Multiple periodic solutions for two classes of nonlinear difference systems involving classical (ϕ1,ϕ2)-Laplacian

Abstract

In this paper, we investigate the existence of multiple periodic solutions for two classes of nonlinear difference systems involving (φ 1 , φ 2 )-Laplacian. First, by using an important critical point theorem due to B. Ricceri, we establish an existence theorem of three periodic solutions for the first nonlinear difference system with (φ 1 , φ 2 )-Laplacian and two parameters. Moreover, for the second nonlinear difference system with (φ 1 , φ 2 )-Laplacian, by using the Clark's Theorem, we obtain a multiplicity result of periodic solutions under a symmetric condition. Finally, two examples are given to verify our theorems.

Existence of periodic solutions for a class of discrete systems with classical or bounded (phi_1,phi_2)-Laplacian

“…By using a critical point theorem due to Ricceri in [20], they obtained that (1.4) has at least three distinct T -periodic solutions, and by using the Clark's theorem, they obtained a multiplicity result of T -periodic solutions if F satisfies a symmetric condition. It is easy to see the differences between those results in [31] and our results below in this paper. Motivated by [14,15,24] and [30], in this paper, we investigate the existence of T -periodic solutions for the following system with classical or bounded (φ 1 , φ 2 )-Laplacian:…”

“…By virtue of an abstract critical point theorem in [21], they presented some existence results of homoclinic solutions for system (1.3). In [31], our second author and Wang investigated the following two classes of nonlinear difference systems with classical (φ 1 , φ 2 )-Laplacian:    µ∆ ρ 1 (t − 1)φ 1 ∆u 1 (t − 1) − µρ 3 (t)φ 3 (u 1 (t)) + ∇ u 1 W t, u 1 (t), u 2 (t) = 0, µ∆ ρ 2 (t − 1)φ 2 ∆u 2 (t − 1) − µρ 4 (t)φ 4 (u 2 (t)) + ∇ u 2 W t, u 1 (t), u 2 (t) = 0, (1.4) and ∆ γ 1 (t − 1)φ 1 ∆u 1 (t − 1) − γ 3 (t)φ 3 (|u 1 (t)|) + ∇ u 1 F t, u 1 (t), u 2 (t) = 0, ∆ γ 2 (t − 1)φ 2 ∆u 2 (t − 1) − γ 4 (t)φ 4 (|u 2 (t)|) + ∇ u 2 F t, u 1 (t), u 2 (t) = 0, where µ ∈ R, ρ i : R → R + , γ i : R → R + and φ i , i = 1, 2, 3, 4 satisfy some reasonable assumptions. By using a critical point theorem due to Ricceri in [20], they obtained that (1.4) has at least three distinct T -periodic solutions, and by using the Clark's theorem, they obtained a multiplicity result of T -periodic solutions if F satisfies a symmetric condition.…”