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On generalizations of the KKM principle on abstract convex spaces. (English) Zbl 1120.47038
Let $\langle D\rangle$ denote the set of all nonempty finite subsets of a set $D$. An abstract convex space $(E,D;\Gamma)$ consists of a nonempty set $E$, a nonempty set $D$, and a multimap $\Gamma:\langle D\rangle\multimap E$ with nonempty values. Let $(E,D;\Gamma)$ be an abstract convex space and $Z$ a set. For a multimap $F: E\multimap Z$ with nonempty values, if a multimap $G: D\multimap Z$ satisfies $$F[\Gamma(A)]\subset G(A):= \bigcup_{y\in A} G(y)\quad\text{for all }A\in\langle D\rangle,$$ then $G$ is called a KKM map with respect to $F$. A KKM map $G: D\multimap E$ is a KKM map with respect to the identity map $1_E$. In this paper, the author introduces abstract convex spaces which are adequate to establish the KKM theory. In these spaces, he generalizes and simplifies some known results in the theory on convex spaces, $G$-convex spaces, and others. The KKM type maps are used to obtain coincidence theorems and fixed point theorems. A number of examples of abstract convex spaces and generalizations of the KKM principle are then discussed by the author.

47H04Set-valued operators