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Zbl 1149.47040
Park, Sehie
Elements of the KKM theory on abstract convex spaces. (English)
[J] J. Korean Math. Soc. 45, No. 1, 1-27 (2008). ISSN 0304-9914

The concept of abstract convex space is introduced as follows: a triple $(E,D,\Gamma)$ is an abstract convex space iff $E$ and $D$ are nonempty sets and $\Gamma$ is a multivalued operator with nonempty values, from the set of all nonempty finite subsets of $D$ to $E$. If $(E,D,\Gamma)$ is an abstract convex space, $Z$ is a set and $F:E\multimap Z$ is a multivalued operator with nonempty values, then a multivalued operator $G:D\multimap Z$ is said to be a KKM map with respect to $F$ if $$F(\Gamma(A))\subset G(A), \text { for each nonempty and finite subset } A\subset D.$$ A KKM theory in this setting is given. Then, as consequences, some applications for particular abstract convex spaces are presented.
[Adrian Petruşel (Cluj-Napoca)]

MSC 2000:: *47H04 Set-valued operators
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
46A16 Non-locally convex linear spaces
46A55 Convexity in topological linear spaces
52A07 Convex sets in topological vector spaces (convex geometry)
54C60 Set-valued maps
54H25 Fixed-point theorems in topological spaces
55M20 Fixed points and coincidences (algebraic topology)

Keywords: KKM operator; fixed point; coincidence point; equilibrium problem; minimax theorem

Cited in: Zbl 1156.47047 Zbl 1144.47040

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