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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a \in Q}$$\end{document} such that s(a) = a for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s \in S}$$\end{document}. 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.