Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

Abstract

we show the existence and multiplicity of positive solutions of the nonlineardiscrete fourth-order boundary value problem ${\Delta }^{4}u\left(t-2\right)=\lambda h\left(t\right)f\left(u\left(t\right)\right)$, $t\in {\mathbb{T}}_2$, $u\left(1\right)=u\left(T+1\right)={\Delta }^{2}u\left(0\right)={\Delta }^{2}u\left(T\right)=0$, where $\lambda >0$, $h:{\mathbb{T}}_2\to (0,\infty )$ is continuous, and $f:\mathbb{R}\to [0,\infty )$ is continuous, $T>4$, ${\mathbb{T}}_2=\left\{\mathrm{2,3},\dots ,T\right\}$. The main tool is the Dancer's global bifurcation theorem.