The existence of countably many positive solutions for some nonlinear nnth order mm-point boundary value problems

In this paper, we consider the existence of countably many positive solutions for nnth-order mm-point boundary value problems consisting of the equation u(n)(t)+a(t)f(u(t))=0,t∈(0,1), with one of the following boundary value conditions: u(0)=∑i=1m−2kiu(ξi),u′(0)=⋯=u(n−2)(0)=0,u(1)=0, and u(0)=0,u′(0)=⋯=u(n−2)(0)=0,u(1)=∑i=1m−2kiu(ξi), where n≥2,ki>0(i=1,2,…,m−2),0<ξ1<ξ2<⋯<ξm−2<1,a(t)∈Lp[0,1] for some p≥1p≥1 and has countably many singularities in [0,12). The associated Green’s function for the nnth order mm-point boundary value problem is first given, and we show that there exist countably many positive solutions using Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.