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Properties of the Fixed Point Set of Contractive Multi-Functions

Published online by Cambridge University Press:  20 November 2018

Helga Schirmer
Helga Schirmer*
Affiliation:
Carleton University, Ottawa, Ontario
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A well known theorem by S. Banach states that a contractive function f: X → X on a complete metric space X has a fixed point, and that this fixed point is unique. This result has a partial extension to multi-functions: every contractive compact-valued multi-function on a complete metric space has a fixed point (see Definition 1 and Theorem 1 below). But simple examples show that this fixed point is no longer unique. We investigate some questions concerned with the properties of the fixed point set Φ of a contractive multi-function φ. Is, e.g., Φ connected if φ is connected-valued? Is Φ convex if φ is convex-valued? The answer is yes if X is the real line (§2), but examples in §3 and §4 show that in general the answer is no.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 169 - 173
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Markin, J. T., A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc. 74 (1968), 639-640.Google Scholar
2. Nadler, S. B. Jr., Multi-valued contractive mappings, Pacific J. Math. 30 (1969), 475-488.Google Scholar