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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0787883-2
Keywords: Ultimately nonexpansive semigroup, recurrence, common fixed point, isometry, affine isometry
Article copyright: © Copyright 1985 American Mathematical Society