Positive solutions for second-order m-point boundary value problems

By using the Schauder fixed point theorem and analysis method, we establish the existence of solutions for the m-point boundary value problemu″(t)+a(t)f(u)=0,u(0)=0, u(1)−∑i=1m−2 kiu(ξi)=b,where b,ki>0 (i=1,2,…,m−2),0<ξ1<ξ2<⋯<ξm−2<1, a(t) is allowed to be singular at t=0,1. Under some conditions, we show that there exists a positive number b∗ such that the problem has at least one positive solution for 0<b<b∗ and no solution for b>b∗.