Lower and Upper Solutions for System of Differential Equations Involving Homeomorphism and Nonlinear Boundary Conditions

Abstract

We study the existence of solution to the system of differential equations (ϕ(u′))′=f(t,u,u′)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi (u'))'=f(t,u,u')$$\\end{document} with nonlinear boundary conditions g(u(0),u,u′)=0,h(u′(1),u,u′)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} g(u(0),u,u')=0, \\quad h(u'(1),u,u')=0, \\end{aligned}$$\\end{document}where f:[0,1]×Rn×Rn→Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:[0,1]\ imes \\mathbb {R}^{n}\ imes \\mathbb {R}^{n}\\rightarrow \\mathbb {R}^{n}$$\\end{document}, g,h:Rn×C([0,1],Rn)×C([0,1],Rn)→Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g,h:\\mathbb {R}^{n}\ imes C([0,1],\\mathbb {R}^{n})\ imes C([0,1],\\mathbb {R}^{n})\\rightarrow \\mathbb {R}^{n}$$\\end{document} are continuous, ϕ:∏i=1n(-ai,ai)→Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi :\\prod _{i=1}^{n}(-a_i,a_i) \\rightarrow \\mathbb {R}^{n}$$\\end{document}, 0<ai≤+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<a_i\\le +\\infty $$\\end{document}, ϕ(s)=ϕ1(s1),⋯,ϕn(sn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi (s)=\\left( \\phi _1(s_1),\\dots ,\\phi _n(s_n)\\right) $$\\end{document} and ϕi:(-ai,ai)→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi _i:(-a_i,a_i)\\rightarrow \\mathbb {R}$$\\end{document} is a one dimensional regular or singular homeomorphism. Our proofs are based on the concept of the lower and upper solutions.