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Mat. Sb., 2005, Volume 196, Number 9, Pages 3–22 (Mi msb1418)  

This article is cited in 5 scientific papers (total in 5 papers)

Inverse function theorem and conditions of extremum for abnormal problems with non-closed range

E. R. Avakova, A. V. Arutyunovb

a Institute of Control Sciences, Russian Academy of Sciences
b Peoples Friendship University of Russia

Abstract: The following two classical problems are considered: the existence and the estimate of a solution of an equation defined by a map $F$ in the neighbourhood of a point $x^*$; necessary conditions for an extremum at $x^*$ of a smooth function under equality-type constraints defined in terms of a non-linear map $F$. If the range of the first derivative of $F$ at $x^*$ is not closed, then one cannot use classical methods of analysis based on inverse function theorems and Lagrange's principle. The results on these problems obtained in this paper are of interest in the case when the range of the first derivative of $F$ at $x^*$ is non-closed; these are a further development of classical results extending them to abnormal problems with non-closed range.

DOI: https://doi.org/10.4213/sm1418

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English version:
Sbornik: Mathematics, 2005, 196:9, 1251–1269

Bibliographic databases:

UDC: 518.9+517.97
MSC: 46A99, 58C25, 49K27
Received: 17.05.2004 and 21.02.2005

Citation: E. R. Avakov, A. V. Arutyunov, “Inverse function theorem and conditions of extremum for abnormal problems with non-closed range”, Mat. Sb., 196:9 (2005), 3–22; Sb. Math., 196:9 (2005), 1251–1269

Citation in format AMSBIB
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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. E. R. Avakov, A. V. Arutyunov, A. F. Izmailov, “Necessary Conditions for an Extremum in a Mathematical Programming Problem”, Proc. Steklov Inst. Math., 256 (2007), 2–25  mathnet  crossref  mathscinet  zmath  elib  elib
    2. 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
    3. E. R. Avakov, A. V. Arutyunov, D. Yu. Karamzin, “An investigation of smooth maps in a neighbourhood of an abnormal point”, Izv. Math., 78:2 (2014), 213–250  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. E. R. Avakov, “Stability theorem and extremum conditions for abnormal problems”, Proc. Steklov Inst. Math., 291 (2015), 4–23  mathnet  crossref  crossref  isi  elib
    5. S. E. Zhukovskii, Ch. T. Ngok, “Suschestvovanie obratnoi funktsii v okrestnosti neregulyarnogo znacheniya”, Vestnik rossiiskikh universitetov. Matematika, 24:126 (2019), 141–149  mathnet  crossref  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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