Existence of positive solutions of four-point BVPs for one-dimensional generalized Lane-Emden systems on whole line

Abstract

This paper is concerned with four-point boundary value problems of the one-dimensional generalized Lane-Emden systems on whole lines. The Green's functions $G(t,s)$ for the problem $-(\rho(t)x'(t))'=0$ with boundary conditions $\lim\limits_{t\to-\infty}x(t)-kx(\xi)= \lim\limits_{t\to +\infty}x(t)-l x(\eta)=0$ and $\lim\limits_{t\to-\infty}x(t)-kx(\xi)= \lim\limits_{t\to +\infty}\rho(t)x'(t)-l \rho(\eta)x'(\eta)=0$ are obtained respectively. We proved that $G(t,s)\ge 0$ under some assumptions which actually generalize a corresponding result in [J. Math. Anal. Appl. 305 (2005) 253-276]. Sufficient conditions to guarantee the existence of positive solutions of this kind of models are established. Examples are given at the end of the paper.