Geometric Properties, Minimax Inequalities, and Fixed Point Theorems on Convex Spaces

Abstract

Abstract. Using a selection theorem, we obtain a very general Ky Fan type geometric property of convex sets and apply it to the existence of maximizable quasiconcave functions, new minimax inequalities, and fixed point theorems for upper hemicontinuous multifunctions. Our results generalize works of Ha, Fan, Jiang, Himmelberg, and many others. IntroductionIn 1961, Fan [Fl] generalized the celebrated Knaster-Kuratowski-Mazurkiewicz theorem (simply, KKM theorem) and gave a number of applications in a sequence of his subsequent papers. For the literature, see [F2]. Fan [Fl] also established an elementary but very basic "geometric" lemma which is equivalent to his KKM theorem. Later, Browder [Br] restated this result in the more convenient form of a fixed point theorem by means of the Brouwer fixed point theorem and the partition of unity argument. Now, it is well known that the Brouwer theorem, the Sperner lemma, the KKM theorem, Fan's geometric lemma, the Fan-Browder fixed point theorem, and many others are equivalent.In many of his works in the KKM theory, Fan actually based his arguments mainly on the geometric property of convex sets. Since then there have appeared numerous applications, various generalizations, or equivalent formulations of Fan's geometric property or the Fan-Browder fixed point theorem.The purpose in the present paper is to establish a very general Fan type geometric property of convex sets and to apply it to the existence of maximizable quasiconcave functions, new minimax inequalities, and fixed point theorems for upper hemicontinuous multifunctions. Our new geometric property generalizes that of Ha [Ha, Theorem 3]. His minimax inequality is also extended. Finally,

Fixed points, intersection theorems, variational inequalities, and equilibrium theorems

“…Lemma 3.1 (see [55]). Let X be a topological space, Y a convex space, and S, T : X Y maps satisfying…”

Fixed‐point and coincidence theorems for set‐valued maps with nonconvex or noncompact domains in topological vector spaces

“…For maps defined on convex sets, there are many variations, generalizations, and applications (see, e.g., [1,10,13,24,25,31,32,34,36,38,41,42,43,46]) of the well-known Fan minimax inequality [20], Hartman-Stampacchia variational inequality [26], and Iohvidov theorem [29].…”

“…In the past decade, there was a renewed interest in the fixed-point and coincidence theory of set-valued maps in topological vector spaces (see, e.g., [2,3,8,9,10,11,12,28,36,37,38,39,40,42,43,44,45,46,47,48]), partially due to new and powerful methods of investigations introduced into it (notably 2 Fixed-point and coincidence theorems based on those introduced by Fan and Browder). Most of the work has centered around the fixed-point and coincidence theory of maps on convex compact sets, but there are also a considerable number of papers devoted to maps on nonconvex and noncompact sets (see, e.g., [8,45]).…”

Fixed points of upper semicontinuous mappings in locally G-convex spaces

“…Since T{y) is G-convex, it follows that }{y) : = 4>ty{y)) C 0(A J ( y ) ) C r J ( y ) C T(y) for each y e Y (as r J ( y ) C T{y)), that is, /(y) € T(y) for each y G Y, and this completes the proof. D Before establishing our fixed point theorem, we also recall the following result which is Lemma 2.1 of Park et al [22] from the Lefschetz-type fixed point theorem for composites of acyclic maps due to Gorniewicz and Granas [11] We now prove the following fixed point theorem in locally G-convex spaces. This is a generalisation of the Fan-Glicksberg type fixed point theorems for upper semicontin-uous set-valued mappings with non-empty closed acyclic values given in several places (for example, see Kirk and Shin [18], Park [20, 2 1 , 22, 23], Tarafdar [27,28,29], Wu [30] and others in locally convex spaces).…”

“…Theorem 2.1 also extends Theorem 3.2 of Tarafdar and Watson [29] and the corresponding Fan-Glicksberg type fixed point theorems given by Wu [30] to locally G-convex spaces which include locally ff-convex space and locally convex if-spaces as special classes. For example, as special cases of Theorem 2.1 we have the following generalisations of Fan-Glicksberg type fixed point theorems in both locally convex H-spaces and locally convex Hausdorff topological vector spaces (for example, see also Wu [30], Park [20,21,22,23] and related references there, Fan [7] and Glicksberg [9] The relationship between hyperconvex metric spaces and nonexpansive mappings is an important one as shown independently by the work of Sine [25] and Soardi [26]. Hyperconvex metric spaces have been used widely and many interesting results for nonexpansive mappings have been established in the framework of hyperconvex structures, for example, see Baillon [2], Goebel and Kirk [10], Khamsi [17], Since [24,25] and others.…”