Elements of the KKM theory on abstract convex spaces

We introduce a new concept of abstract convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an abstract convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of abstract convex topological spaces, we obtain Fan-Browder type fixed point theorems, almost fixed point theorems for multimaps, mutual relations between the map classes K and B, variational inequalities, the von Neumann type minimax theorems, and the Nash equilibrium theorems.