On a theorem of Cellina for set valued functions
type: Item
instance of: scholarly article
title: On a theorem of Cellina for set valued functions
zbMATH Open document ID: 0648.54015
DOI: 10.1216/RMJ-1988-18-1-37
published in: Rocky Mountain Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: \textit{A. Cellina} proved [Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., VIII. Ser. 48, 412-416 (1970; Zbl 0198.280)] the following fundamental result about approximating a multifunction by continuous single valued functions. Cellina's Theorem. Let X and Y be normed linear spaces, and let C be a convex subset of X. Suppose \(\rho\) denotes the box metric in \(X\times Y\). Let \(\Gamma\) be a convex valued multifunction from C to Y. If \(\Gamma\) is Hausdorff upper semicontinuous and has totally bounded values, then for each \(\epsilon >0\) there exists a continuous function \(f: C\to Y\) for which \(h_{\rho}(f,\Gamma)<\epsilon\). The author proves the following extension of Cellina's result. Theorem. Let (X,d) be a metric space and let Y be a normed linear space. Suppose F is a continuous multifunction from X into Y with totally bounded starshaped values. If \(\Gamma\) maps isolated points to singletons, then for each \(\epsilon >0\), there exists a continuous function \(f: X\subset Y\) for which \(h_{\rho}(f,\Gamma)\leq \epsilon\). The author also proves the following sharp converse to Cellina's result. Theorem. Let X be a locally compact metric space and let Y be a complete metric space. Suppose \(\Gamma\) is a multifunction from X to Y with closed graph, and for each \(\epsilon >0\), there exists a continuous function \(f: X\subset Y\) for which \(h_{\rho}(f,\Gamma)<\epsilon\). Then \(\Gamma\) is Hausdorff upper semicontinuous and has totally bounded values (equivalently, \(\Gamma\) is upper semicontinuous and has compact values).
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 52A05
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 4058398
zbMATH Keywords: normed linear spaces
zbMATH Keywords: totally bounded values
zbMATH Keywords: totally bounded starshaped values
zbMATH Keywords: locally compact metric space
zbMATH Keywords: closed graph
author: Gerald Beer
reviewed by: Thomas B. Muenzenberger
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4058398