Optimal existence criteria for symmetric positive solutions to a singular three-point boundary value problem

In this paper we are concerned with the existence of symmetric positive solutions of the following singular second order three-point boundary value problem u″(t)+h(t)f(t,u(t))=0,00,α+β>γ/2α,β,γ>0,α+β>γ/2, h:(0,1)→[0,∞)h:(0,1)→[0,∞) is symmetric on (0,1)(0,1) and may be singular at t=0t=0 and t=1t=1. First, the Green’s function for associated linear boundary value problem is constructed, and some useful properties of the Green’s function are obtained. Then by applying the fixed-point index theory, we establish some optimal criteria for the existence of one or two symmetric positive solutions which involve the principal eigenvalue of a related linear operator. Finally we illustrate our results by several examples, none of which can be handled using the existing results.