Concave iteration semigroups of linear continuous set-valued functions

Andrzej Smajdor; Wilhelmina Smajdor

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 2272-2282
  • ISSN: 2391-5455

Abstract

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Let F t: t ≥ 0 be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and G ( x ) = lim s → 0 F 0 x - F s x F 0 x - F s x - s - s for x ∈ K.

How to cite

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Andrzej Smajdor, and Wilhelmina Smajdor. "Concave iteration semigroups of linear continuous set-valued functions." Open Mathematics 10.6 (2012): 2272-2282. <http://eudml.org/doc/269048>.

@article{AndrzejSmajdor2012,
abstract = {Let F t: t ≥ 0 be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and \[G(x) = \mathop \{\lim \}\limits \_\{s \rightarrow 0\} \{\{\left( \{F^0 \left( x \right) - F^s \left( x \right)\} \right)\} \mathord \{\left\bad. \{\vphantom\{\{\left( \{F^0 \left( x \right) - F^s \left( x \right)\} \right)\} \{\left( \{ - s\} \right)\}\}\} \right. \hspace\{0.0pt\}\} \{\left( \{ - s\} \right)\}\}\] for x ∈ K.},
author = {Andrzej Smajdor, Wilhelmina Smajdor},
journal = {Open Mathematics},
keywords = {Concave iteration semigroup; Derivatives of set-valued functions; Riemann integral of set-valued functions; concave iteration semigroup; derivatives of set-valued functions},
language = {eng},
number = {6},
pages = {2272-2282},
title = {Concave iteration semigroups of linear continuous set-valued functions},
url = {http://eudml.org/doc/269048},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Andrzej Smajdor
AU - Wilhelmina Smajdor
TI - Concave iteration semigroups of linear continuous set-valued functions
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 2272
EP - 2282
AB - Let F t: t ≥ 0 be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and \[G(x) = \mathop {\lim }\limits _{s \rightarrow 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord {\left\bad. {\vphantom{{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \hspace{0.0pt}} {\left( { - s} \right)}}\] for x ∈ K.
LA - eng
KW - Concave iteration semigroup; Derivatives of set-valued functions; Riemann integral of set-valued functions; concave iteration semigroup; derivatives of set-valued functions
UR - http://eudml.org/doc/269048
ER -

References

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