Positive and Monotone Solutions of an m-point Boundary Value Problem

Abstract

We study the second-order ordinary differential equation y''(t) = -ƒ(t, y(t), y'(t)), 0 ≤ t ≤ 1, subject to the multi-point boundary conditions αy(0) ± βy' (0) = 0, y(1) = Σm-2i=1 αiy(ξi). We prove the existence of a positive solution (and monotone in some cases) under superlinear and/or sublinear growth rate in ƒ. Our approach is based on an analysis of the corresponding vector field on the (y, y') face-plane and on Kneser's property for the solution's funnel.