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Izv. Akad. Nauk SSSR Ser. Mat., 1980, Volume 44, Issue 3, Pages 483–509 (Mi izv1696)  

This article is cited in 8 scientific papers (total in 8 papers)

Continuity of a multivalued mapping connected with the problem of minimizing a functional

V. I. Berdyshev


Abstract: Let $X$ and $U$ be locally convex spaces, $\varphi(x,u)$ a proper convex lower semicontinuous functional on $X\times U$ and $t=t(u)\geqslant\inf\{\varphi(x,u)\colon x\in X\}$. This paper gives conditions for the multivalued mapping
$$ \Phi_t\colon u\in U\to \Phi_t(u)=\{x\in X\colon\varphi(x,u)\leqslant t\} $$
to be uniformly continuous and satisfy a Lipschitz condition, and determines the relation of $\Phi_t$ with other multivalued mappings, in particular, with a metric projection. On the basis of the functional conjugate to $\varphi$ a mapping conjugate to $\Phi_t$ is introduced and a condition for its upper semicontinuity is presented. The problem of minimizing a homogeneous convex functional on a convex set is considered.
Bibliography: 21 titles.

Full text: PDF file (2388 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1981, 16:3, 431–456

Bibliographic databases:

UDC: 519.3.81
MSC: Primary 46A05, 46A20, 46A55; Secondary 49A27
Received: 10.04.1978

Citation: V. I. Berdyshev, “Continuity of a multivalued mapping connected with the problem of minimizing a functional”, Izv. Akad. Nauk SSSR Ser. Mat., 44:3 (1980), 483–509; Math. USSR-Izv., 16:3 (1981), 431–456

Citation in format AMSBIB
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\by V.~I.~Berdyshev
\paper Continuity of a~multivalued mapping connected with the problem of minimizing a~functional
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1980
\vol 44
\issue 3
\pages 483--509
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=582157}
\zmath{https://zbmath.org/?q=an:0468.90084|0443.90100}
\transl
\jour Math. USSR-Izv.
\yr 1981
\vol 16
\issue 3
\pages 431--456
\crossref{https://doi.org/10.1070/IM1981v016n03ABEH001317}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Marinov, “Stability estimates of continuous selections for metric almost-projections”, Math. Notes, 55:4 (1994), 367–371  mathnet  crossref  mathscinet  zmath  isi
    2. M. B. Lignola, J. Morgan, “Topological existence and stability for stackelberg problems”, J Optim Theory Appl, 84:1 (1995), 145  crossref  mathscinet  zmath  isi
    3. A. V. Marinov, “The Lipschitz constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness”, Izv. Math., 62:2 (1998), 313–318  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. P. V. Al'brecht, “Differentiable operators of nearly best approximation”, Izv. Math., 63:4 (1999), 631–647  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. “Vitalii Ivanovich Berdyshev”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S1–S9  mathnet  crossref  isi
    6. Balashov, MV, “Uniform convexity and the splitting problem for selections”, Journal of Mathematical Analysis and Applications, 360:1 (2009), 307  crossref  isi  elib
    7. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730  mathnet  crossref  mathscinet
    8. A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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