Existence of periodic solutions for a class of (ϕ1,ϕ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\phi _{1},\phi _{2})$\end{document}-Laplacian difference system with asymptotically (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-linear conditions

In this paper, we consider a (ϕ1,ϕ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\phi _{1},\phi _{2})$\end{document}-Laplacian system as follows: {Δϕ1(Δu(t−1))+∇uF(t,u(t),v(t))=0,Δϕ2(Δv(t−1))+∇vF(t,u(t),v(t))=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \textstyle\begin{cases} \Delta \phi _{1} (\Delta u(t-1) )+\nabla _{u} F(t,u(t),v(t))=0, \\ \Delta \phi _{2} (\Delta v(t-1) )+\nabla _{v} F(t,u(t),v(t))=0, \end{cases}\displaystyle \end{aligned}$$ \end{document} where F(t,u(t),v(t))=−K(t,u(t),v(t))+W(t,u(t),v(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F(t,u(t),v(t))=-K(t,u(t),v(t))+W(t,u(t),v(t))$\end{document} is T-periodic in t. By using the mountain pass theorem, we obtain that the (ϕ1,ϕ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\phi _{1},\phi _{2})$\end{document}-Laplacian system has at least one periodic solution if W is asymptotically (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-linear at infinity. Our results improve and extend some known works.