Existence of three positive solutions for mm-point boundary-value problems with one-dimensional pp-Laplacian

In this paper, we consider the multipoint boundary value problem for the one-dimensional pp-Laplacian (ϕp(u′))′+q(t)f(t,u(t),u′(t))=0,t∈(0,1), subject to the boundary conditions: u(0)=0,u(1)=∑i=1m−2aiu(ξi), where ϕp(s)=|s|p−2s,p>1,ξi∈(0,1)ϕp(s)=|s|p−2s,p>1,ξi∈(0,1) with 0<ξ1<ξ2<⋯<ξm−2<10<ξ1<ξ2<⋯<ξm−2<1 and ai∈[0,1),0≤∑i=1m−2ai<1. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem. The interesting point is that the nonlinear term ff explicitly involves a first-order derivative.