Properties of the Fixed Point Set of Contractive Multi-Functions
Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 169-173
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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.
Schirmer, Helga. Properties of the Fixed Point Set of Contractive Multi-Functions. Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 169-173. doi: 10.4153/CMB-1970-036-4
@article{10_4153_CMB_1970_036_4,
author = {Schirmer, Helga},
title = {Properties of the {Fixed} {Point} {Set} of {Contractive} {Multi-Functions}},
journal = {Canadian mathematical bulletin},
pages = {169--173},
year = {1970},
volume = {13},
number = {2},
doi = {10.4153/CMB-1970-036-4},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-036-4/}
}
TY - JOUR AU - Schirmer, Helga TI - Properties of the Fixed Point Set of Contractive Multi-Functions JO - Canadian mathematical bulletin PY - 1970 SP - 169 EP - 173 VL - 13 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-036-4/ DO - 10.4153/CMB-1970-036-4 ID - 10_4153_CMB_1970_036_4 ER -
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