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

Andrzej Smajdor. "Hukuhara's differentiable iteration semigroups of linear set-valued functions." Annales Polonici Mathematici 83.1 (2004): 1-10. <http://eudml.org/doc/280771>.

@article{AndrzejSmajdor2004,
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 $\{F^\{t\}: t ≥ 0\}$ of continuous linear set-valued functions $F^\{t\}: K → cc(K)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $Φ(t,x) = F^\{t\}(x)$ is a solution of the problem $D_\{t\}Φ(t,x) = Φ(t,G(x)) := ⋃ \{Φ(t,y): y ∈ G(x)\}$, Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_\{t\}Φ(t,x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G(x) := lim_\{s → 0+\} (F^\{s\}(x) - x)/s$ for x ∈ K.},
author = {Andrzej Smajdor},
journal = {Annales Polonici Mathematici},
keywords = {linear set-valued functions; iterations; Hukuhara's derivative; Banach space; differentiable iteration semigroup},
language = {eng},
number = {1},
pages = {1-10},
title = {Hukuhara's differentiable iteration semigroups of linear set-valued functions},
url = {http://eudml.org/doc/280771},
volume = {83},
year = {2004},
}

TY - JOUR
AU - Andrzej Smajdor
TI - Hukuhara's differentiable iteration semigroups of linear set-valued functions
JO - Annales Polonici Mathematici
PY - 2004
VL - 83
IS - 1
SP - 1
EP - 10
AB - 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 ${F^{t}: t ≥ 0}$ of continuous linear set-valued functions $F^{t}: K → cc(K)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $Φ(t,x) = F^{t}(x)$ is a solution of the problem $D_{t}Φ(t,x) = Φ(t,G(x)) := ⋃ {Φ(t,y): y ∈ G(x)}$, Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_{t}Φ(t,x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G(x) := lim_{s → 0+} (F^{s}(x) - x)/s$ for x ∈ K.
LA - eng
KW - linear set-valued functions; iterations; Hukuhara's derivative; Banach space; differentiable iteration semigroup
UR - http://eudml.org/doc/280771
ER -