Kakutani-type fixed point theorems: A survey

Abstract

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 such that s(a) = a for each s ∈ 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. Mathematics Subject Classification (2010).Primary 47-03, 47H10, 54H25; Secondary 37B05.Keywords. The Markov-Kakutani fixed point theorem, Kakutani's fixed point theorem, Ryll-Nardzewski's fixed point theorem, Furstenberg's fixed point theorem.

Set Invariant Means and Set Fixed Point Properties

“…Fixed point property for semigroups is one of the interesting concepts related to the semigroups theory that investigated by many authors, see [5]- [9]. Set-fixed point property for discrete semigroups is defined in [1, Definition 4.5].…”

Around the nonlinear Ryll-Nardzewski theorem

“…The fundamental tool for our results is the following consequence of Furstenberg's structure theorem [9], extended from metrizable to arbitrary compact affine systems by Namioka (see [17,Theorem 4.1], [18,Theorem 4.1]).…”

The Hyers theorem via the Markov–Kakutani fixed point theorem

“…One of the most celebrated results of the theory of common fixed points is a theorem proved independently by Markov [8] and Kakutani [7] (see also [9] and [10]). …”