Duality in fixed point theory of multivalued mappings with applications

Abstract

Bohnenblust-Karlin [2] extended the fixed point theorem of Kakutani [lo] for multivalued mappings on Banach Spaces. Ky Fan [3] and Glicksberg [9] extended independently the same to locally convex linear topological spaces. Ky Fan [3] applied his result to a number of applications in series of subsequent papers [4], [5], [6], [7], and [8]. A particular case of one of these results [3] and [8] includes a theorem of von Neumann [ 1 l] which in turn, implies the fundamental theorem of game theory. In [l] Browder proved a simple but powerful fixed point theorem and obtained various new results as well as simpler proofs of the basic results of Ky Fan [3] and [8]. In this paper we consider the dual situation. Since each fixed point of a multivalued mapping is a fixed point of its inverse (multivalued) mapping, it is possible to make a dual statement of a fixed point theorem to yield a fixed point of the inverse mapping. In the present paper, we consider dual statements for multivalued mappings specifically related to two basic fixed point theorems: namely, Browder’s frxed point theorem [l] and the fixed point theorem of Ky Fan [3]. We also apply these dual theorems to a number of applications which are dual to those in [3] and [8]. In particular, we obtain a theorem dual to the theorem of von Neumann. We hope this duality principle (see section 1) will yield new results in other areas. The authors are thankful to Professor Ky Fan for valuable suggestions.