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

Abstract

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} $$\\begin{aligned} \ extstyle\\begin{cases} \\Delta \\phi _{1} (\\Delta u(t-1) )+\ abla _{u} F(t,u(t),v(t))=0, \\\\ \\Delta \\phi _{2} (\\Delta v(t-1) )+\ abla _{v} F(t,u(t),v(t))=0, \\end{cases}\\displaystyle \\end{aligned}$$ \\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.