Necessary and Sufficient Condition for the Existence of Solutions to a Discrete Second-Order Boundary Value Problem

Abstract

This paper is concerned with the existence of solutions for the discrete second-order boundary value problem ${\Delta }^{2}u(t-1)+{\lambda }_{1}u\left(t\right)+g\left(\Delta u\right(t\left)\right)=f\left(t\right)$, $t\in \{\mathrm{1,2},\dots ,T\}$, $u\left(0\right)=u(T+1)=0$, where $T>1$ is an integer, $f:\{1,\dots ,T\}\to \mathbb{R}$, $g:\mathbb{R}\to \mathbb{R}$ is bounded and continuous, and ${\lambda }_1$ is the first eigenvalue of the eigenvalue problem ${\Delta }^{2}u(t-1)+\lambda u\left(t\right)=0$, $t\in \mathbb{T}$, $u\left(0\right)=u(T+1)=0$.