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

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)=βu(η),\ u(T)=α\int_{0}^ηu(s)ds, where $0<\eta <T$, $0<\alpha <\frac{2T}{{\eta }^2}$, $0\le \beta <\frac{2T-\alpha {\eta }^2}{\alpha {\eta }^2-2\eta +2T}$ 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.

arXiv comment12 pages, (v3) new section added (examples)