On existence and multiplicity of positive solutions to singular multi-point boundary value problems

The existence and multiplicity of positive solutions are established for the multi-point boundary value problem −(p(t)u′(t))′+F(t,u(t))=0,0<t<1,u(0)=∑j=1maju(xj),w(1)=∑j=1mbjw(xj), where w(t):=p(t)u′(t), aj,bj∈[0,+∞) with 0<∑j=1maj<1 and ∑j=1mbj<1, xj∈(0,1) with 0<x1<x2<⋯<xm<1, under certain conditions on p and F. The arguments are based upon the positivity of the Green's function and the Krasnosel'skii fixed point theorem.