Existence uniqueness for a class of Nonlinear Discrete Boundary Value Problems

TL;DR

This paper develops a monotone iterative method supported by upper and lower solutions to establish the existence and uniqueness of solutions for nonlinear discrete boundary value problems, including a maximum principle based on Green's function.

Contribution

It introduces new existence results and a maximum principle for linear discrete boundary value problems, leading to unique solutions for specific nonlinear problems.

Findings

Established a maximum principle for linear discrete boundary problems.

Proved existence of unique solutions under certain conditions.

Developed a monotone iterative method for nonlinear problems.

Abstract

A monotone iterative method is proposed to solve nonlinear discrete boundary value problems with the support of upper and lower solutions. We establish some new existence results. Under some sufficient conditions, we establish maximum principle for linear discrete boundary value problem, which relies on Green's function and its constant sign. We then use it to establish existence of unique solution for the nonlinear discrete boundary value problem $\Delta^2 y(t-1)= f(t, y(t)),~t\in[1, T]$, $y(0)=0,~y(T+1)=0$.