Skip to main content
SpringerLink
Log in

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

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aoyama, K., Kohsaka, F.: Fixed point theorems for \(\alpha \)-nonexpansive mappings in Banach spaces. Nonlinear Anal. 74, 4387–4391 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Byren, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl 20, 120–130 (2004)

    MathSciNet  Google Scholar 

  3. Das, G., Debata, J.P.: Fixed points of quasi-nonexpansive mappings. Indian J. Pure. Appl. Math. 17, 1263–1269 (1986)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fukhar-ud-din, H.: Existence and approximation of fixed points in convex metric spaces. Carpathian J. Math. 30, 175–185 (2014)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/s40840-016-0310-x

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ishikawa, S.: Fixed point by a new iteration method. Proc. Am. Math. Soc. 44, 147–150 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Khan, S.H., Takahashi, W.: Approximating common fixed points of two asymptotically nonexpansive mappings. Sci. Math. Jpn. 53, 143–148 (2001)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kuhfittig, P.K.F.: Common fixed points of nonexpansive mappings by iteration. Pac. J. Math. 97, 137–139 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mann, R.W.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  19. Moosaei, M.: On fixed points of fundamentally nonexpansive mappings in Banach spaces. Int. J. Nonlinear Anal. Appl. 7, 219–224 (2016)

    MATH  Google Scholar 

  20. Naraghirad, E.: Approximation of common fixed points of nonlinear mappings satisfying jointly demiclosedness principle in Banach spaces. Mediterr. J. Math. 14, 162 (2017)

    Article  MATH  Google Scholar 

  21. Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shimizu, T., Takahashi, W.: Fixed points of multivalued mappings in certain convex metric spaces. Topol. Methods Nonlinear Anal. 8, 197–203 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 341, 1088–1095 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Takahashi, W.: A convexity in metric spaces and nonexpansive mappings. Kodai Math. Sem. Rep. 22, 142–149 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  25. Takahashi, W.: Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 11, 79–88 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Takahashi, W., Tamura, T.: Convergence theorems for a pair of nonexpansive mappings. J. Nonlinear Convex Anal. 5, 45–58 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    H. Fukhar-ud-din & A. R. Khan

  2. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan

    H. Fukhar-ud-din

Authors
  1. H. Fukhar-ud-din
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. R. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Fukhar-ud-din.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fukhar-ud-din, H., Khan, A.R. Approximation of Common Fixed Point of Two Quasi-nonexpansive Mappings in Convex Metric Spaces. Mediterr. J. Math. 15, 77 (2018). https://doi.org/10.1007/s00009-018-1121-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1121-0

Keywords

Mathematics Subject Classification

Search

Navigation