The Markhoff-Kakutani theorem for compact affine maps (Q2702968)

At first the authors prove the following fixed point theorem: Let \(X\) be a nonempty convex set of a Hausdorff topological vector space and \({\mathcal F}\) a commuting family of continuous affine selfmappings of \(X\), such that there exists a compact map \(f\in{\mathcal F}\). Then \({\mathcal F}\) has a common fixed point. This is a generalization of the classical Markov-Kakutani fixed point theorem: The underlying topological vector space is not supposed to be locally convex and the assumption of the compactness of the domain \(X\) is replaced by that of the map \(f\in{\mathcal F}\). Furthermore, the authors generalize their own result: They don't require the compactness of a map \(f\in{\mathcal F}\) but only of a condensing map or a generalized or ultimately condensing map. Other similar results are given.