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On the structure of the fixed-point set of a nonexpansive mapping in a Banach space

Author: Ronald E. Bruck
Journal: Proc. Amer. Math. Soc. 61 (1976), 16-18
MSC: Primary 47H10
MathSciNet review: 0428125
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Abstract: If $ C$ is a closed convex subset of a reflexive, strictly convex Banach space $ E$, and $ T:C \to E$ is a nonexpansive mapping which has a fixed-point in the interior of $ C$, then there exists a nonexpansive mapping $ {T^{\ast}}:E \to E$ whose fixed-point set in $ C$ is the fixed-point set of $ T$.

References [Enhancements On Off] (What's this?)

  • [1] R. E. Bruck, Jr., A characterization of Hilbert space, Proc. Amer. Math. Soc. 43 (1974), 173-175. MR 49 #5788. MR 0341038 (49:5788)
  • [2] -, Properties of fixed point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179 (1973), 251-262. MR 48 #2843. MR 0324491 (48:2843)
  • [3] D. G. de Figueiredo and L. A. Karlovitz, On the extension of contractions on normed spaces, Proc. Sympos. Pure Math., vol. 18, Part 1, Amer. Math. Soc., Providence, R. I., 1970, pp. 95-104. MR 43 #877. MR 0275120 (43:877)

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Keywords: Nonexpansive mapping, nonexpansive retraction
Article copyright: © Copyright 1976 American Mathematical Society

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