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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
DOI: https://doi.org/10.1090/S0002-9939-08-09390-8
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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  • 1. S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181.
  • 2. R. Caccioppoli, Un teorema generale sull'esistenza di elementi uniti in una transformazione funzionale, Rend. Accad. Naz. Lincei, 11 (1930), 794-799.
  • 3. J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215 (1976), 241-251. MR 0394329 (52:15132)
  • 4. J. Caristi and W. A. Kirk, Geometric fixed point theory and inwardness conditions, Lecture Notes in Math., Vol. 490, pp. 74-83, Springer, Berlin, 1975. MR 0399968 (53:3806)
  • 5. Lj. B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267-273. MR 0356011 (50:8484)
  • 6. M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74-79. MR 0133102 (24:A2936)
  • 7. J. Jachymski, An iff fixed point criterion for continuous self-mappings on a complete metric space, Aequationes Math., 48 (1994), 163-170. MR 1295089 (96c:54069)
  • 8. R. Kannan, Some results on fixed points - II, Amer. Math. Monthly, 76 (1969), 405-408. MR 0257838 (41:2487)
  • 9. W. A. Kirk, Contraction mappings and extensions, in Handbook of Metric Fixed Point Theory (W. A. Kirk and B. Sims, Eds.), 2001, pp. 1-34, Kluwer Academic Publishers, Dordrecht. MR 1904272 (2003f:54096)
  • 10. -, Fixed points of asymptotic contractions, J. Math. Anal. Appl., 277 (2003), 645-650. MR 1961251 (2003k:47093)
  • 11. S. Leader, Equivalent Cauchy sequences and contractive fixed points in metric spaces, Studia Math., 76 (1983), 63-67. MR 728197 (85c:54081)
  • 12. J. Matkowski, Integrable solutions of functional equations, Diss. Math., 127, Warsaw, 1975. MR 0412650 (54:772)
  • 13. A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326-329. MR 0250291 (40:3530)
  • 14. P. V. Subrahmanyam, Remarks on some fixed point theorems related to Banach's contraction principle, J. Math. Phys. Sci., 8 (1974), 445-457. MR 0358749 (50:11208)
  • 15. T. Suzuki, Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl., 253 (2001), 440-458. MR 1808147 (2002f:49038)
  • 16. -, Several fixed point theorems concerning $ \tau$-distance, Fixed Point Theory Appl., 2004, no. 3, 195-209. MR 2096951
  • 17. -, Some notes on Meir-Keeler contractions and L-functions, Bull. Kyushu Inst. Technol., 53 (2006), 1-13. MR 2237618 (2007g:54068)
  • 18. -, A definitive result on asymptotic contractions, J. Math. Anal. Appl., 335 (2007), 707-715. MR 2340349

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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: https://doi.org/10.1090/S0002-9939-08-09390-8
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.

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