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

[BIBTex]

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,Rn) is said to be decomposable if for every A∈F and f∈K, g∈K one has fχA+gχT\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.