Abstract
We consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation {−∇[ϕ(△u(t))]=λa(t,u(t))+μb(t,u(t)),t∈T,u(1)=u(N)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\nabla [\\phi (\\triangle u(t))]=\\lambda a(t,u(t))+\\mu b(t,u(t)), \\quad t\\in \\mathbb{T}, \\\\ u(1)=u(N)=0, \\end{cases} $$\\end{document} where lambda ,mu geq 0, mathbb{T}={2,ldots ,N-1} with N>3, phi (s)=s/sqrt{1-s^{2}}. The function f:=lambda a(t,s)+mu b(t,s) is either sublinear, or superlinear, or sub-superlinear near s=0. Applying the topological method, we prove the existence of either one or two, or three positive solutions.
Highlights
Let Z denote the integer set for N ∈ Z with N > 3, T := {2, . . . , N – 1}, T := {1, . . . , N}
We prove the existence of either one or two, or three positive solutions
1 Introduction Let Z denote the integer set for N ∈ Z with N > 3, T := {2, . . . , N – 1}, T := {1, . . . , N}
Summary
Let Z denote the integer set for N ∈ Z with N > 3, T := {2, . . . , N – 1}, T := {1, . . . , N}. Let Z denote the integer set for N ∈ Z with N > 3, T := {2, . We are concerned with existence and multiplicity of positive solutions for the quasilinear difference boundary value problem. Where λ, μ ≥ 0, u(t) = u(t + 1) – u(t) is the forward difference operator, ∇u(t) √= u(t) – u(t – 1) is the backward difference operator, φ : (–1, 1) → R is given by φ(s) = s/ 1 – s2, a, b : T × R → R are continuous functions. Ma, Wei, and Chen [12] studied the existence and multiplicity of positive solutions for the quasilinear two-point boundary value problem.
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