Approximation from the exterior of multifunctions with connected values
type: Item
instance of: scholarly article
title: Approximation from the exterior of multifunctions with connected values
zbMATH Open document ID: 0826.26012
DOI: 10.1007/BF01033068
author: Carlo Benassi
author: Andrea Gavioli
published in: Set-Valued Analysis
publication date: 1995-11-19T00:00:00Z
review text: Let \(({\mathcal M}, d)\) be a metric space and \(\mathcal K\) be the family of all compact connected subsets of \(\mathbb{R}^n\). Equip \(\mathcal K\) with the Hausdorff metric \(h\). Then one can define the set \(\text{Lip}({\mathcal M}, {\mathcal K})\) of Lipschitz (multi-) functions \(f: ({\mathcal M}, d)\to ({\mathcal K}, h)\). Theorem. For every upper semicontinuous multi-function \(F: ({\mathcal M}, d)\to ({\mathcal K}, h)\) there exists a sequence \(\{F_n\}_{n\in \mathbb{N}}\subset \text{Lip}({\mathcal M}, {\mathcal K})\) such that \[ F(t)\leq F_{n+ 1}(t)\leq F_n(t)\tag{a} \] for all \(t\in {\mathcal M}\) and \(n\in \mathbb{N}\); \[ \lim_{n\to+ \infty} h(F_n(t), F(t))= 0\tag{b} \] for every \(t\in {\mathcal M}\). A function \(G: {\mathcal M}\to {\mathcal K}\) is said to be upper semicontinuous if for every \(t\in {\mathcal M}\) and \(\varepsilon> 0\) there is an open set \(U\ni t\) such that \(F(t')\) for every \(t'\in U\) lies in the \(\varepsilon\)- neighbourhood of \(F(t)\).
Mathematics Subject Classification ID: 26E25
Mathematics Subject Classification ID: 28B20
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 720903
zbMATH Keywords: upper semicontinuity
zbMATH Keywords: approximation by Lipschitz function
zbMATH Keywords: multi-function
reviewed by: Yuri A. Brudnyi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 720903