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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 $\left \| {T(x + \varepsilon (y - x)) - T(x)} \right \| \leqslant \alpha \varepsilon \left \| {x - y} \right \|$. 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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Keywords: Contraction mapping, directional contraction, fixed point theorem
Article copyright: © Copyright 1977 American Mathematical Society