Convergence of minimal and approximate minimal elements of sets in partially ordered vector spaces
type: Item
instance of: scholarly article
title: Convergence of minimal and approximate minimal elements of sets in partially ordered vector spaces
zbMATH Open document ID: 0997.90100
author: Pierre Loridan
author: Jacqueline Morgan
author: Roberto Raucci
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-01-27T00:00:00Z
review text: Let \(Z\) be a real normed vector space partially ordered by a closed convex pointed cone \(C\) with nonempty interior, and let \(A\) be a nonempty set in \(Z\). In the first part of the paper the authors point out conditions of minimal character ensuring that the set \(\operatorname {Min}A= \{a\in A\mid (a-C)\cap (A\setminus\{a\})= \emptyset\}\) satisfies one of the following inclusions: \[ \begin{aligned} \operatorname {Min}A &\subseteq \text{Lim inf Min }A_n,\\ \operatorname {Min}A &\subseteq \text{Lim sup Min }A_n,\\ \text{Lim sup Min }A_n &\subseteq \operatorname {Min} A. \end{aligned} \] In the second part of the paper they associate with any \(\varepsilon\in C\) the set \(\varepsilon\)-\(S(A)= (\varepsilon+ \operatorname {Min}A-C)\cap A\) of restricted \(\varepsilon\)-minimal elements of \(A\) and the set \(\varepsilon\)-\(\widehat{S}(A)= (\varepsilon+ \operatorname {Min}A- \operatorname {int}C)\cap A\) of strict restricted \(\varepsilon\)-minimal elements of \(A\). Concerning these sets they also reveal conditions of minimal character such that one of the following inclusions holds: \[ \begin{aligned} \varepsilon\text{-} \widehat{S}(A) &\subseteq \text{Lim inf} (\varepsilon\text{-} \widehat{S}(A_n)+ C),\\ \varepsilon\text{-} \widehat{S}(A) &\subseteq \text{Lim sup} (\varepsilon\text{-} \widehat{S}(A_n)+ C),\\ \text{Lim sup } \varepsilon\text{-} S(A_n) &\subseteq \varepsilon\text{-} S(A). \end{aligned} \]
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 46B99
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 90C29
zbMATH DE Number: 1384932
zbMATH Keywords: minimal elements
zbMATH Keywords: approximate minimal elements
reviewed by: Wolfgang W. Breckner
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6579
OpenAlex ID: W1971928711
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DOI: 10.1006/JMAA.1999.6579
description: scientific article; zbMATH DE number 1384932