Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

The contraction principle for mappings on a metric space with a graph


Author: Jacek Jachymski
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
Published electronically: December 5, 2007
MathSciNet review: 2367109
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [DS58] N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Interscience Publishers, Ltd., London, 1958. MR 0117523 (22:8302)
  • [E61] M. Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7-10. MR 0120625 (22:11375)
  • [G-BL06] T. Gnana Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379-1393. MR 2245511 (2007c:47067)
  • [GD03] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. MR 1987179 (2004d:58012)
  • [Ja01] J. Jachymski, Order-theoretic aspects of metric fixed point theory, Handbook of Metric Fixed Point Theory (eds., W. A. Kirk and B. Sims), 613-641, Kluwer Acad. Publ., Dordrecht, 2001. MR 1904289 (2003f:54094)
  • [Jo97] R. Johnsonbaugh, Discrete Mathematics, Prentice-Hall, Inc., New Jersey, 1997.
  • [KR67] R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511-520. MR 0212457 (35:3328)
  • [L95] A. Lasota, From fractals to stochastic differential equations, Chaos--the interplay between stochastic and deterministic behaviour (Karpacz, 1995), 235-255, Lecture Notes in Phys. 457, Springer, Berlin, 1995. MR 1452617 (98j:28005)
  • [NPR-L07] J. J. Nieto, R. L. Pouso and R. Rodríguez-López, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc. 135 (2007), 2505-2517. MR 2302571
  • [NR-L05] J. J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223-239. MR 2212687 (2006m:47097)
  • [NR-L07] J. J. Nieto and R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, English Ser. (2007), 2205-2212.
  • [PR06] A. Petruşel and I. A. Rus, Fixed point theorems in ordered $ L$-spaces, Proc. Amer. Math. Soc. 134 (2006), 411-418. MR 2176009 (2006g:47097)
  • [RR04] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443. MR 2053350 (2005a:47112)
  • [R04] I. A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004), 259-261. MR 2050229 (2005c:41034)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H10, 05C40, 54H25

Retrieve articles in all journals with MSC (2000): 47H10, 05C40, 54H25


Additional Information

Jacek Jachymski
Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
Email: jachym@p.lodz.pl

DOI: https://doi.org/10.1090/S0002-9939-07-09110-1
Keywords: Fixed point, Picard operator, partial order, connected graph, Bernstein operator
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society