New versions of the Fan-Browder fixed point theorem and existence of economic equilibria
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We introduce a generalized form of the Fan-Browder fixed point theorem and apply it to game-theoretic and economic equilibrium existence problem under the more generous restrictions. Consequently, we state some of recent results of Urai (2000) in more general and efficient forms. 1. Introduction 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, Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 149–158 2000 Mathematics Subject Classification: 54H25, 47H10, 46A16, 46A55, 91B50 URL: http://dx.doi.org/10.1155/S1687182004308089
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Fan-Browder fixed point theorem and economic equilibria
we show that a number of Urai’s results [12] (e.g., Theorem 1 for the case (K∗ ), Theorem 2 for the case (NK∗ ), Theorem 3 for the case (K∗ ), Theorem 19, and their Corollaries) can be stated in more generalized and efficient forms. 2. Preliminaries A multimap (or simply, a map) F : X Y is a function from a set X into the power set 2Y of the set Y ; that is, a function with the values F(x) ⊂ Y for x ∈ X and the fibers F − (y) = {x ∈ X | y ∈ F(x)} for y ∈ Y . For A ⊂ X, let F(A) := {F(x) | x ∈ A}. For a set D, let D denote the set of nonempty finite subsets of D. Let X be a subset of a vector space and D a nonempty subset of X. We call (X,D) a convex space if coD ⊂ X and X has a topology that induces the Euclidean to
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