Approximation of Common Fixed Point of Two Quasi-nonexpansive Mappings in Convex Metric Spaces



We use a simple iterative algorithm for two quasi-nonexpansive mappings to approximate their common fixed point through \( \triangle \)-convergence and strong convergence of the algorithm. Our results are new in the literature of metrical fixed point theory and are also valid in CAT(0) spaces.


Convex metric space quasi-nonexpansive mapping jointly demiclosed principle common fixed point iterative algorithm convergence 

Mathematics Subject Classification

47H09 47H10 


  1. 1.
    Aoyama, K., Kohsaka, F.: Fixed point theorems for \(\alpha \)-nonexpansive mappings in Banach spaces. Nonlinear Anal. 74, 4387–4391 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Byren, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl 20, 120–130 (2004)MathSciNetGoogle Scholar
  3. 3.
    Das, G., Debata, J.P.: Fixed points of quasi-nonexpansive mappings. Indian J. Pure. Appl. Math. 17, 1263–1269 (1986)MathSciNetMATHGoogle Scholar
  4. 4.
    Diaz, J.B., Metcalf, F.T.: On the structure of the set of subsequential limit points of successive approximations. Bull. Am. Math. Soc. 73, 516–519 (1967)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fukhar-ud-din, H.: Existence and approximation of fixed points in convex metric spaces. Carpathian J. Math. 30, 175–185 (2014)MathSciNetMATHGoogle Scholar
  6. 6.
    Fukhar-ud-din, H.: One step iterative scheme for a pair of nonexpansive mappings in a convex metric space. Hacet. J. Math. Stat. 44, 1023–1031 (2015)MathSciNetMATHGoogle Scholar
  7. 7.
    Fukhar-ud-din, H.: Convergence of Ishikawa type iteration process of three quasi-nonexpansive mappings in a convex metric space. Annale Univ. Ovidius Constanta Mathematica 23(2), 83–92 (2015)MathSciNetMATHGoogle Scholar
  8. 8.
    Fukhar-ud-din, H., Saleh, K.: One-step iterations for a finite family of generalized nonexpansive mappings in CAT(0) spaces. Bull. Malays. Math. Sci. Soc. 41, 597–608 (2018). MathSciNetGoogle Scholar
  9. 9.
    Ghoncheh, S.J.H., Razani, A.: Fixed point theorems for some generalized nonexpansive mappings in ptolemy spaces. Fixed Point Theory Appl. 2014, 76 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ishikawa, S.: Fixed point by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khan, A.R., Khamsi, M.A., Fukhar-ud-din, H.: Strong convergence of a general iteration scheme in CAT\(\left(0\right) \) -spaces. Nonlinear Anal. 74, 783–791 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Khan, S.H., Fukhar-ud-din, H.: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Anal. 61, 1295–1301 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Khan, S.H., Fukhar-ud-din, H.: Convergence theorems for two finite families of some generalized nonexpansive mappings in hyperbolic spaces. J. Nonlinear Sci. Appl. 10, 734–743 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Khan, S.H., Takahashi, W.: Approximating common fixed points of two asymptotically nonexpansive mappings. Sci. Math. Jpn. 53, 143–148 (2001)MathSciNetMATHGoogle Scholar
  15. 15.
    Khan, S.H., Abbas, M., Khan, A.R.: Common fixed points of two nonexpansive mappings by a new one-step iteration process. Iran. J. Sci. Tech., Trans. A 33(A3), 249–257 (2009)MathSciNetGoogle Scholar
  16. 16.
    Kohsaka, F., Takahashi, W.: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. (Basel) 91(2), 166–177 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kuhfittig, P.K.F.: Common fixed points of nonexpansive mappings by iteration. Pac. J. Math. 97, 137–139 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mann, R.W.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Moosaei, M.: On fixed points of fundamentally nonexpansive mappings in Banach spaces. Int. J. Nonlinear Anal. Appl. 7, 219–224 (2016)MATHGoogle Scholar
  20. 20.
    Naraghirad, E.: Approximation of common fixed points of nonlinear mappings satisfying jointly demiclosedness principle in Banach spaces. Mediterr. J. Math. 14, 162 (2017)CrossRefMATHGoogle Scholar
  21. 21.
    Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Shimizu, T., Takahashi, W.: Fixed points of multivalued mappings in certain convex metric spaces. Topol. Methods Nonlinear Anal. 8, 197–203 (1996)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 341, 1088–1095 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Takahashi, W.: A convexity in metric spaces and nonexpansive mappings. Kodai Math. Sem. Rep. 22, 142–149 (1970)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Takahashi, W.: Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 11, 79–88 (2010)MathSciNetMATHGoogle Scholar
  26. 26.
    Takahashi, W., Tamura, T.: Convergence theorems for a pair of nonexpansive mappings. J. Nonlinear Convex Anal. 5, 45–58 (1995)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of MathematicsThe Islamia University of BahawalpurBahawalpurPakistan

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