A fixed point theorem of Markov-Kakutani type for a commuting family of convex multivalued maps

摘要

Let Γ be a commuting family of upper semicontinuous convex multivalued maps of K into itself with nonempty closed values, where K is a nonempty compact convex subset of a locally convex Hausdorff topological vector space E. We then show that the Markov-Kakutani fixed-point theorem holds; that is, there exists at least one point x ∈ K such that x ∈ u(x) for all u in Γ.