%0 Journal Article
%T Evolution of the Minimax Inequality of Ky Fan
%A Sehie Park
%J Journal of Operators
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/124962
%X There are quite a few generalizations or applications of the 1984 minimax inequality of Ky Fan compared with his original 1972 minimax inequality. In a certain sense, the relationship between the 1984 inequality and several hundreds of known generalizations of the original 1972 inequality has not been recognized for a long period. Hence, it would be necessary to seek such relationship. In this paper, we give several generalizations of the 1984 inequality and some known applications in order to clarify the close relationship among them. Some new types of minimax inequalities are added. 1. Introduction The KKM theory is originated from the Knaster-Kuratowski-Mazurkiewicz (KKM for short) theorem of 1929 [1]. Since then, it has been found a large number of results which are equivalent to the KKM theorem; see [2, 3]. Typical examples of the most remarkable and useful equivalent formulations are Ky Fan's KKM lemma of 1961 [4] and his minimax inequality of 1972 [5]. The inequality and its various generalizations are very useful tools in various fields of mathematical sciences. Since 1961, Ky Fan showed that the KKM theorem provides the foundation for many of the modern essential results in diverse areas of mathematical sciences. Actually, a milestone in the history of the KKM theory was erected by Fan in 1961 [4]. His 1961 KKM Lemma (or the Fan-KKM theorem) extended the KKM theorem to arbitrary topological vector spaces and had been applied to various problems in his subsequent papers [5每10]. Recall that, at the beginning, the basic theorems in the KKM theory and their applications were established for convex subsets of topological vector spaces mainly by Fan in 1961每1984 [4每10]. A number of intersection theorems and their applications to various equilibrium problems followed. In our previous review [11], we recalled Fan's contributions to the KKM theory based on his celebrated 1961 KKM lemma, and introduced relatively recent applications of the lemma due to other authors in the twenty-first century. Then, the KKM theory was extended to convex spaces by Lassonde in 1983 [12] and to -spaces (or H-spaces) by Horvath in 1983每1993 [13每16] and others. Since 1993, the theory has been extended to generalized convex (G-convex) spaces in a sequence of papers of the present author and others; see [2]. Since 2006, the main theme of the theory has become abstract convex spaces in the sense of Park [17每30]. The basic theorems in the theory have numerous applications to various equilibrium problems in nonlinear analysis and other fields. In our previous review [30], we
%U http://www.hindawi.com/journals/joper/2013/124962/