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Existence of positive solutions of four-point BVPs for one-dimensional generalized Lane-Emden systems on whole line. (English) Zbl 1336.34043
Summary: 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 \(\underset{t \to - \infty}{\lim}\, x(t) - kx(\xi) =\underset{t\to +\infty}{\lim}\, x(t) - lx(\eta) = 0\) and \(\underset{t \to -\infty}{\lim}\, x(t)-kx(\xi) = \underset{t \to + \infty}{\lim}\, \rho (t)x'(t) - lp(\eta )x'(\eta ) = 0\) are obtained respectively. We proved that \(G(t, s) \geq 0\) under some assumptions which actually generalize a corresponding result in [B. Liu, J. Math. Anal. Appl. 305, No. 1, 253–276 (2005; Zbl 1073.34075)]. 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.
MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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