Existence of three positive solutions for image-point boundary-value problems with one-dimensional image-Laplacian Original Research Article

In this paper, we consider the multipoint boundary value problem for the one-dimensional pp-Laplacian View the MathML source(ϕp(u′))′+q(t)f(t,u(t),u′(t))=0,t∈(0,1), Turn MathJax on subject to the boundary conditions: View the MathML sourceu(0)=0,u(1)=∑i=1m−2aiu(ξi), Turn MathJax on 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 View the MathML sourceai∈[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.