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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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: A lemma of Ludvik Janos is used to show that if a nonexpansive self-map $T$ of a compact set $X$ is contractive on $\Delta ’X$, the boundary of $X$ in $\overline {{\text {co}}} X$, then $T$ has a fixed point in $X$. It is further proven that if $T(\Delta ’X) \cap \Delta ’X = \emptyset$, or if $T$ maps any point $y$ of $X$ away from $\Delta ’X$, then $T$ has a fixed point in $X$.


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Keywords: Fixed point, nonexpansive map, strictly convex, boundary in convex hull
Article copyright: © Copyright 1981 American Mathematical Society