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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 3, Pages 189–216 (Mi izv8450)  

This article is cited in 4 scientific papers (total in 4 papers)

Continuous selection for set-valued mappings

I. G. Tsar'kov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study properties of set-valued mappings $F$ admitting a continuous selection $f$ which is a continuous $\epsilon$-selection (from the set of $\epsilon$-closest points) for the images $F(x)$ $(x\in X)$. This is interpreted as an $\epsilon$-selection for continuously varying sets in a space with continuously varying norms. We deduce new fixed-point theorems from the results obtained. We also study geometric-topological properties of sets all of whose $r$-neighbourhoods possess a continuous $\epsilon$-selection for every $\epsilon>0$. We obtain a characterization of such sets.

Keywords: $\epsilon$-selection, continuous selection for set-valued mappings, $\overset{ _\circ}{B}$-infinite connectedness, $\overset{ _\circ}{B}$-approximative infinite connectedness, $\overset{ _\circ}{B}$-neighbourhood infinite connectedness, fixed-point theorems.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00022-a
This paper was written with the financial support of the Russian Foundation for Basic Research (grant no. 13-01-00022-a).


DOI: https://doi.org/10.4213/im8450

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English version:
Izvestiya: Mathematics, 2017, 81:3, 645–669

Bibliographic databases:

UDC: 517.982.256
MSC: 54C60, 54C65, 54H25
Received: 21.02.2016
Revised: 11.04.2016

Citation: I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. RAN. Ser. Mat., 81:3 (2017), 189–216; Izv. Math., 81:3 (2017), 645–669

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. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859  mathnet  crossref  crossref  adsnasa  isi  elib
    2. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579  mathnet  crossref  crossref  adsnasa  isi  elib
    3. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734  mathnet  crossref  crossref  isi  elib
    4. I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238  mathnet  crossref  crossref  isi  elib
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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