# A Global Bifurcation Theorem for Convex-Valued Differential Inclusions

### S. Domachowski

University of Gdansk, Poland### J. Gulgowski

University of Gdansk, Poland

## Abstract

\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.

## Cite this article

S. Domachowski, J. Gulgowski, A Global Bifurcation Theorem for Convex-Valued Differential Inclusions. Z. Anal. Anwend. 23 (2004), no. 2, pp. 275–292

DOI 10.4171/ZAA/1198