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

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.

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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

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
\Bibitem{Ber80} \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 \mathnet{http://mi.mathnet.ru/izv1696} \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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981MK41200001} 

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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
2. M. B. Lignola, J. Morgan, “Topological existence and stability for stackelberg problems”, J Optim Theory Appl, 84:1 (1995), 145
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
4. P. V. Al'brecht, “Differentiable operators of nearly best approximation”, Izv. Math., 63:4 (1999), 631–647
5. “Vitalii Ivanovich Berdyshev”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S1–S9
6. Balashov, MV, “Uniform convexity and the splitting problem for selections”, Journal of Mathematical Analysis and Applications, 360:1 (2009), 307
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
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
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