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Elements of the KKM theory on abstract convex spaces. (English) Zbl 1149.47040
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.

47H04Set-valued operators
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
46A16Non-locally convex linear spaces
46A55Convex sets in topological linear spaces; Choquet theory
52A07Convex sets in topological vector spaces (convex geometry)
54C60Set-valued maps (general topology)
54H25Fixed-point and coincidence theorems in topological spaces
55M20Fixed points and coincidences (algebraic topology)
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