Existence and non-existence of positive solutions of four-point BVPs for second order ordinary differential equations on whole line

Yuji Liu, Xiaohui Yang

This paper is concerned with four-point boundary value problems of second order singular differential equations on whole lines. The Green's function $G(t,s)$ for the problem $$\begin{array}{l} -(\rho(t)x'(t))'=0,imimits_{to-ıfty}\rho(t)x'(t)-kx(\xi)=imimits_{to +ıfty}\rho(t)x'(t)+l x(\eta)=0 \end{array} $$ is obtained. We proved that $G(t,s)\ge 0$ under some assumptions which actually generalizes a corresponding result in [Appl. Math. Comput. 217(2)(2010) 811-819]. Sufficient conditions to guarantee the existence and non-existence of positive solutions are established.