Existence of multiple positive solutions for one-dimensional p-Laplacian

In this paper we consider the multipoint boundary value problem for one-dimensional p-Laplacian(ϕp(u′))′+f(t,u)=0,t∈(0,1), subject to the boundary value conditions:ϕp(u′(0))=∑i=1n−2aiϕp(u′(ξi)),u(1)=∑i=1n−2biu(ξi), where ϕp(s)=|s|p−2sϕp(s)=|s|p−2s, p>1p>1, ξi∈(0,1)ξi∈(0,1) with 0<ξ1<ξ2<⋯<ξn−2<10<ξ1<ξ2<⋯<ξn−2<1, and ai,biai,bi satisfy ai,bi∈[0,∞]ai,bi∈[0,∞], 0<∑i=1n−2ai<1, and ∑i=1n−2bi<1. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple (at least three) positive solutions to the above boundary value problem.