Fixed points and boundaries
Author:
Eric Chandler
Journal:
Proc. Amer. Math. Soc. 82 (1981), 398-400
MSC:
Primary 47H10; Secondary 47H09
DOI:
https://doi.org/10.1090/S0002-9939-1981-0612728-7
MathSciNet review:
612728
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Abstract | References | Similar Articles | Additional Information
Abstract: A lemma of Ludvik Janos is used to show that if a nonexpansive self-map of a compact set
is contractive on
, the boundary of
in
, then
has a fixed point in
. It is further proven that if
, or if
maps any point
of
away from
, then
has a fixed point in
.
- [1] Eric Chandler and Gary Faulkner, Fixed points in nonconvex domains, Proc. Amer. Math. Soc. 80 (1980), no. 4, 635–638. MR 587942, https://doi.org/10.1090/S0002-9939-1980-0587942-9
- [2] Michael Edelstein, On non-expansive mappings of Banach spaces, Proc. Cambridge Philos. Soc. 60 (1964), 439–447. MR 164222
- [3] H. Freudenthal and W. Hurewicz, Dehnungen, Verkurzungen, Isometrien, Fund. Math. 26 (1936), 120-122.
- [4] Ludvik Janos and J. L. Solomon, A fixed point theorem and attractors, Proc. Amer. Math. Soc. 71 (1978), no. 2, 257–262. MR 482716, https://doi.org/10.1090/S0002-9939-1978-0482716-2
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1981-0612728-7
Keywords:
Fixed point,
nonexpansive map,
strictly convex,
boundary in convex hull
Article copyright:
© Copyright 1981
American Mathematical Society