Multiple positive solutions for some multi-point boundary value problems with pp-Laplacian

This paper deals with the existence of multiple positive solutions for the quasilinear second-order differential equation(φp(u′(t)))′+a(t)f(t,u(t))=0,t∈(0,1),subject to one of the following boundary conditions:φp(u′(0))=∑i=1m-2aiφp(u′(ξi)),u(1)=∑i=1m-2biu(ξi),oru(0)=∑i=1m-2aiu(ξi),φp(u′(1))=∑i=1m-2biφp(u′(ξi)),where φp(s)=|s|p-2s,p>1,0<ξ1<ξ2<⋯<ξm-2<1φp(s)=|s|p-2s,p>1,0<ξ1<ξ2<⋯<ξm-2<1, and ai,biai,bi satisfy ai,bi∈[0,∞)ai,bi∈[0,∞), (i=1,2,…,m-2)(i=1,2,…,m-2), 0<∑i=1m-2ai<1, 0<∑i=1m-2bi<1. Using the five functionals fixed point theorem, we provide sufficient conditions for the existence of multiple (at least three) positive solutions for the above boundary value problems.