survey the recent progress in usage of the critical point theory to study the existence of multiple periodic and subharmonic solutions in second order difference equations and discrete Hamiltonian systems with variational structure. Next, we propose a new topological method, based on the application of the equivariant version of the Brouwer degree to study difference equations without an extra assumption on variational structure. New result on the existence of multiple periodic solutions in difference systems (without assuming that they are their variational) satisfying a Nagumo-type condition is obtained. Finally, we also put forward a new direction for further investigations.

Infinitely Many Positive Solutions for a Coupled Discrete Boundary Value Problem

“…In recent years, variational methods have been employed to study difference equation and various results have been obtained. See, for instance, [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Global Behavior of a Discrete Anticompetitive System in the Plane

“…Difference equations or systems have been attracting more and more attention by authors since these models display some complicated character comparing with its analogue differential equations; see [1][2][3][4][5][6][7][8][9][10][11]. In [12], the authors gave a discussion on the following rational system in the plane :…”

“…System (1) contains, as special cases, a large number of equations whose dynamics have not been thoroughly understood yet and there still exists research aspect to be further studied. Several global asymptotic results for some special cases of (1) have been obtained in [13][14][15][16][17][18]. As a special anticompetitive system of (1), the difference system…”

Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions

“…Therefore, it is of both practical importance and theoretical significance to study the existence of homoclinic orbits of this kind of discrete systems. Since the first variational result about periodic solutions of difference equations was obtained by Guo and Yu [7] in 2003, critical point theory has been widely used in studying the existence of some special solutions of discrete systems, for example, see [3,20].…”