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The KKM principle in abstract convex spaces: equivalent formulations and applications. (English) Zbl 1214.47042
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.
MSC:
47H04Set-valued operators
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
46A16Non-locally convex linear spaces
46A55Convex sets in topological linear spaces; Choquet theory
49J27Optimal control problems in abstract spaces (existence)
49J35Minimax problems (existence)
52A07Convex sets in topological vector spaces (convex geometry)
54C60Set-valued maps (general topology)