Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: NEW VERSIONS OF THE FAN-BROWDER FIXED POINT THEOREM AND EXISTENCE OF ECONOMIC EQUILIBRIA | NEW VERSIONS OF THE FAN-BROWDER FIXED POINT THEOREM AND EXISTENCE OF ECONOMIC EQUILIBRIA SEHIE PARK Received 18 August 2003 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
