On measurable multifunctions with applications to random multivalued equations. (English) Zbl 0634.28005

Papageorgiou, Nikolaos S.

Math. Jap. 32, 437-464 (1987).

Let (\(\Omega\),\({\mathcal A},\mu)\) be a complete, finite measure space, let X be a separable, reflexive Banach space and let \(F: \Omega \times X\to 2^ X\setminus \{\emptyset \}\) be a set-valued function with closed and convex values such that \((i)\quad F(\cdot,x)\) is measurable, \((ii)\quad F(\omega,\cdot)\) is upper semicontinuous (with respect to the weak topology) and \((iii)\quad F(\cdot,\cdot)\) is separable (an analogue to the definition of a separable process). Under these assumptions the author shows that \(F(\cdot,\cdot)\) is \({\mathcal A}\times {\mathcal B}(X)\)- measurable, where \({\mathcal B}(X)\) are the Borel-sets on X.
Now suppose that (\(\Omega\),\({\mathcal A})\) is a measurable space with \({\mathcal A}^ a \)Souslin family and X is a Souslin metric space. Then the measurability of \(\limsup_ nF_ n\) and \(\liminf_ nF_ n\) is proved for a sequence of closed valued, measurable set-valued functions \(F_ n: \Omega \to 2^ X\setminus \{\emptyset \}.\) Other similar results are shown.
The measurability results are used to obtain new implicit function theorems [cf. J. C. Himmelberg, Fundam. Math. 87, 53-72 (1975; Zbl 0296.28003)] and to prove existence of solutions for implicit random differential inclusions and random functional-differential inclusions.

Reviewer: W.J.A.Stich

MSC:

28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections
60H25	Random operators and equations (aspects of stochastic analysis)
49J27	Existence theories for problems in abstract spaces

Keywords:

measurable multifunctions; random operator equations; random multivalued equations; measurable set-valued functions; implicit function theorems; implicit random differential inclusions; random functional-differential inclusions

Citations:

Zbl 0296.28003