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

Abstract

In this paper a new fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values is established in the setting of an abstract convex structure -called a locally G-convex space, which generalises usual convexity such as locally convex //-spaces, locally convex spaces (locally //-convex spaces), hyperconvex metric spaces and locally convex topological spaces. Our fixed point theorem includes corresponding Fan-Glicksberg type fixed point theorems in locally convex //-spaces, locally convex spaces, hyperconvex metric space and locally convex spaces in the existing literature as special cases.

Coincidences and fixed points in locally G-convex spaces

“…The emphasis of this work is to study multifunctions with Gconvex values instead of acyclic values. Therefore the results established here are proved by different means and they do not compare with the results in [18,4].…”

“…These spaces generalise the notion of //-convexity (see Definition 1 below) as well as hyperconvexity. We refer to [18,6] for further discussion on the relations between these concepts of convexity.…”

“…It should be noted that Yuan [18] has generalised the Fan-Glicksberg fixed point theorem for multifunctions with acyclic values, and in Ding and Tarafdar [4], a coincidence point theorem has been proved (in H-spaces) for a pair of multifunctions, one of which has acyclic values. The emphasis of this work is to study multifunctions with Gconvex values instead of acyclic values.…”

A Maximal Element Theorem inFWC-Spaces and Its Applications

“…Let ( X , d ) be a hyperconvex metric space in [ 15 ]. Then by Lemma 2.3 of Yuan [ 27 ], we know that ( X , d ) is an H -space and thus, ( X , d ) is an FWC -space.…”