Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green’s function

摘要

Consider positive solutions and multiple positive solutions for a discrete nonlinear third-order boundary value problem {Δ3u(t?1)=a(t)f(t,u(t)),t∈[1,T?2]Z,Δu(0)=u(T)=0,Δ2u(η)?αΔu(T?1)=0,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ extstyleegin{cases} Delta ^{3}u(t-1)=a(t)f(t,u(t)), quad tin [1,T-2]_{mathbb{Z}}, Delta u(0)=u(T)=0,qquad Delta ^{2} u(eta )-lpha Delta u(T-1)=0, end{cases} $$end{document} which has the sign-changing Green’s function. Here T8documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$T8$end{document} is a positive integer, [1,T?1]Z={1,2,…,T?2}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$[1,T-1]_{mathbb{Z}}={1,2,dots ,T-2}$end{document}, α∈[0,1T?1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$lpha in [0, rac{1}{T-1})$end{document}, a:[0,T?2]Z→(0,+∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$a:[0,T-2]_{mathbb{Z}}o (0,+infty )$end{document}, f:[1,T?2]Z×[0,∞)→[0,∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$f:[1,T-2]_{ mathbb{Z}}imes [0,infty )o [0,infty )$end{document} is continuous.