The purpose of this paper is to extend Himmelberg's fixed-point theorem, replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypotheses, 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 mentioned above. ᮊ
A B S T R A C TBy generalizing the classical Knaster-Kuratowski-Mazurkiewicz Theorem, we obtain a result that provides su¢cient conditions to ensure the non-emptiness of several kinds of choice functions. This result generalizes well-known results on the existence of maximal elements for binary relations (Bergstrom, 1975;Walker, 1977;Tian, 1993), on the non-emptiness of nonbinary choice functions (Nehring, 1996;Llinares and Sánchez, 1999) and on the non-emptiness of some classical solutions for tournaments (top cycle and uncovered set) on non-…nite sets.
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, 25, 291-306]), B 0 -simplicial convexity and L-spaces [H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano and J.V. LLinares (1998). Abstract convexity and fixed points. Journal of Mathematical Analysis and Applications, 222, 138-150]. Moreover, in the context of mc-spaces, a characterization result of nonempty finite intersection, in the line with the KnasterKuratowski-Mazurkiewicz Lemma, some consequences of it and some generalizations of Browder's existence of continuous selection and fixed point theorem are presented.
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