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


Asymptotic normal structure and fixed points of nonexpansive mappings

Authors: J.-B. Baillon and R. Schöneberg
Journal: Proc. Amer. Math. Soc. 81 (1981), 257-264
MSC: Primary 47H10; Secondary 46B20, 47H09
MathSciNet review: 593469
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Abstract: A mapping $ f$ defined on a subset $ X$ of a Banach space $ E$ and taking values in $ E$ is said to be nonexpansive if $ \left\vert {f(x) - f(y)} \right\vert \leqslant \left\vert {x - y} \right\vert$ for all $ x,y \in X$. In this paper we introduce a promising new geometric property of Banach spaces and show that it yields via a minor modification of known arguments a new fixed point theorem for nonexpansive mappings which includes Kirk's famous result as well as a recent result of Karlovitz. We also discuss in detail a situation not covered by this result.

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PII: S 0002-9939(1981)0593469-1
Keywords: Nonexpansive mappings, normal structure, asymptotic normal structure
Article copyright: © Copyright 1981 American Mathematical Society