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 Mat. Zametki, 2018, Volume 104, Issue 5, Pages 745–754 (Mi mz11821)

New Criteria for the Existence of a Continuous $\varepsilon$-Selection

I. G. Tsar'kov

Lomonosov Moscow State University

Abstract: We study sets admitting a continuous selection of near-best approximations and characterize those sets in Banach spaces for which there exists a continuous $\varepsilon$-selection for each $\varepsilon>0$. The characterization is given in terms of $P$-cell-likeness and similar properties. In particular, we show that a closed uniqueness set in a uniformly convex space admits a continuous $\varepsilon$-selection for each $\varepsilon>0$ if and only if it is $\mathring{B}$-approximately trivial. We also obtain a fixed point theorem.

Keywords: continuous $\varepsilon$-selection, fixed point.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00295-a This work was supported by the Russian Foundation for Basic Research under grant 16-01-00295-a.

DOI: https://doi.org/10.4213/mzm11821

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English version:
Mathematical Notes, 2018, 104:5, 727–734

Bibliographic databases:

UDC: 517.982.256
Revised: 06.02.2018

Citation: I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Mat. Zametki, 104:5 (2018), 745–754; Math. Notes, 104:5 (2018), 727–734

Citation in format AMSBIB
\Bibitem{Tsa18} \by I.~G.~Tsar'kov \paper New Criteria for the Existence of a Continuous $\varepsilon$-Selection \jour Mat. Zametki \yr 2018 \vol 104 \issue 5 \pages 745--754 \mathnet{http://mi.mathnet.ru/mz11821} \crossref{https://doi.org/10.4213/mzm11821} \elib{http://elibrary.ru/item.asp?id=36361320} \transl \jour Math. Notes \yr 2018 \vol 104 \issue 5 \pages 727--734 \crossref{https://doi.org/10.1134/S0001434618110147} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000454546800014} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85059230186} 

• http://mi.mathnet.ru/eng/mz11821
• https://doi.org/10.4213/mzm11821
• http://mi.mathnet.ru/eng/mz/v104/i5/p745

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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. I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008
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