Periodic solutions of a 2nth-order nonlinear difference equation

Abstract

In this paper, a 2nth-order nonlinear difference equation is considered. Using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of periodic solutions. Results obtained complement or improve the existing ones.

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

“…By using the critical point theory, Guo and Yu [23] established sufficient conditions on the existence of periodic solutions of second-order nonlinear difference equations. Compared to first-order or second-order difference equations, the study of higher-order equations has received considerably less attention (see, for example, [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references contained therein). Peil and Peterson [26] in 1994 studied the asymptotic behavior of solutions of 2nth-order difference equation…”

“…Migda [28] in 2004 studied an mth-order linear difference equation. Cai, Yu [24] in 2007 and Zhou, Yu, Chen [25] in 2010 obtained some criteria for the existence of periodic solutions of the following difference equation…”

Existence of periodic solutions of second order nonlinear p-Laplacian difference equations

“…Critical point theory is also an important tool to deal with problems of differential equations [19,22]. Because of applications in many areas for difference equations [1,16,20], recently, a few authors have gradually paid attention to applying critical point theory to deal with periodic solutions of discrete systems, see [2,3,[10][11][12][26][27][28]. A great deal of work has also been done in the study of the existence of solutions to discrete boundary value problems with the p-Laplacian operator.…”

“…However, to our best knowledge, results obtained in the literature on the periodic solutions of (1.1) are very scarce. Since f in (1.1) depends on u n+1 and u n−1 , the traditional ways of establishing the functional in [2,3,[10][11][12][26][27][28] are inapplicable to our case. The main purpose of this paper is to give some sufficient conditions for the existence of periodic solutions to second order nonlinear p-Laplacian difference equations.…”

Nonexistence and existence results for a 2 $$n$$ n th-order discrete mixed boundary value problem

“…For the general background of difference equations, one can refer to monographs [1,11,21,28]. Since the last decade, there has been much progress on the qualitative properties of difference equations, which included results on stability and attractivity and results on oscillation and other topics, see [1,[16][17][18]22,[33][34][35].…”