Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 18, pp. 1-16.
Title: Positive and monotone solutions of an m-point boundary-value problem
Author: Panos K. Palamides (Naval Academy of Greece)
Abstract:
We study the second-order ordinary differential equation
$$ y''(t)=-f(t,y(t),y'(t)),\quad 0\leq t\leq 1, $$
subject to the multi-point boundary conditions
$$
\alpha y(0)\pm \beta y'(0)=0,\quad y(1)=\sum_{i=1}^{m-2}\alpha_iy(\xi_i)\,.
$$
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.
Submitted January 10, 2002. Published February 18, 2002.
Math Subject Classifications: 34B10, 34B18, 34B15.
Key Words: multipoint boundary value problems; positive monotone
solution; vector field; sublinear; superlinear;
Kneser's property; solution's funel.