A metric approach to asymptotic analysis.
type: Item
instance of: scholarly article
title: A metric approach to asymptotic analysis.
zbMATH Open document ID: 1038.49025
published in: Bulletin des Sciences Mathématiques
publication date: 2003-12-14T00:00:00Z
review text: Having a subset \(E\) of the normed vector space \(X\), the \(\varepsilon\)-enlargement of \(E\) is the set \(C_\varepsilon(E):=\{x\in X\mid d(x,E)<\varepsilon\| x\| \}\cup\{0\}\). The subsets \(E,F\) of \(X\) are apart if there is some \(\varepsilon>0\) such that \(C_\varepsilon(E)\cap C_\varepsilon(F)\) is bounded. Moreover, the subset \(C\) of \(X\) is said to be a firm (outer) asymptotic approximation of \(E\subset X\) if for every \(\varepsilon>0\) there exists \(r>0\) such that \(E\setminus rU_X\subset C_\varepsilon(C)\); a cone \(C\) is a firm asymptotic cone of \(E\) if \(C\) is a firm asymptotic approximation of \(E\). In the present paper the author accomplishes a comprehensive study of these notions and gives applications to boundedness properties of multifunctions, closedness of the sum of two sets, existence of minimum points and of fixed points.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 2017286
zbMATH Keywords: apart sets
zbMATH Keywords: asymptotic cone
zbMATH Keywords: asymptotic function
zbMATH Keywords: boundedness
zbMATH Keywords: closedness
zbMATH Keywords: convergence
zbMATH Keywords: expansive map
zbMATH Keywords: fixed-point
zbMATH Keywords: separation
author: Jean-Paul Penot
reviewed by: Constantin Zălinescu
MaRDI profile type: Publication
full work available at URL: j.bulsci.2003.08.001
OpenAlex ID: W1985937190
cites work: STABILITY OF THE SOLUTION SET OF NON-COERCIVE VARIATIONAL INEQUALITIES
cites work: On the Dieudonné theorem in reflexive Banach spaces
cites work: On the closedness of the algebraic difference of closed convex sets.
cites work: Set-valued analysis
cites work: Noncoercive Optimization Problems
cites work: How to deal with the unbounded in optimization: Theory and algorithms
cites work: General existence theorems for unilateral problems in continuum mechanics
cites work: Continuity of the Fenchel correspondence and continuity of polarities
cites work: Q4495123
cites work: Q5588318
cites work: Q4137621
cites work: L'image de la limite supérieure d'une famille d'ensembles est-elle égale à la limite supérieure de la famille des images ?
cites work: Sur la séparation des ensembles convexes
cites work: Q3281164
cites work: Q4046801
cites work: Theory of Fréchet cones
cites work: Regular Points of Lipschitz Functions
cites work: Generalizations of Slater's constraint qualification for infinite convex programs
cites work: Q5526485
cites work: Q5732050
cites work: Recession maps and applications
cites work: Recessively compact sets: properties and uses
cites work: Convergence of asymptotic directions
cites work: Contrôle dans les inéquations variationelles elliptiques
cites work: On regularity conditions in mathematical programming
cites work: Compact nets, filters, and relations
cites work: Differentiability of Relations and Differential Stability of Perturbed Optimization Problems
cites work: Preservation of persistence and stability under intersections and operations. I: Persistence
cites work: The Cosmic Hausdorff Topology, the Bounded Hausdorff Topology and Continuity of Polarity
cites work: Duality for radiant and shady programs
cites work: Continuity of usual operations and variational convergences
cites work: Q4026301
cites work: Q5753754
cites work: Recession cones and asymptotically compact sets
DOI: 10.1016/J.BULSCI.2003.08.001
description: scientific article; zbMATH DE number 2017286