Extremal Selections of Multifunctions Generating a Continuous Flow

TL;DR

This paper proves the existence of unique continuous solutions to certain differential equations generated by multifunctions satisfying a Lipschitz selection property, extending classical results to non-convex, compact-valued multifunctions.

Contribution

It establishes the existence of unique Carathéodory solutions for differential equations driven by multifunctions with the Lipschitz selection property, including non-convex cases.

Findings

Existence of measurable selections from extremal points of multifunctions.

Unique solutions depend continuously on initial conditions.

Lipschitz multifunctions with compact values satisfy the key property.

Abstract

Let $F:[0,T]\times\R^n\mapsto 2^{\R^n}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if $F$ satisfies the following Lipschitz Selection Property: \begin{itemize} \item[{(LSP)}] {\sl For every $t,x$, every $y\in \overline{co} F(t,x)$ and $\varepsilon>0$, there exists a Lipschitz selection $\phi$ of $\overline{co}F$, defined on a neighborhood of $(t,x)$, with $|\phi(t,x)-y|<\varepsilon$.} \end{itemize} then there exists a measurable selection $f$ of $ext F$\ such that, for every $x_0$, the Cauchy problem $$ \dot x(t)=f(t,x(t)),\qquad\qquad x(0)=x_0 $$ has a unique Caratheodory solution, depending continuously on $x_0$. We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class, for which (LSP) holds, consists of those continuous multifunctions $F$ whose values are compact and have convex closure with nonempty interior.