Existence of periodic solutions for a class of discrete systems with classical or bounded (φ1, φ2)-Laplacian

In this paper, we investigate the existence of periodic solutions for the nonlinear discrete system with classical or bounded (φ1, φ2)-Laplacian: {Δϕ1(Δu1(t−1))+∇u1F(t,u1(t),u2(t))=0,Δϕ2(Δu2(t−1))+∇u2F(t,u1(t),u2(t))=0. By using the saddle point theorem, we obtain that system with classical ( ϕ 1 , ϕ 2 )-Laplacian has at least one periodic solution when F has (p, q)-sublinear growth, and system with bounded ( ϕ 1 , ϕ 2 )-Laplacian has at least one periodic solution when F has ( p , q )-sublinear growth. By using the least action principle, we obtain that system with classical or bounded ( ϕ 1 , ϕ 2 )-Laplacian has at least one periodic solution when F has a growth like Lipschitz condition.