A Quest for Simple and Unified Proofs in Regularity Theory: Perturbation Stability

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F23%3A43970880" target="_blank" >RIV/49777513:23520/23:43970880 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.heldermann.de/JCA/JCA30/JCA303/jca30037.htm" target="_blank" >https://www.heldermann.de/JCA/JCA30/JCA303/jca30037.htm</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

A Quest for Simple and Unified Proofs in Regularity Theory: Perturbation Stability

Popis výsledku v původním jazyce

Ioffe’s criterion and various reformulations of it have become a standard tool in proving theorems guaranteeing metric regularity of a (set-valued) mapping. First, we demonstrate that one should always use directly the so-called general criterion which follows, for example, from Ekeland’s variational principle, and that there is no need to make a detour through the slope-based consequences of this general statement. Second, we argue that when proving perturbation stability results, in the spirit of Lyusternik-Graves theorem, there is no need to employ the concept of a lower semicontinuous envelope even in the case of an incomplete target space. The gist is to use the “correct” function to which Ekeland’s variational principle is applied; namely, the distance function to the graph of the set-valued mapping under consideration. This approach originates in the notion of graphical regularity introduced by L. Thibault, which is equivalent to the property of metric regularity. Our criteria cover also both metric subregularity and metric semiregularity, which are weaker properties obtained by fixing one of the points in the definition of metric regularity.

Název v anglickém jazyce

A Quest for Simple and Unified Proofs in Regularity Theory: Perturbation Stability

Popis výsledku anglicky

Ioffe’s criterion and various reformulations of it have become a standard tool in proving theorems guaranteeing metric regularity of a (set-valued) mapping. First, we demonstrate that one should always use directly the so-called general criterion which follows, for example, from Ekeland’s variational principle, and that there is no need to make a detour through the slope-based consequences of this general statement. Second, we argue that when proving perturbation stability results, in the spirit of Lyusternik-Graves theorem, there is no need to employ the concept of a lower semicontinuous envelope even in the case of an incomplete target space. The gist is to use the “correct” function to which Ekeland’s variational principle is applied; namely, the distance function to the graph of the set-valued mapping under consideration. This approach originates in the notion of graphical regularity introduced by L. Thibault, which is equivalent to the property of metric regularity. Our criteria cover also both metric subregularity and metric semiregularity, which are weaker properties obtained by fixing one of the points in the definition of metric regularity.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF20-11164L" target="_blank" >GF20-11164L: Regularita zobrazení a aplikace</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

JOURNAL OF CONVEX ANALYSIS

ISSN

0944-6532

e-ISSN

0944-6532

Svazek periodika

30

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

22

Strana od-do

771-792

Kód UT WoS článku

001115506800002

EID výsledku v databázi Scopus

2-s2.0-85174290108