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 - (ρ(t)x'(t))'=0 with boundary conditions lim_t→-∞x(t) - kx(ξ) = lim_t→+∞x(t) - lx(μ) = 0 and lim_t→-∞x(t) - kx(ξ) = lim_t→+∞ρ(t)x'(t) - lρ(μ)x'(μ) = 0 are obtained respectively. We proved that G(t,s)≥ 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.

Tbilisi Mathematical Journal, Vol. 8(2) (2015), pp. 257-280