Electron. J. Diff. Eqns., Vol. 2002(2002), No. 18, pp. 1-16.

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

Panos K. Palamides

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.

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Panos K. Palamides
Naval Academy of Greece,
Piraeus 183 03, Greece
and
Department of Mathematics, Univ. of Ioannina,
451 10 Ioannina, Greece
e-mail:ppalam@otenet.gr

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