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

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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 (ρ(t)x′(t)) = 0 with boundary conditions limtx(t)kx(ξ)=limt+x(t)lx(η)=0 and limtx(t)kx(ξ)=limt+ρ(t)x(t)lp(η)x(η)=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.

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