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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 $\left(E,D,{\Gamma }\right)$ 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 $\left(E,D,{\Gamma }\right)$ is an abstract convex space, $Z$ is a set and $F:E⊸Z$ is a multivalued operator with nonempty values, then a multivalued operator $G:D⊸Z$ is said to be a KKM map with respect to $F$ if

$F\left({\Gamma }\left(A\right)\right)\subset G\left(A\right),\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{each}\phantom{\rule{4.pt}{0ex}}\text{nonempty}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{finite}\phantom{\rule{4.pt}{0ex}}\text{subset}\phantom{\rule{4.pt}{0ex}}A\subset D·$

A KKM theory in this setting is given. Then, as consequences, some applications for particular abstract convex spaces are presented.

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