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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] L. M. Blumenthal, Theory and applications of distance geometry, Clarendon Press, Oxford, 1953. MR 0054981 (14:1009a)
  • [2] R. D. Holmes, Fixed points for local radial contractions, Proc. Seminar on Fixed Point Theory and Its Applications (Dalhousie Univ., 1975), Academic Press, New York, 1976, pp. 79-89. MR 0448327 (56:6634)
  • [3] E. Rakotch, A note on $ \alpha $-locally contractive mappings, Bull. Res. Council Israel 40 (1962), 188-191. MR 0146799 (26:4319)
  • [4] W. Rinow, Die innere Geometrie der metrischen Räume, Springer-Verlag, Berlin, 1961. MR 0123969 (23:A1290)

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Keywords: Local contraction, fixed point theorem, contraction mapping principle
Article copyright: © Copyright 1978 American Mathematical Society

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