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,

A New Generalization of Browder Fixed Point Theorem with Applications

“…By Theorem 1.2 in [6], G has a continuous selection f : By Theorem 1.2 in [6], G has a continuous selection f :…”