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Trudy Inst. Mat. i Mekh. UrO RAN, 2010, Volume 16, Number 1, Pages 30–39 (Mi timm525)  

This article is cited in 3 scientific papers (total in 3 papers)

On implicit function theorems at abnormal points

A. V. Arutyunovab

a Peoples Friendship University of Russia
b South Mathematical Institute of VSC RAS

Abstract: We consider the equation $F(x,\sigma)=0$, $x\in K$, in which $\sigma$ is a parameter and $x$ is an unknown variable taking values in a specified convex cone $K$ lying in a Banach space $X$. This equation is investigated in a neighborhood of a given solution $(x_*,\sigma_*)$, where Robinson's constraint qualification may be violated. We introduce the 2-regularity condition, which is considerably weaker than Robinson's constraint qualification; assuming that it is satisfied, we obtain an implicit function theorem for this equation. The theorem is a generalization of the known implicit function theorems even in the case when the cone $K$ coincides with the whole space $X$.

Keywords: implicit function theorem, abnormal point, Robinson's constraint qualification, 2-regularity, 2-regularity with respect to a cone.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2010, 271, suppl. 1, S18–S27

Bibliographic databases:

Document Type: Article
UDC: 518.9+517.97
Received: 24.12.2009

Citation: A. V. Arutyunov, “On implicit function theorems at abnormal points”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 1, 2010, 30–39; Proc. Steklov Inst. Math. (Suppl.), 271, suppl. 1 (2010), S18–S27

Citation in format AMSBIB
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\by A.~V.~Arutyunov
\paper On implicit function theorems at abnormal points
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2010
\vol 16
\issue 1
\pages 30--39
\mathnet{http://mi.mathnet.ru/timm525}
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\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2010
\vol 271
\issue , suppl. 1
\pages S18--S27
\crossref{https://doi.org/10.1134/S0081543810070023}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79953236186}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Arutyunov, “Smooth abnormal problems in extremum theory and analysis”, Russian Math. Surveys, 67:3 (2012), 403–457  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Zhukovskii S.E., Mingaleeva Z.T., “Suschestvovanie i nepreryvnost neyavnoi funktsii v okrestnosti anormalnoi tochki”, Vestnik moskovskogo universiteta. seriya 15: vychislitelnaya matematika i kibernetika, 2 (2012), 10–15  mathscinet  elib
    3. Gfrerer H., Mordukhovich B.S., “Robinson Stability of Parametric Constraint Systems Via Variational Analysis”, SIAM J. Optim., 27:1 (2017), 438–465  crossref  mathscinet  zmath  isi  scopus
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