Symmetric Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem

Abstract

In this paper, we consider the following second-order three-point boundary value problem

$$ \begin{array}{*{20}c} {{{u}\ifmmode{''}\else$''$\fi{\left( t \right)} + a{\left( t \right)}f{\left( {u{\left( t \right)}} \right)} = 0,}} & {{0 < t < 1,}} \\ {{u{\left( 0 \right)} - u{\left( 1 \right)} = 0,}} & {{{u}\ifmmode{'}\else$'$\fi{\left( 0 \right)} - {u}\ifmmode{'}\else$'$\fi{\left( 1 \right)} = u{\left( {1/2} \right)},}} \\ \end{array} $$

where a : (0, 1) → [0, ∞) is symmetric on (0, 1) and may be singular at t = 0 and t = 1, f : [0, ∞) → [0, ∞) is continuous. By using Krasnoselskii’s fixed point theorem in a cone, we get some existence results of positive solutions for the problem. The associated Green’s function for the three-point boundary value problem is also given.