New Versions of the Fan-browder Fixed Point Theorem and Existence of Economic Equilibria

In 1961, using his own generalization of the Knaster-Kuratowski-Mazurkiewicz (simply, KKM) theorem, Fan [2] established an elementary but very basic “geometric” lemma for multimaps and gave several applications. In 1968, Browder [1] obtained a fixed point theorem which is the more convenient form of Fan’s lemma. With this result alone, Browder carried through a complete treatment of a wide range of coincidence and fixed point theory, minimax theory, variational inequalities, monotone operators, and game theory. Since then, this result is known as the Fan-Browder fixed point theorem, and there have appeared numerous generalizations and new applications. For the literature, see Park [7, 8, 9]. Recently, Urai [12] reexamined fixed point theorems for set-valued maps from a unified viewpoint on local directions of the values of a map on a subset of a topological vector space to itself. Some basic fixed point theorems were generalized by Urai so that they could be applied to game-theoretic and economic equilibrium existence problem under some generous restrictions. However, in view of the recent development of the KKM theory, we found that some (not all) of Urai’s results can be stated in a more general and efficient way. In fact, compact convex subsets of Hausdorff topological vector spaces that appeared in some of Urai’s results can be replaced by mere convex spaces with finite open (closed) covers. Moreover, Urai’s main tools are the partition of unity argument on such covers, where the Hausdorff compactness is essential, and the Brouwer fixed point theorem. In the present paper, we introduce a generalized form of the Fan-Browder fixed point theorem, which is the main tool of our work. Using this theorem instead of Urai’s tools,