Fixed point theorems in locally G-convex spaces

Abstract

It is well-known that the Brouwer /xed point theorem, the Sperner lemma, the Knaster–Kuratowski–Mazurkiewicz theorem (simply, the KKM principle), and many results in nonlinear analysis are equivalent. In particular, it was shown in [13] that the KKM principle implies the Brouwer theorem. In this paper, we show that the KKM theorem implies far-reaching generalizations of the Brouwer theorem including well-known /xed point theorems due to Schauder, Tychono9, Kakutani, Himmelberg, and many others. For the literature, see [17,26]. In a recent work, Tarafdar [34] obtained a /xed point theorem for a continuous compact multimap T :X (X with closed H -convex values, where X is a locally H -convex uniform space. In the present paper, we show that his theorem holds for u.s.c. maps instead of continuous maps and for the class of generalized convex spaces (or G-convex spaces) containing properly that of H -spaces and many other types of spaces. Our main result (Theorem 2) is applied to various /xed point theorems for LG-spaces, LC-spaces, hyperconvex spaces, and normed vector spaces. Section 2 deals with a new KKM theorem for generalized convex spaces. In Section 3, we obtain our main /xed point theorem for LG-spaces and some of its simple consequences, especially, a generalization of the Himmelberg theorem. Section 4 deals with lower semicontinuous multimaps and >-maps de/ned on paracompact LC-spaces. In fact, the selection theorems due to Ben-El-Mechaiekh and Oudadess [3] and Park [22] are used to deduce new /xed point theorems for such multimaps. In Section 5, our