Existence of periodic solutions for a class of \((\phi _{1},\phi _{2})\)-Laplacian difference system with asymptotically \((p,q)\)-linear conditions

In this paper, we consider a $({\varphi }_1,{\varphi }_2)$ $(\phi _{1},\phi _{2})$ -Laplacian system as follows: $\{\begin{array}{c}\mathrm{\Delta }{\varphi }_1(\mathrm{\Delta }u(t-1))+{\mathrm{\nabla }}_{u}F(t,u(t),v(t))=0,\hfill \\ \mathrm{\Delta }{\varphi }_2(\mathrm{\Delta }v(t-1))+{\mathrm{\nabla }}_{v}F(t,u(t),v(t))=0,\hfill \end{array}$ \(\begin{aligned} \textstyle\begin{cases} \Delta \phi _{1} (\Delta u(t-1) )+\nabla _{u} F(t,u(t),v(t))=0, \\ \Delta \phi _{2} (\Delta v(t-1) )+\nabla _{v} F(t,u(t),v(t))=0, \end{cases}\displaystyle \end{aligned}\) where $F(t,u(t),v(t))=-K(t,u(t),v(t))+W(t,u(t),v(t))$ $F(t,u(t),v(t))=-K(t,u(t),v(t))+W(t,u(t),v(t))$ is T-periodic in t. By using the mountain pass theorem, we obtain that the $({\varphi }_1,{\varphi }_2)$ $(\phi _{1},\phi _{2})$ -Laplacian system has at least one periodic solution if W is asymptotically $(p,q)$ $(p,q)$ -linear at infinity. Our results improve and extend some known works.