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On mappings contractive in the sense of Kannan


Author: Ludvik Janos
Journal: Proc. Amer. Math. Soc. 61 (1976), 171-175
MSC: Primary 54H25; Secondary 54E40
DOI: https://doi.org/10.1090/S0002-9939-1976-0425936-3
MathSciNet review: 0425936
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f:X \to X$ be a continuous compact mapping of a metric space $ (X,d)$ into itself with the property that $ x,y \in X$ and $ x \ne y$ implies $ d(f(x),f(y)) < \tfrac{1} {2}[d(x,f(x)) + d(y,f(y))]$. It is shown that under these conditions $ f$ has a unique fixed point and, moreover, $ f$ is a Banach contraction relative to a suitable remetrization of the space $ X$. A similar result concerning condensing mappings is also obtained.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0425936-3
Keywords: Fixed points, Banach contraction principle, contractions, self-maps contractive in the sense of Kannan and Edelstein, remetrization, Kuratowski's and Hausdorff's measures of noncompactness, condensing mapping
Article copyright: © Copyright 1976 American Mathematical Society

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