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 Mat. Zametki, 2002, Volume 71, Issue 4, Pages 611–632 (Mi mz372)

Multi-Valued Mappings of Bounded Generalized Variation

V. V. Chistyakov

N. I. Lobachevski State University of Nizhni Novgorod

Abstract: We study the mappings taking real intervals into metric spaces and possessing a bounded generalized variation in the sense of Jordan–Riesz–Orlicz. We establish some embeddings of function spaces, the structure of the mappings, the jumps of the variation, and the Helly selection principle. We show that a compact-valued multi-valued mapping of bounded generalized variation with respect to the Hausdorff metric has a regular selection of bounded generalized variation. We prove the existence of selections preserving the properties of multi-valued mappings that are defined on the direct product of an interval and a topological space, have a bounded generalized variation in the first variable, and are upper semicontinuous in the second variable.

DOI: https://doi.org/10.4213/mzm372

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English version:
Mathematical Notes, 2002, 71:4, 556–575

Bibliographic databases:

UDC: 517.518.24+515.124
Revised: 09.02.2001

Citation: V. V. Chistyakov, “Multi-Valued Mappings of Bounded Generalized Variation”, Mat. Zametki, 71:4 (2002), 611–632; Math. Notes, 71:4 (2002), 556–575

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz372
• https://doi.org/10.4213/mzm372
• http://mi.mathnet.ru/eng/mz/v71/i4/p611

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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. Balcerzak, M, “On Helly's principle for metric semigroup valued by mappings of two real variables”, Bulletin of the Australian Mathematical Society, 66:2 (2002), 245
2. Chistyakov, VV, “The optimal form of selection principles for functions of a real variable”, Journal of Mathematical Analysis and Applications, 310:2 (2005), 609
3. V. V. Chistyakov, “A Pointwise Selection Principle for Functions of a Single Variable with Values in a Uniform Space”, Siberian Adv. Math., 16:3 (2006), 15–41
4. A. A. Vasil'eva, “Multivalent Maps with Second-Order Modulus of Continuity”, Math. Notes, 82:5 (2007), 708–712
5. Yu. V. Tretyachenko, V. V. Chistyakov, “Selection Principle for Pointwise Bounded Sequences of Functions”, Math. Notes, 84:3 (2008), 396–406
6. Chistyakov, VV, “Modular metric spaces, I: Basic concepts”, Nonlinear Analysis-Theory Methods & Applications, 72:1 (2010), 1
7. Yu. V. Tret'yachenko, “A generalization of the Helly theorem for functions with values in a uniform space”, Russian Math. (Iz. VUZ), 54:5 (2010), 35–46
8. Mielke A., Roubicek T., Rate-Independent Systems, Applied Mathematical Sciences, 193, Springer, 2015, 1–660
9. Chistyakov V.V. Chistyakova S.A., “The Joint Modulus of Variation of Metric Space Valued Functions and Pointwise Selection Principles”, Studia Math., 238:1 (2017), 37–57
10. Chistyakov V.V., “Asymmetric Variations of Multifunctions With Application”, J. Math. Anal. Appl., 478:2 (2019), 421–444
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