Skip to main content
Fixed Point Theory and Algorithms for Sciences and Engineering
Download PDF
  • Research Article
  • Open access
  • Published:

An Extension of Gregus Fixed Point Theorem

Fixed Point Theory and Applications volume 2007, Article number: 078628 (2007) Cite this article

Abstract

Let be a closed convex subset of a complete metrizable topological vector space and a mapping that satisfies for all , where , , , , , and . Then has a unique fixed point. The above theorem, which is a generalization and an extension of the results of several authors, is proved in this paper. In addition, we use the Mann iteration to approximate the fixed point of .

[1234567891011121314151617181912345678910111213141516171819]

References

  1. Greguš M Jr.: A fixed point theorem in Banach space. Bollettino. Unione Matematica Italiana. A. Serie V 1980,17(1):193–198.

    MATH  Google Scholar 

  2. Murthy PP, Cho YJ, Fisher B: Common fixed points of Greguš type mappings. Glasnik Matematički. Serija III 1995,30(50)(2):335–341.

    MathSciNet  Google Scholar 

  3. Mukherjee RN, Verma V: A note on a fixed point theorem of Greguš. Mathematica Japonica 1988,33(5):745–749.

    MATH  MathSciNet  Google Scholar 

  4. Olaleru JO: A generalization of Greguš fixed point theorem. Journal of Applied Sciences 2006,6(15):3160–3163.

    Article  Google Scholar 

  5. Chidume CE: Geometric properties of Banach spaces and nonlinear iterations. Research Monograph, International Centre for Theoretical Physics, Trieste, Italy, in preparation

  6. Kaewcharoen A, Kirk WA: Nonexpansive mappings defined on unbounded domains. Fixed Point Theory and Applications 2006, 2006: 13 pages.

    Article  MathSciNet  Google Scholar 

  7. Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965,72(9):1004–1006. 10.2307/2313345

    Article  MATH  MathSciNet  Google Scholar 

  8. Kannan R: Some results on fixed points. III. Fundamenta Mathematicae 1971,70(2):169–177.

    MATH  MathSciNet  Google Scholar 

  9. Wong CS: On Kannan maps. Proceedings of the American Mathematical Society 1975,47(1):105–111. 10.1090/S0002-9939-1975-0358468-0

    Article  MATH  MathSciNet  Google Scholar 

  10. Chatterjea SK: Fixed-point theorems. Comptes Rendus de l'Académie Bulgare des Sciences 1972, 25: 727–730.

    MATH  MathSciNet  Google Scholar 

  11. Olaleru JO: On weighted spaces without a fundamental sequence of bounded sets. International Journal of Mathematics and Mathematical Sciences 2002,30(8):449–457. 10.1155/S0161171202011857

    Article  MATH  MathSciNet  Google Scholar 

  12. Schaefer HH, Wolff MP: Topological Vector Spaces, Graduate Texts in Mathematics. Volume 3. 2nd edition. Springer, New York, NY, USA; 1999:xii+346.

    Book  Google Scholar 

  13. Adasch N, Ernst B, Keim D: Topological Vector Spaces, Lecture Notes in Mathematics. Volume 639. Springer, Berlin; 1978:i+125.

    Google Scholar 

  14. Berinde V: On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Mathematica Universitatis Comenianae. New Series 2004,73(1):119–126.

    MATH  MathSciNet  Google Scholar 

  15. Köthe G: Topological Vector Spaces. I, Die Grundlehren der mathematischen Wissenschaften. Volume 159. Springer, New York, NY, USA; 1969:xv+456.

    Google Scholar 

  16. Rhoades BE: Comments on two fixed point iteration methods. Journal of Mathematical Analysis and Applications 1976,56(3):741–750. 10.1016/0022-247X(76)90038-X

    Article  MATH  MathSciNet  Google Scholar 

  17. Hardy GE, Rogers TD: A generalization of a fixed point theorem of Reich. Canadian Mathematical Bulletin 1973, 16: 201–206. 10.4153/CMB-1973-036-0

    Article  MATH  MathSciNet  Google Scholar 

  18. Goebel K, Kirk WA, Shimi TN: A fixed point theorem in uniformly convex spaces. Bollettino. Unione Matematica Italiana. Serie IV 1973, 7: 67–75.

    MATH  MathSciNet  Google Scholar 

  19. Olaleru JO: On the convergence of Mann iteration scheme in locally convex spaces. Carpathian Journal of Mathematics 2006,22(1–2):115–120.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of Lagos, P.O. Box 31, Lagos, Nigeria

    JO Olaleru & H Akewe

Authors
  1. JO Olaleru
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H Akewe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JO Olaleru.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Olaleru, J., Akewe, H. An Extension of Gregus Fixed Point Theorem. Fixed Point Theory Appl 2007, 078628 (2007). https://doi.org/10.1155/2007/78628

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/78628

Keywords