Existence of positive solutions for a three-point integral boundary-value problem

TL;DR

This paper investigates the existence of positive solutions for a three-point integral boundary value problem using Krasnosel'skii's fixed-point theorem, establishing conditions for one or two solutions.

Contribution

It introduces new existence results for positive solutions of a specific three-point integral boundary value problem using fixed-point theory.

Findings

Existence of at least one positive solution under certain conditions.

Existence of two positive solutions under additional constraints.

Conditions on parameters for solution existence are explicitly derived.

Abstract

In this paper, by using the Krasnosel'skii's fixed-point theorem, we study the existence of at least one or two positive solutions to the three-point integral boundary value problem {equation*} \label{eq-1} {gathered} {u^{\prime \prime}}(t)+a(t)f(u(t))=0,\ 0<t<T, u(0)={\beta}u(\eta),\ u(T)={\alpha}\int_{0}^{\eta}u(s)ds, {gathered} {equation*} where $0<{\eta}<T$, $0<{\alpha}< \frac{2T}{{\eta}^{2}}$, $0\leq{\beta}<\frac{2T-\alpha\eta^{2}}{\alpha\eta^{2}-2\eta+2T}$ are given constants.