The Fibonacci–Mann iteration for monotone asymptotically pointwise nonexpansive mappings

  • Buthinah A. Bin DehaishEmail author


In this work, we investigate the convergence of the Fibonacci–Mann iteration associated with a monotone asymptotic pointwise nonexpansive mapping defined in a modular function space. The first main result deals with the modular convergence of such iteration when the mapping is assumed to be compact. Relaxing the compactness assumption, we obtain a \(\rho \)-a.e. convergence of the iteration. These two results are similar to the main conclusions of the original work of Schu.


Asymptotically pointwise nonexpansive mapping fibonacci sequence fixed point Mann iteration process modular function spaces monotone Lipschitzian mapping uniform convexity 

Mathematics Subject Classification

Primary 46B20 47E10 



  1. 1.
    Alfuraidan, M.R., Bachar, M., Khamsi, M.A.: Fixed points of monotone \(\rho \)-asymptotically nonexpansive mappings in modular function spaces. J. Nonlinear Convex Anal. 18–4, 565–573 (2017)MathSciNetGoogle Scholar
  2. 2.
    Alfuraidan, M.R., Khamsi, M.A.: Fibonacci–Mann iteration for monotone asymptotic nonexpansive mappings. Bull. Aust. Math. Soc. 96, 307–316 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bachar, M., Khamsi, M.A.: Recent contributions to fixed point theory of monotone mappings. J. Fixed Point Theory Appl. 19(3), 1953–1976 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bin Dehaish, B.A.: On monotone asymptotic pointwise nonexpansive mappings in modular function spaces (submitted) Google Scholar
  5. 5.
    Bin Dehaish, B.A., Khamsi, M.A.: Fibonacci–Man iteration for monotone asymptotic nonexpansive mappings in modular spaces (submitted) Google Scholar
  6. 6.
    Bin Dehaish, B.A., Khamsi, M.A.: Monotone asymptotic pointwise contractions. Filomat 31(11), 3291–3294 (2017). MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bruck, R.E., Kuczumow, T., Reich, S.: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 65, 169–179 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Carl, S., Heikkilä, S.: Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory. Springer, Berlin (2011)CrossRefGoogle Scholar
  9. 9.
    Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. Wiley, New York (2001)CrossRefGoogle Scholar
  11. 11.
    Khamsi, M.A., Kozlowski, W.M.: On asymptotic pointwise nonexpansive mappings in modular function spaces. J. Math. Anal. Appl. 380, 697–708 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khamsi, M.A., Kozlowski, W.M.: Fixed Point Theory in Modular Function Spaces. Birkhauser, New York (2015)CrossRefGoogle Scholar
  13. 13.
    Khamsi, M.A., Kozlowski, W.M., Reich, S.: Fixed point theory in modular function spaces. Nonlinear Anal. 14, 935–953 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Khamsi, M.A., Kozlowski, W.M., Shutao, C.: Some geometrical properties and fixed point theorems in Orlicz spaces. J. Math. Anal. Appl. 155(2), 393–412 (1991)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004–1006 (1965)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kirk, W.A.: Fixed points of asymptotic contractions. J. Math. Anal. Appl. 277, 645–650 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kirk, W.A.: Asymptotic pointwise contractions, Plenary Lecture, The 8th International Conference on Fixed Point Theory and Its Applications. Chiang Mai University, Thailand, July 16–22 (2007)Google Scholar
  18. 18.
    Kozlowski, W.M.: Modular Function Spaces, Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 122. Dekker, New York (1988)Google Scholar
  19. 19.
    Krasnoselskii, M.A., Rutickii, Ya B.: Convex functions and Orlicz spaces. P. Nordhoff Ltd, Groningen (1961)Google Scholar
  20. 20.
    Milnes, H.W.: Convexity of Orlicz spaces. Pac. J. Math. 7, 1451–1486 (1957)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Orlicz, W.: Über konjugierte Exponentenfolgen. Studia Math. 3, 200–211 (1931)CrossRefGoogle Scholar
  22. 22.
    Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132(5), 1435–1443 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Reich, S., Zaslavski, A.J.: A convergence theorem for asymptotic contractions. J. Fixed Point Theory Appl. 4, 27–33 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Reich, S., Zaslavski, A.J.: Monotone contractive mappings. J. Nonlinear Var. Anal. 1, 391–401 (2017)zbMATHGoogle Scholar
  25. 25.
    Schu, J.: Weak and strong convergence to fixed points of \(\rho \)-asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 153–159 (1991)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Shutao, C.: Geometry of Orlicz spaces. Dissert. Math. 356 (1996)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsFaculty of Science University of JeddahJeddahSaudi Arabia

Personalised recommendations