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

A remark on directional contractions


Authors: W. A. Kirk and William O. Ray
Journal: Proc. Amer. Math. Soc. 66 (1977), 279-283
MSC: Primary 47H10
MathSciNet review: 0454755
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Abstract: Let X be a Banach space and D a convex subset of X. A mapping $ T:D \to X$ is called a directional contraction if there exists a constant $ \alpha \in (0,1)$ such that corresponding to each $ x,y \in D$ there exists $ \varepsilon = \varepsilon (x,y) \in (0,1]$ for which $ \left\Vert {T(x + \varepsilon (y - x)) - T(x)} \right\Vert \leqslant \alpha \varepsilon \left\Vert {x - y} \right\Vert$. Tests for lipschitzianness are obtained which yield the fact that if a closed mapping is a directional contraction, then it must be a global contraction, and sufficient conditions are given under which a nonclosed directional contraction $ T:D \to D$ always has a fixed point.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0454755-8
PII: S 0002-9939(1977)0454755-8
Keywords: Contraction mapping, directional contraction, fixed point theorem
Article copyright: © Copyright 1977 American Mathematical Society