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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point


Author: Tomonari Suzuki
Journal: Proc. Amer. Math. Soc. 136 (2008), 4089-4093
MSC (2000): Primary 54H25
Published electronically: June 4, 2008
MathSciNet review: 2425751
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09390-8
PII: S 0002-9939(08)09390-8
Keywords: Fixed point, successive approximations, Banach-Caccioppoli contraction principle, Leader mapping
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.