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Proceedings of the American Mathematical Society

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Local contractions in metric spaces


Authors: Thakyin Hu and W. A. Kirk
Journal: Proc. Amer. Math. Soc. 68 (1978), 121-124
MSC: Primary 54E40; Secondary 54H25
MathSciNet review: 0464180
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Abstract: It is shown that a theorem of E. Rakotch for locally contractive mappings can be deduced from Banach's contraction mapping theorem, and a counterexample to an assertion of R. D. Holmes concerning local radial contractions is given.


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

  • [1] Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
  • [2] R. D. Holmes, Fixed point for local radial contractions, Fixed point theory and its applications (Proc. Sem., Dalhousie Univ., Halifax, N. S., 1975) Academic Press, New York, 1976, pp. 79–89. MR 0448327
  • [3] E. Rakotch, A note on 𝛼-locally contractive mappings, Bull. Res. Council Israel Sect. F 10F (1962), 188–191 (1962). MR 0146799
  • [4] Willi Rinow, Die innere Geometrie der metrischen Räume, Die Grundlehren der mathematischen Wissenschaften, Bd. 105, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0123969

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0464180-2
Keywords: Local contraction, fixed point theorem, contraction mapping principle
Article copyright: © Copyright 1978 American Mathematical Society