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Zeitschrift für Analysis und ihre Anwendungen


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Volume 23, Issue 2, 2004, pp. 275–292
DOI: 10.4171/ZAA/1198

Published online: 2004-06-30

A Global Bifurcation Theorem for Convex-Valued Differential Inclusions

S. Domachowski[1] and J. Gulgowski[2]

(1) University of Gdansk, Poland
(2) University of Gdansk, Poland

\newcommand\f{\phi} We prove a global bifurcation theorem for convex-valued completely continuous maps. Basing on this theorem we prove an existence theorem for convex-valued differential inclusions with Sturm-Liouville boundary conditions $$u''(t) \in \f(t,u(t),u'(t))\ \ \hbox{for a.e.}\ \ t \in (a,b)$$ $$l(u) = 0$$ The assumptions refer to the appropriate asymptotic behaviour of $\f(t,x,y)$ for $|x| + |y|$ close to $0$ and to $+\infty$, and they are independent from the well known Bernstein-type conditions. In the last section we give a set of examples of $\f$ satisfying the assumptions of the given theorem but not satisfying the Bernstein conditions.

Keywords: Differential inclusions, Sturm-Liouville boundary conditions, global bifurcation

Domachowski S., Gulgowski J.: A Global Bifurcation Theorem for Convex-Valued Differential Inclusions. Z. Anal. Anwend. 23 (2004), 275-292. doi: 10.4171/ZAA/1198