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Proceedings of the American Mathematical Society

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

 
 

 

A common fixed-point theorem in reflexive locally uniformly convex Banach spaces


Authors: Michael Edelstein and Mo Tak Kiang
Journal: Proc. Amer. Math. Soc. 94 (1985), 411-415
MSC: Primary 47H10; Secondary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1985-0787883-2
MathSciNet review: 787883
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Abstract: Let $X$ be a reflexive locally uniformly convex Banach space and $G$ an ultimately nonexpansive commutative semigroup of continuous self-maps of $X$. If there exists a point $x$ in $X$ recurrent under $G$ such that $G(x)$ is bounded, then $G$ has a common fixed point in $\overline {{\text {co}}} (G(x))$. If $X$ is a Hilbert space then there is exactly one such point in $\overline {{\text {co}}} (G(x))$.


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Keywords: Ultimately nonexpansive semigroup, recurrence, common fixed point, isometry, affine isometry
Article copyright: © Copyright 1985 American Mathematical Society