Existence of multiple positive solutions for a nonlocal boundary value problem

Abstract

Sufficient conditions are given for the existence of multiple positive solutions of a boundary value problem of the form $x''(t)+q(t)f(x(t))=0$, $t\in [0,1]$, $x(0)=0$ and $x(1)=\int_{\alpha}^{\beta}x(s)dg(s)$, where $0< \alpha < \beta < 1$. A weaker boundary value problem is used to get information on the corresponding integral operator. Then the results follow by applying the Krasnosel'skiĭ fixed point theorem on a suitable cone.