Positive Solutions of Nonlinear Three-Point Integral Boundary-Value Problems for Second-Order Differential Equations

TL;DR

This paper establishes the existence of positive solutions for a nonlinear second-order three-point integral boundary value problem using fixed point theorems, covering cases where the nonlinearity is superlinear or sublinear.

Contribution

It extends the theory of boundary value problems by applying Krasnoselskii's fixed point theorem to nonlinear problems with three-point integral conditions.

Findings

Existence of positive solutions under superlinear conditions.

Existence of positive solutions under sublinear conditions.

Application of Krasnoselskii's fixed point theorem in cones.

Abstract

We investigate the existence of positive solutions to the nonlinear second-order three-point integral boundary value problem \label{eq-1} {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, 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. We show the existence of at least one positive solution if $f$ is either superlinear or sublinear by applying Krasnoselskii's fixed point theorem in cones.