Abstract
In this paper, we obtain the global structure of positive solutions for nonlinear discrete simply supported beam equation \t\t\tΔ4u(t−2)=λf(t,u(t)),t∈T,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=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} $$\\begin{aligned}& \\Delta ^{4}u(t-2)= \\lambda f\\bigl(t,u(t)\\bigr),\\quad t\\in \\mathbb{T}, \\\\& u(1)=u(T+1)=\\Delta ^{2}u(0)=\\Delta ^{2}u(T)=0, \\end{aligned}$$ \\end{document} with fin C(mathbb{T}times [0,infty ),[0,infty )) satisfying local linear growth condition and f(t,0)=0 uniformly for tin mathbb{T}, where mathbb{T}={2,ldots,T}, lambda >0 is a parameter. The main results are based on the global bifurcation theorem.
Highlights
It is well known that the fourth-order two-point boundary value problem u(4)(t) = f t, u(t), t ∈ (0, 1), (1.1)
U(0) = u(1) = u (0) = u (1) = 0, appears in the theory of hinged beams [2, 3], so the existence and multiplicity of positive solutions for (1.1) and its discrete analog have been studied by many authors; see, for example, [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]
In 2002, Zhang et al [10] and He et al [11] studied the existence of positive solutions for the following discrete analog: 4u(t – 2) = λh(t)f u(t), t ∈ [2, T]Z, (1.2)
Summary
Difference equations usually describe the evolution of certain phenomena over the course of time, which often occur in numerous settings and forms, both in mathematics and in its applications to economics, statistics, biology, numerical computing, electrical circuit analysis, and other fields; see [1]. Ma and Lu [13] applied the Dancer’s global bifurcation theorem to obtain some new results on the existence and multiplicity of generalized positive solutions of discrete supported beam equation (1.4) with λ = 1. In these papers, they assumed that the nonlinearity f ∈ C([2, T]Z × [0, ∞), [0, ∞)) satisfies f (t, u) ≥ 0 on [2, T]Z × [0, ∞) and f (t, u) > 0 on [2, T]Z × (0, ∞).
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