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Mathematics of the USSR-Sbornik, 1977, Volume 32, Issue 4, Pages 401–412
DOI: https://doi.org/10.1070/SM1977v032n04ABEH002394
(Mi sm2918)
 

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

Stability of a minimization problem under perturbation of the set of admissible elements

V. I. Berdyshev
References:
Abstract: Let $F$ be a continuous real functional on the space $X$. Continuity of the operator $\mathcal F$ from $2^X$ into itself is considered, where $\mathcal F(M)=\bigl\{x\in M:F(x)=\inf F(M)\bigr\}$ for each $M\in 2^X$. In particular, in the case of a normed space $X$ the following is proved. Write
$$ AB=\sup_{x\in A}\inf_{y\in B}\|x-y\|,\qquad h(A,B)=\max\{AB,BA\},\qquad(A,B\subset X), $$
and let $\mathcal M$ be the totality of all closed convex sets in $X$. A set $M\subset X$ is called approximately compact if every minimizing sequence in $M$ contains a subsequence converging to an element of $M$.
Suppose $X$ is reflexive, $F$ is convex and the set $\bigl\{x\in X:F(x)\leqslant r\bigr\}$ is bounded for $r>\inf F(X)$ and contains interior points. Then the following assertions are equivalent:
a) $M_\alpha,M\in\mathcal M$, $h(M_\alpha,M)\to0\Rightarrow\mathcal F(M_\alpha)\mathcal F(M)\to0$,
b) every set $M\in\mathcal M$ is approximately compact.
Bibliography: 15 titles.
Received: 25.10.1976
Bibliographic databases:
UDC: 519.3
MSC: 49A25, 49A30
Language: English
Original paper language: Russian
Citation: V. I. Berdyshev, “Stability of a minimization problem under perturbation of the set of admissible elements”, Math. USSR-Sb., 32:4 (1977), 401–412
Citation in format AMSBIB
\Bibitem{Ber77}
\by V.~I.~Berdyshev
\paper Stability of a~minimization problem under perturbation of the set of admissible elements
\jour Math. USSR-Sb.
\yr 1977
\vol 32
\issue 4
\pages 401--412
\mathnet{http://mi.mathnet.ru//eng/sm2918}
\crossref{https://doi.org/10.1070/SM1977v032n04ABEH002394}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=493666}
\zmath{https://zbmath.org/?q=an:0368.90118}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1977GL81400001}
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  • https://doi.org/10.1070/SM1977v032n04ABEH002394
  • https://www.mathnet.ru/eng/sm/v145/i4/p467
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    Abstract page:458
    Russian version PDF:127
    English version PDF:17
    References:70
     
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