Hukuhara's differentiable iteration semigroups of linear set-valued functions
Volume 83 / 2004
Abstract
Let $K$ be a closed convex cone with nonempty interior in a real Banach space and let $cc(K)$ denote the family of all nonempty convex compact subsets of $K$. A family $\{F^{t}: t \geq 0 \}$ of continuous linear set-valued functions $F^{t}: K \to cc(K)$ is a differentiable iteration semigroup with $F^{0}(x) = \{x \}$ for $x \in K$ if and only if the set-valued function ${\mit\Phi}(t,x) = F^{t}(x)$ is a solution of the problem $$ D_{t} {\mit\Phi}(t,x) = {\mit\Phi}(t,G(x)) := \bigcup \{ {\mit\Phi}(t,y): y \in G(x) \},\quad\ {\mit\Phi}(0,x) = \{ x \}, $$ for $x \in K$ and $t \geq 0$, where $D_{t} {\mit\Phi}(t,x)$ denotes the Hukuhara derivative of ${\mit\Phi}(t,x)$ with respect to $t$ and $ G(x) := \lim_{s \to 0+} (F^{s}(x) - x)/{s} $ for $x \in K.$
