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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Remarks on pseudo-contractive mappings


Author: W. A. Kirk
Journal: Proc. Amer. Math. Soc. 25 (1970), 820-823
MSC: Primary 47.85; Secondary 46.00
MathSciNet review: 0264481
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Abstract: Let $ X$ be a Banach space, $ D \subset X$. A mapping $ U:D \to X$ is said to be pseudo-contractive if for all $ u,v \in D$ and all $ r > 0$, $ \vert\vert u - v\vert\vert \leqq \vert\vert(1 + r)(u - v) - r(U(u) - U(v))\vert\vert$. This concept is due to F. E. Browder, who showed that $ U:X \to X$ is pseudo-contractive if and only if $ I - U$ is accretive. In this paper it is shown that if $ X$ is a uniformly convex Banach, $ B$ a closed ball in $ X$, and $ U$ a Lipschitzian pseudo-contractive mapping of $ B$ into $ X$ which maps the boundary of $ B$ into $ B$, then $ U$ has a fixed point in $ B$. This result is closely related to a recent theorem of Browder.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0264481-X
PII: S 0002-9939(1970)0264481-X
Keywords: Fixed point theory, uniformly convex Banach spaces, pseudo-contractive mappings, nonexpansive mappings, accretive mappings
Article copyright: © Copyright 1970 American Mathematical Society