Hukuhara's differentiable iteration semigroups of linear set-valued functions

Andrzej Smajdor¹

¹ Department of Mathematics Pedagogical University Podchorążych 2 30-084 Kraków, Poland

Annales Polonici Mathematici, Tome 83 (2004) no. 1, pp. 1-10

Résumé

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 $x \in K.$

Détail

DOI : 10.4064/ap83-1-1

Keywords: closed convex cone nonempty interior real banach space denote family nonempty convex compact subsets family geq continuous linear set valued functions differentiable iteration semigroup only set valued function mit phi solution problem mit phi mit phi bigcup mit phi quad mit phi geq where mit phi denotes hukuhara derivative mit phi respect lim