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.References
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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