# Relaxation theorem for set-valued functions with decomposable values

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

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

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topAndrzej 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

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

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