Remarks on the extension of random contractions
type: Item
instance of: scholarly article
title: Remarks on the extension of random contractions
zbMATH Open document ID: 0962.54014
DOI: 10.1016/S0362-546X(98)00356-3
author: Nazeer Shahzad
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-06-11T00:00:00Z
review text: Let (\(\Omega\),\(\Sigma\)) be a measurable space with \(\Sigma\) a \(\sigma\)-algebra of subsets of \(\Omega\) and let \(F: \Omega \rightarrow 2^{X}\smallsetminus \{ \emptyset \}\) be a multifunction. We will say that \(F(.)\) is separable if there exists a countable set \(D\subset X\) such that \(F(\omega)= \overline{ F(\omega)\cap D}\), for all \(\omega \in \Omega\). Let \(\text{Gr}F\) be a graph of \(F(.)\). Then \(T:\text{Gr}F\rightarrow 2^{Y}\smallsetminus \{ \emptyset \}\) is a random operator with stochastic domain \(F(.)\) if for all \(x\in X\) and all \(U\subset Y\) open, \(\{ \omega\in \Omega : T(\omega ,x)\cap U\neq \emptyset , x\in F(\omega)\} \in \Sigma\). A function \(f\) (defined and assuming its values in metric spaces) is contraction if it is a nonexpansive mapping. The results contained in this paper are connected with the existence of the extension of some contraction functions. In particular, the main theorem of this paper is the following Let (\(\Omega\),\(\Sigma\)) be a measurable space with \(\Sigma\) a Souslin family. Let \(X\) and \(Y\) be separable complete metric spaces such that the pair (\(X,Y\)) has Kirzbraun property. Let \(F:\Omega \rightarrow 2^{X}\smallsetminus \{ \emptyset \}\) be a separable weakly measurable multifunction. Then every random contraction \(f:\text{Gr}F\rightarrow Y\) with stochastic domain \(F(.)\) can be extended to a random contraction on all \(\Omega \times X\).
Mathematics Subject Classification ID: 54C20
zbMATH DE Number: 1512515
zbMATH Keywords: random contraction
zbMATH Keywords: metric space
zbMATH Keywords: extension of map
zbMATH Keywords: Kirzbraun property
zbMATH Keywords: property (K)
reviewed by: Ryszard Pawlak
MaRDI profile type: Publication
full work available at URL: s0362-546x(98)00356-3
OpenAlex ID: W2005284081
Wikidata QID: Q126789838
description: scientific article; zbMATH DE number 1512515