Positive solutions of a nonlinear three-point eigenvalue problem with integral boundary conditions

TL;DR

This paper investigates the existence of positive solutions for a nonlinear three-point eigenvalue boundary value problem involving integral conditions, using fixed point theorems and Green's functions.

Contribution

It provides new sufficient conditions for positive solutions of a nonlinear BVP with integral boundary conditions using Krasnoselskii's fixed point theorem.

Findings

Established eigenvalue intervals for the problem.

Derived sufficient conditions for positive solutions.

Applied fixed point theory to nonlinear boundary value problems.

Abstract

In this paper, we study the existence of positive solutions of a three-point integral boundary value problem (BVP) for the following second-order differential equation \begin{equation*} \begin{gathered} {u^{\prime \prime }}(t)+\lambda a(t)f(u(t))=0,\ \ 0<t<1, \\ u^{\prime}(0)=0, \ u(1)={\alpha}\int_{0}^{\eta}u(s)ds, \end{gathered} \end{equation*} where $\lambda>0$ is a parameter, $0<{\eta}<1$, $0<{\alpha}< \frac{1}{{\eta}}$. By using the properties of the Green's function and Krasnoselskii's fixed point theorem on cones, the eigenvalue intervals of the nonlinear boundary value problem are considered, some sufficient conditions for the existence of at least one positive solutions are established.