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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point
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by Tomonari Suzuki PDF
Proc. Amer. Math. Soc. 136 (2008), 4089-4093 Request permission

Abstract:

If $(X, d)$ is a complete metric space and $T : X \to X$ is a contraction mapping, then the conclusion of the Banach-Caccioppoli contraction principle is that the sequence of successive approximations of $T$ starting from any point of the space converges to a unique fixed point. In this paper, we obtain a sufficient and necessary condition of the above conclusion in terms of the so-called strong Leader mappings.
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Additional Information
  • Tomonari Suzuki
  • Affiliation: Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550, Japan
  • Email: suzuki-t@mns.kyutech.ac.jp
  • Received by editor(s): August 20, 2007
  • Received by editor(s) in revised form: October 12, 2007
  • Published electronically: June 4, 2008
  • Additional Notes: The author was supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 4089-4093
  • MSC (2000): Primary 54H25
  • DOI: https://doi.org/10.1090/S0002-9939-08-09390-8
  • MathSciNet review: 2425751