Discussiones Mathematicae Differential Inclusions 16(1)
(1996) 91-97

## RELAXATION THEOREM FOR SET-VALUED FUNCTIONS WITH DECOMPOSABLE VALUES

Andrzej Kisielewicz

*Institute of Mathematics, Technical University *

*Podgórna 50, 65-246 Zielona Góra, Poland*

*
*
## Abstract

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

**Keywords:** set-valued function, continuous selection, fixed
point,decomposability, set-valued stochastic processes.

**1991 Mathematics Subject Classification:** 54C65, 54C60.

## References

[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. |

[3] |
A. Kisielewicz, *Selection theorem for set-valued function with
decomposable values*, Comm. Math., **34** (1994), 123-135. |