A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions

In this paper, we concert with the existence of positive solution for the following nonlinear singular differential system with four-point boundary conditions-x″=f(t,y),-y″=g(t,x),αx(0)-βx′(0)=δx(1)+γx′(1)=0,y(0)=ay(ξ1),y(1)=by(ξ2),where 0<ξ1<ξ2<1,α,β,δ,γ,a,b0<ξ1<ξ2<1,α,β,δ,γ,a,b are nonnegative constants such that ρ=βγ+αγ+αδ>0ρ=βγ+αγ+αδ>0. By structuring upper and lower solution and using Schauder fixed point theorem, a necessary and sufficient condition for the existence of positive solutions is established. An example is worked out to illustrate our main result.