Abstract
We introduce a generalized form of the Fan-Browder fixed point theorem and apply 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.
Highlights
In 1961, using his own generalization of the Knaster-Kuratowski-Mazurkiewicz (KKM) theorem, Fan [2] established an elementary but very basic “geometric” lemma for multimaps and gave several applications
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
Summary
In 1961, using his own generalization of the Knaster-Kuratowski-Mazurkiewicz (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. 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 this result is known as the Fan-Browder fixed point theorem, and there have appeared numerous generalizations and new applications. 150 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
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