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An Extension of Gregus Fixed Point Theorem

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 .

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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.

    MathSciNet  Article  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

    MATH  MathSciNet  Article  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

    MATH  MathSciNet  Article  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

    MATH  MathSciNet  Article  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

    MATH  MathSciNet  Article  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

    MATH  MathSciNet  Article  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 

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Correspondence to JO Olaleru.

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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

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

Keywords

  • Point Theorem
  • Differential Geometry
  • Computational Biology