The purpose of this paper is to extend Himmelberg's fixed point theorem replacing the usual convexity in topological vector spaces by an abstract topological notion of convexity which generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypothese, of a fixed point for a compact approachable map and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentionned above.

Abstract Convexity and Fixed Points

Abstract convexity and fixed points

Any multimap from a Hausdorff compact space to a generalized convex space has a continuous selection whenever it has nonempty generalized convex values and open fibers. Some related new results are obtained as well as generalizations of many known results. Furthermore, applications to fixed point theorems and existence of equilibria are given.

Continuous Selection Theorems in Generalized Convex Spaces

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. In this paper, we clearly derive a sequence of a dozen statements which characterize the KKM spaces and equivalent formulations of the partial KKM principle. As their applications, we add more than a dozen statements including generalized formulations of von Neumann minimax theorem, von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.

The KKM principle in abstract convex spaces: Equivalent formulations and applications

The aim of this article is to analyze the relationship between various notions of abstract convexity structures that we find in the literature, in connection with the problem of the existence of continuous selections and fixed points of correspondences. We focus mainly on the notion of mc -spaces, which was introduced in [J.V. LLinares (1998). Unified treatment of the problem of the existence of maximal elements in binary relations: a characterization. Journal of Mathematical Economics , 29 , 285-302], and its relationship with c -spaces [Ch.D. Horvath (1991). Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications , 156 , 341-357], simplicial convexity [R. Bielawski (1987). Simplicial convexity and its applications. Journal of Mathematical Analysis and Applications , 127 , 155-171], order convexity (used in [Ch.D. Horvath and J.V. LLinares (1996). Maximal elements and fixed points for binary relations on topological ordered spaces. Journal of Mathematical Economics , ...

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Juan Vicente Llinares Ciscar

Papers

Maximal elements and fixed points for binary relations on topological ordered spaces