26565 articles – 20400 references  [version française]
Detailed view Article in peer-reviewed journal
Journal of Mathematical Analysis and applications 152, 1 (1990) 46-60
Fixed points for Kakutani factorizable multifunctions
Marc Lassonde1

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 multi-function 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.
1:  LAMIA - Laboratoire de Mathématiques Informatique et Applications