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

Result code in 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>
Result on the web
<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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Result language
angličtina
Original language name
A Quest for Simple and Unified Proofs in Regularity Theory: Perturbation Stability
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GF20-11164L" target="_blank" >GF20-11164L: Regularity properties of mappings and applications</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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