On positive solutions of an mm-point nonhomogeneous singular boundary value problem

This paper is concerned with the existence, nonexistence and multiplicity of positive solutions for the following second order mm-point nonhomogeneous singular boundary value problem u″(t)+a(t)f(t,u)=0,t∈(0,1),u(0)=0,u(1)−∑i=1m−2kiu(ξi)=b, where b>0,ki>0(i=1,2,…,m−2),0<ξ1<ξ2<⋯<ξm−2<1,∑i=1m−2kiξi<1,a(t) may be singular at t=0t=0 and/or t=1t=1. We show that, under suitable conditions, there exists a positive number b∗b∗ such that the above problem has at least two positive solutions for 0b∗b>b∗ by using the Krasnosel’skii–Guo fixed point theorem, the upper–lower solutions method and topological degree theory.