A Global Bifurcation Theorem for Convex-Valued Differential Inclusions

Abstract

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

The assumptions refer to the appropriate asymptotic behaviour of $\phi (t,x,y)$ 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 $\phi$ satisfying the assumptions of the given theorem but not satisfying the Bernstein conditions.