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On Kannan maps


Author: Chi Song Wong
Journal: Proc. Amer. Math. Soc. 47 (1975), 105-111
DOI: https://doi.org/10.1090/S0002-9939-1975-0358468-0
MathSciNet review: 0358468
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Abstract | References | Additional Information

Abstract: Let $ K$ be a (nonempty) weakly compact convex subset of a Banach space $ B$. Let $ T$ be a self map on $ K$ such that for all $ x,y$ in $ K,\vert\vert T(x) - T(y)\vert\vert \leqslant (\vert\vert x - T(x)\vert\vert + \vert\vert y - T(y)\vert\vert)/2$. It is proved without the continuity of $ T$ and Zorn's lemma that $ T$ has a fixed point if and only if $ \inf \{ \vert\vert x - T(x)\vert\vert:x \in K\} = 0$. A characterization of the existence of fixed points for such $ T$ is obtained in terms of close-to-normal structure. As consequences, the following results are obtained: (i) $ T$ has a unique fixed point if $ B$ is locally uniformly convex or more generally if $ B$ has the property $ {\mathbf{A}}$: For any sequence $ \{ {x_n}\} $ in $ B,\{ {x_n}\} $ converges to a point $ x$ in $ B$ if it converges weakly to $ x$ and $ \{ \vert\vert x\vert\vert\} $ converges to $ \vert\vert x\vert\vert$; (ii) $ T$ has a unique fixed point if $ B$ is separable.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0358468-0
Keywords: Close-to-normal structure, diametral point, fixed point, locally uniformly convex Banach space, separable Banach space, weakly compact convex set
Article copyright: © Copyright 1975 American Mathematical Society

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