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Zbl 1214.47042
Park, Sehie
The KKM principle in abstract convex spaces: equivalent formulations and applications. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 4, 1028-1042 (2010). ISSN 0362-546X

This paper is a valuable contribution to KKM theory. Precisely, the author shows that a sequence of a dozen statements characterize the KKM spaces and are equivalent formulations of the partial KKM principle. As applications, the author adds more than a dozen statements including generalized formulations of the von Neumann minimax theorem, the von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.
[Long Wei (Jiangxi)]

MSC 2000:: *47H04 Set-valued operators
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
54H25 Fixed-point theorems in topological spaces
46A16 Non-locally convex linear spaces
46A55 Convexity in topological linear spaces
49J27 Optimal control problems in abstract spaces (existence)
49J35 Minimax problems (existence)
52A07 Convex sets in topological vector spaces (convex geometry)
54C60 Set-valued maps

Keywords: abstract convex space; $G$-convex spaces; KKM space; (partial) KKM principle; minimax inequality; minimax theorem; Nash equilibrium point; variational inequality

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