Abstract convexity and fixed points. (English) Zbl 0986.54054

Summary: 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.


54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
54E15 Uniform structures and generalizations
54C60 Set-valued maps in general topology
52A01 Axiomatic and generalized convexity
47H04 Set-valued operators
Full Text: DOI


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