Kakutani-type fixed point theorems: a survey (Q626551)

The contents of this paper is best described by quoting its summary: ``A Kakutani-type fixed point theorem refers to a theorem of the following kind: Given a group or semigroup \(S\) of continuous affine transformations \(s:Q\rightarrow Q\), 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 \(a \in Q\) such that \(s(a) = a\) for each \(s \in S\). In [Proc.\ Imp.\ Acad.\ Jap.\ 14, 27--31 (1938; Zbl 0019.29704)], \textit{S.\,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.''