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

Abstract

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),ax(0)-ßx'(0)=dx(1)+?x'(1)=0,-y''=g(t,x),y(0)=ay(?1),y(1)=by(?2),where 0<?1<?2<1,a,ß,d,?,a,b are nonnegative constants such that ?=ß?+a?+ad>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.