Kakutani-type fixed point theorems: A survey

A Kakutani-type fixed point theorem refers to a theorem of the following kind: Given a group or semigroup S of continuous affine transformations s : Q → Q, where Q is a nonempty compact convex subset of a Hausdorff locally convex linear topological space, then under suitable conditions S has a common fixed point in Q, i.e., a point a ? Q{a \in Q} such that s(a) = a for each s ? S{s \in S}. In 1938, Kakutani gave two conditions under each of which a common fixed point of S in Q exists. They are (1) the condition that S be a commutative semigroup, and (2) the condition that S be an equicontinuous group. The present survey discusses subsequent generalizations of Kakutani’s two theorems above.