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 Vestnik YuUrGU. Ser. Mat. Model. Progr., 2016, Volume 9, Issue 1, Pages 20–31 (Mi vyuru299)

Mathematical Modelling

On fixed point theory and its applications to equilibrium models

D. A. Serkovab

a Ural Federal University named after the first President of Russia B.N. Yeltsin, Yekaterinburg, Russian Federation
b Krasovskii Institute of Mathematics and Mechanics

Abstract: For a given set and a given (generally speaking, multivalued) mapping of this set into itself, we study the problem on the existence of fixed points of this mapping, i.e., of points contained in their images. We assume that the given set is nonempty and the given mapping is defined on the entire set. In these conditions, we give the description (redefinition) of the set of fixed points in the set-theoretic terms. This general idea is concretized for cases where the set is endowed with a topological structure and the mapping has additional properties associated with this structure. In particular, we provide necessary and sufficient conditions for the existence of fixed points of mappings with closed graph in Hausdorff topological spaces as well as in metric spaces. An example illustrating the possibilities and advantages of the proposed approach is given. The immediate applications of these results to the search of equilibrium states in game problems are also given: we describe the sets of saddle points in the minimax problem (an analogue of the Fan theorem) and of Nash equilibrium points in the game with many participants in cases where the sets of strategies of players are Hausdorff spaces or metrizable topological spaces.

Keywords: multivalued mapping; fixed point; saddle point; Nash equilibrium.

DOI: https://doi.org/10.14529/mmp160102

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Bibliographic databases:

UDC: 517.952+517.977
MSC: 47H10, 54C10, 54E45, 91B50
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Citation: D. A. Serkov, “On fixed point theory and its applications to equilibrium models”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 9:1 (2016), 20–31

Citation in format AMSBIB
\Bibitem{Ser16} \by D.~A.~Serkov \paper On fixed point theory and its applications to equilibrium models \jour Vestnik YuUrGU. Ser. Mat. Model. Progr. \yr 2016 \vol 9 \issue 1 \pages 20--31 \mathnet{http://mi.mathnet.ru/vyuru299} \crossref{https://doi.org/10.14529/mmp160102} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000371348500002} \elib{http://elibrary.ru/item.asp?id=25717230} 

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• http://mi.mathnet.ru/eng/vyuru/v9/i1/p20

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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. D. A. Serkov, “Ob odnom podkhode k analizu mnozhestva istinnosti: razmykanie predikata”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:4 (2016), 525–534
2. D. A. Serkov, “Unlocking of predicate: application to constructing a non-anticipating selection”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 283–291
3. D. A. Serkov, “K postroeniyu mnozhestva istinnosti predikata”, Izv. IMI UdGU, 50 (2017), 45–61
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