Existence of periodic solutions for a 2nth-order nonlinear difference equation

Abstract

The authors consider the 2nth-order difference equationwhere f : Z × R → R is a continuous function in the second variable, f (t + T , z) = f (t, z) for all (t, z) ∈ Z × R, r t+T = r t for all t ∈ Z, and T a given positive integer. By the Linking Theorem, some new criteria are obtained for the existence and multiplicity of periodic solutions of the above equation.

Existence of solutions to boundary value problems for a higher-dimensional difference system

“…In 2007, Cai and Yu [2] established some criteria for the existence of periodic solutions of a 2n-th order difference equation…”

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…”

“…However, to the best of our knowledge, the results on homoclinic solutions of higher-order nonlinear difference equations are scarce in the literature [10,11,15,20,24,27]. Furthermore, since (1) contains both advance and retardation, there are very few manuscripts dealing with this subject.…”

Periodic solutions of a 2nth-order nonlinear difference equation

“…Only since 2003, the critical point theory (the powerful tool used to study the existence of periodic solutions of differential equations) has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. In particular, Yu and his collaborators considered second-order nonlinear difference equations [18,19,28], fourth-order [7] and 2nth-order [6] superlinear difference equations.…”