On single-valuedness of set-valued maps satisfying linear inclusions
type: Item
instance of: scholarly article
title: On single-valuedness of set-valued maps satisfying linear inclusions
zbMATH Open document ID: 1163.26353
author: Kazimierz Nikodem
author: Dorian Popa
published in: Banach Journal of Mathematical Analysis
publication date: 2008-12-09T00:00:00Z
full work available at URL: https://eudml.org/doc/55508
full work available at URL: v3n1.htm
review text: Let \(X,Y\) be vector spaces, \({\mathcal P}_0(Y)\) the family of all nonempty subsets of \(Y\) and \(F:X\to {\mathcal P}_0(Y)\). The main result of the paper says that if \(F\) satisfies \(\alpha F(x)+\beta F(y)\subset F(\gamma x+\delta y\)), \(x,y\in X\), where \(\alpha ,\beta ,\gamma ,\delta \) are non-zero reals, and \(F(x_0)\) is a singleton for some \(x_0\in X\), then \(F\) is single-valued of the form \(F(x)=a(x)+c\), where \(a:X\to Y\) is additive and \(c\in Y\) is a constant. The authors also give two results on the single-valuedness of convex processes and \((\alpha ,\beta )\)-convex processes. The presented theorems generalize many earlier results.
reviewed by: Andrzej Nowak
Mathematics Subject Classification ID: 26E25
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 26A51
zbMATH DE Number: 5379948
zbMATH Keywords: Set-valued map
zbMATH Keywords: linear inclusion
zbMATH Keywords: single-valuedness
MaRDI profile type: Publication
OpenAlex ID: W2142569380
DOI: 10.15352/BJMA/1240336422
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description: scientific article; zbMATH DE number 5379948