Existence of periodic solutions for second-order nonlinear difference equations

Abstract

By using the critical point method, the existence of periodic solutions for second-order nonlinear difference equations is obtained. The proof is based on the Saddle Point Theorem in combination with variational technique. The problem is to solve the existence of periodic solutions of second-order nonlinear difference equations. One of our results obtained complements the result in the literature.

Multiple periodic solutions for the second-order nonlinear difference equations

“…By virtue of the minimax methods with variational techniques, the solvability conditions on multiple periodic solutions are proved for difference equation when β = δ + 1. In particular, our results complement and generalize the results in [6] and [18].…”

“…which implies that such a function F satisfies condition (F3) of Theorem 3.1 but does not satisfy the corresponding condition of Theorem 3.2 in [6] and the corresponding condition of Theorem 1.3 in [18]. Moreover, our conclusion complements the results of Theorem 3.2 in [6] and Theorem 1.3 in [18].…”

“…In [20] the authors were interested in the results on the existence of positive solutions. However, to the best of our knowledge, when δ = 1 and ∇F(n, x) = 0, besides [6] and [18], in the literature there are no results on the solvability condition of periodic solutions for equation (1). By employing the mountain pass lemma, in [6] the authors proved that there are at least two nontrivial periodic solutions of equation (1) under the following conditions:…”

“…Later, by virtue of the saddle-point theorem, in [18] the authors obtained the periodic solution of equation (1) under condition (a) and the following assumptions:…”

“…Hence it is natural for us to consider the case β = δ + 1. In the present papere, the motivation comes from the recent papers [6,7,14,18]. By virtue of the minimax methods with variational techniques, the solvability conditions on multiple periodic solutions are proved for difference equation when β = δ + 1.…”