Advances in Periodic Difference Equations with Open Problems

Abstract

In this paper, we review some recent results on the dynamics of semi-dynamical systems generated by the iteration of a periodic sequence of continuous maps. In particular, we state several open problems focused on the structure of periodic orbits, forcing between periodic orbits, sharing periodic orbits, folding and unfolding periodic systems, and on applications of periodic systems.

Periodic Solutions with Minimal Period for Fourth-Order Nonlinear Difference Equations

“…Because of their importance, many literature and monographs deal with their existence and uniqueness problems; see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

Existence and multiple solutions for second-order p-Laplacian difference equations

“…In this paper, we investigate the solutions of boundary value problems (1.1) with (1.2) for a second-order p-Laplacian difference equation [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. By using the critical point theory [26][27][28], the existence and multiple results are obtained.…”

Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

“…On the other hand, difference equations [9] are closely related to differential equations in the sense that a differential equation model is often derived from a difference equation, and numerical solutions of a differential equation have to be obtained by discretizing the differential equation. Therefore, the study of homoclinic orbits [10][11][12][13][14][15][16][17][18][19][20][21] of difference equation is meaningful.…”

“…Difference equations containing both advance and retardation have many applications in theory and practice [9][10][11]22]. We may think of (1) as a discrete analogue of the following 2nth-order functional differential equation…”