Positive solutions for a three-point boundary value problem

The second-order three-point boundary value problem x″(t)+β2x(t)=h(t)f(t,x(t)),t∈(0,1),x′(0)=0,x(η)=x(1), is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where η∈(0,1)η∈(0,1) is a constant, h(t)h(t) is allowed to be singular at t=0t=0 and t=1t=1. The existence of positive solutions is studied by means of fixed point index theory.