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Proceedings of the American Mathematical Society

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

 

 

Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions


Author: W. A. Kirk
Journal: Proc. Amer. Math. Soc. 50 (1975), 143-149
MSC: Primary 47H10
DOI: https://doi.org/10.1090/S0002-9939-1975-0380527-7
MathSciNet review: 0380527
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Abstract: Let $ K$ be a bounded closed convex subset of a Banach space $ X$ with $ \operatorname{int} K \ne \emptyset $, and suppose $ K$ has the fixed point property with respect to nonexpansive self-mappings (i.e., mappings $ U:K \to K$ such that $ \vert\vert U(x) - U(y)\vert\vert \leq \vert\vert x - y\vert\vert,x,y \in K)$. Let $ T:K \to X$ be nonexpansive and satisfy

$\displaystyle \inf \{ \vert\vert x - T(x)\vert\vert:x \in {\text{ boundary }}K,T(x) \notin K\} > 0.$

It is shown that if in addition, either (i) $ T$ satisfies the Leray-Schauder boundary condition: there exists $ z \in \operatorname{int} K$ such that $ T(x) - z \ne \lambda (x - z)$ for all $ x \in {\text{ boundary }}K,\lambda < 1$, or (ii) $ \inf \{ \vert\vert x - T(x)\vert\vert:x \in K\} = 0$, is satisfied, then $ T$ has a fixed point in $ K$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0380527-7
Keywords: Nonexpansive mapping, fixed point theorem, Leray-Schauder boundary condition
Article copyright: © Copyright 1975 American Mathematical Society