Positive and monotone solutions of an m-point boundary-value problem

• Main Author: Panos K. Palamides
• Format: Article
• Language: English
• Published: Texas State University 2002-02-01
• Series: Electronic Journal of Differential Equations
• Subjects: multipoint boundary value problems; positive monotone solution; vector field; sublinear; superlinear; Kneser's property; solution's funel.
• Online Access: http://ejde.math.txstate.edu/Volumes/2002/18/abstr.html

We study the second-order ordinary differential equation $$ y''(t)=-f(t,y(t),y'(t)),quad 0leq tleq 1, $$ subject to the multi-point boundary conditions $$ alpha y(0)pm eta y'(0)=0,quad y(1)=sum_{i=1}^{m-2}alpha_iy(xi_i),. $$ We prove the existence of a positive solution (and monotone in some cases) under superlinear and/or sublinear growth rate in $f$. 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.