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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The contraction principle for mappings on a metric space with a graph
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by Jacek Jachymski
Proc. Amer. Math. Soc. 136 (2008), 1359-1373
DOI: https://doi.org/10.1090/S0002-9939-07-09110-1
Published electronically: December 5, 2007

Abstract:

We give some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph. This extends and subsumes many recent results of other authors which were obtained for mappings on a partially ordered metric space. As an application, we present a theorem on the convergence of successive approximations for some linear operators on a Banach space. In particular, the last result easily yields the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the space $C[0,1]$.
References
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Bibliographic Information
  • Jacek Jachymski
  • Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
  • Email: jachym@p.lodz.pl
  • Received by editor(s): December 12, 2006
  • Received by editor(s) in revised form: February 13, 2007
  • Published electronically: December 5, 2007
  • Additional Notes: $^*$ Professor Andrzej Lasota passed away on December 28, 2006.

  • Dedicated: To the memory of Professor Andrzej Lasota$^*$
  • Communicated by: Joseph A. Ball
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 1359-1373
  • MSC (2000): Primary 47H10; Secondary 05C40, 54H25
  • DOI: https://doi.org/10.1090/S0002-9939-07-09110-1
  • MathSciNet review: 2367109