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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A remark on directional contractions
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by W. A. Kirk and William O. Ray PDF
Proc. Amer. Math. Soc. 66 (1977), 279-283 Request permission

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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 66 (1977), 279-283
  • MSC: Primary 47H10
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0454755-8
  • MathSciNet review: 0454755