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A fixed point property of $ l\sb 1$-product spaces


Authors: Tadeusz Kuczumow, Simeon Reich and Malgorzata Schmidt
Journal: Proc. Amer. Math. Soc. 119 (1993), 457-463
MSC: Primary 47H10; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1993-1155601-9
MathSciNet review: 1155601
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Abstract: Let $ {X_1}$ and $ {X_2}$ be Banach spaces, and let $ {X_1} \times {X_2}$ be equipped with the $ {l_1}$-norm. If the first space $ {X_1}$ is uniformly convex in every direction, then $ {X_1} \times {X_2}$ has the fixed point property for nonexpansive mappings (FPP) if and only if $ \mathbb{R} \times {X_2}$ (with the $ {l_1}$-norm) does. If $ {X_1}$ is merely strictly convex, $ (\mathbb{R} \times {X_2})$ has the FPP, and $ {C_i} \subset {X_i}$ are weakly compact and convex with the FPP (for $ i = 1,2$), then the fixed point set of every nonexpansive mapping $ T:{C_1} \times {C_2} \to {C_1} \times {C_2}$ is a nonexpansive retract of $ {C_1} \times {C_2}$.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1155601-9
Keywords: Nonexpansive mappings, nonexpansive retracts, fixed points, the semi-Opial property
Article copyright: © Copyright 1993 American Mathematical Society

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