FIXED-POINTS FOR KAKUTANI FACTORIZABLE MULTIFUNCTIONS

A multifunction Γ is called a Kakutani multifunction if there exist two nonempty convex sets X and Y, each in a Hausdorff topological vector space, such that Γ: X → Y is upper semi-continuous with nonempty compact convex values. We prove the following extension of the Kakutani fixed point theorem: Let Γ: X → X be a multifunction from a simplex X into itself; if Γ can be factorized by an arbitrary finite number of Kakutani multifunctions, then Γ has a fixed point. The proof relies on a simplicial approximation technique and the Brouwer fixed point theorem. Extensions to infinite-dimensional spaces and applications to game theory are given. © 1990.