Homoclinic orbits for second order self-adjoint difference equations

Abstract

In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equationswithout any periodicity assumptions on p(t), q(t) and f , providing that f (t, x) grows superlinearly both at origin and at infinity or is an odd function with respect to x ∈ R, and satisfies some additional assumptions.

Existence of homoclinic orbits for a higher order difference system

“…So homoclinic orbits have been extensively studied since the time of Poincaré, see [7,8,17] and the references therein. Recently, Ma and Guo [14,15] have found that the trajectories which are asymptotic to a constant state as the time variable |k| → ∞ also exists in discrete dynamical systems [2-6, 11-15, 20-24, 26]. These trajectories are also called homoclinic orbits (or homoclinic solutions).…”

“…In many studies (see, e.g., [3,4,9,11,12,14,15,23]) of second order difference equations, the following classical Ambrosetti-Rabinowitz condition is assumed (AR) there exists a constant β > 2 such that 0 < βF(k, u) uf(k, u) for all k ∈ Z and u ∈ R \ {0}.…”

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

“…Therefore, the study of homoclinic orbits [10][11][12][13][14][15][16][17][18][19][20][21] of difference equation is meaningful.…”

“…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…”

“…In many studies (see e.g. [16,18,19,22,23]) of second order difference equations, the following classical Ambrosetti-Rabinowitz condition is assumed.…”