Upper and Lower Solution Method for Regular Discrete Second-Order Single-Variable BVPs

TL;DR

This paper develops an upper and lower solution method for regular discrete second-order boundary value problems, establishing existence of positive solutions using fixed point theory and auxiliary problems.

Contribution

It introduces a novel approach employing upper and lower solutions combined with Brouwer Fixed Point Theorem for discrete BVPs.

Findings

Existence of positive solutions under certain conditions.

Construction of auxiliary problems with modified nonlinearities.

Application of Brouwer Fixed Point Theorem to discrete BVPs.

Abstract

This paper investigates the existence of positive solutions for regular discrete second-order single-variable boundary value problems with mixed boundary conditions, including a nonhomogeneous Dirichlet boundary condition, of the form: \begin{equation*} u^{\Delta \Delta}(t-1)+h(t,\ u(t),\ u^{\Delta}(t-1))=0 \mbox{ for }t\in[1,\ T+1];~u^{\Delta}(0)=0;~u(T+2)=g(T+2) \end{equation*} where h is continuous on $[1, T + 1] \times \mathbb{R}^2$ and $g: [0, T + 2] \to \mathbb{R}^+$ is continuous. Using the concept of upper and lower solutions, we establish conditions under which the boundary value problem admits at least one positive solution. Our approach involves constructing an auxiliary problem with a modified nonlinearity and applying Brouwer Fixed Point Theorem to a carefully defined solution operator. We prove that any solution to this auxiliary problem that remains within the bounds of the upper and lower solutions is equivalent to a solution of the original problem.