Relaxation theorem for set-valued functions with decomposable values

Andrzej Kisielewicz

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1996)

  • Volume: 16, Issue: 1, page 91-97
  • ISSN: 1509-9407

Abstract

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Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has f χ A + g χ T ∈ K . Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.

How to cite

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Andrzej Kisielewicz. "Relaxation theorem for set-valued functions with decomposable values." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 16.1 (1996): 91-97. <http://eudml.org/doc/275996>.

@article{AndrzejKisielewicz1996,
abstract = {Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has $fχ_A + gχ_\{T\} ∈ K$. Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.},
author = {Andrzej Kisielewicz},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {set-valued function; continuous selection; fixed point; decomposability; set-valued stochastic processes; decomposable set},
language = {eng},
number = {1},
pages = {91-97},
title = {Relaxation theorem for set-valued functions with decomposable values},
url = {http://eudml.org/doc/275996},
volume = {16},
year = {1996},
}

TY - JOUR
AU - Andrzej Kisielewicz
TI - Relaxation theorem for set-valued functions with decomposable values
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1996
VL - 16
IS - 1
SP - 91
EP - 97
AB - Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has $fχ_A + gχ_{T} ∈ K$. Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.
LA - eng
KW - set-valued function; continuous selection; fixed point; decomposability; set-valued stochastic processes; decomposable set
UR - http://eudml.org/doc/275996
ER -

References

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  1. [1] N. Dunford, J.T. Schwartz, Linear Operators I, Int. Publ. INC., New York 1967. 
  2. [2] F. Hiai and H. Umegaki, Integrals, conditional expections and martingals of multifunctions, J. Multivariate Anal., 7 (1977), 149-182. Zbl0368.60006
  3. [3] A. Kisielewicz, Selection theorem for set-valued function with decomposable values, Comm. Math., 34 (1994), 123-135. Zbl0824.54013

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