Skip to main content
SpringerLink
Log in

Mean convergence theorems using hybrid methods to find common fixed points for noncommutative nonlinear mappings in Hilbert spaces

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This paper considers approximation methods to find common fixed points for two general nonlinear mappings that contain generalized hybrid mappings or normally 2-generalized hybrid mappings. First, we combine Nakajo and Takahashi’s hybrid method with a mean iteration method and prove three strong convergence theorems that approximate common fixed points for nonlinear mappings. We then develop Takahashi, Takeuchi and Kubota’s shrinking projection method and prove three strong convergence theorems. Nonlinear mappings are not necessarily continuous or commutative.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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., Iemoto, S., Kohsaka, F., Takahashi, W.: Fixed point and ergodic theorems for \(\lambda \)-hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 11, 335–343 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Atsushiba, S.: Fixed point property for normally 2-generalized hybrid mappings. J. Nonlinear Var. Anal. 3(2), 149–158 (2019)

    MathSciNet  MATH  Google Scholar 

  3. Atsushiba, S., Takahashi, W.: Approximating common fixed points of two nonexpansive mappings in Banach spaces. Bull. Aust. Math. Soc. 57, 117–127 (1998)

    Article  MathSciNet  Google Scholar 

  4. Baillon, J.B.: Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert. C. R. Acad. Sci. Ser. A-B 280, 1511–1514 (1975)

    MathSciNet  MATH  Google Scholar 

  5. Cholamjiak, P., Cholamjiak, W.: Fixed point theorems for hybrid multivalued mappings in Hilbert spaces. J. Fixed Point Theory Appl. 18, 673–688 (2016)

    Article  MathSciNet  Google Scholar 

  6. Cholamjiak, W., Suantai, S., Cho, Y.J.: Fixed points for nonspreading-type multi-valued mappings: existence and convergence results. Ann. Acad. Rom. Sci. Ser. Math. Apll 10(2), 838–844 (2018)

    MathSciNet  Google Scholar 

  7. Hojo, M., Kondo, A., Takahashi, W.: Weak and strong convergence theorems for commutative normally 2-generalized hybrid mappings in Hilbert spaces. Linear Nonlinear Anal. 4, 117–134 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Hojo, M., Takahashi, S., Takahashi, W.: Attractive point and ergodic theorems for two nonlinear mappings in Hilbert spaces. Linear Nonlinear Anal. 3, 275–286 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Hojo, M., Takahashi, W.: Weak and strong convergence theorems for two commutative nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal. 18, 1519–1533 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Hojo, M., Takahashi, W., Termwuttipong, I.: Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces. Nonlinear Anal. 75, 2166–2176 (2012)

    Article  MathSciNet  Google Scholar 

  11. Iemoto, S., Takahashi, W.: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71, 2082–2089 (2009)

    Article  MathSciNet  Google Scholar 

  12. Igarashi, T., Takahashi, W., Tanaka, K.: Weak convergence theorems for nonspreading mappings and equilibrium problems, in Nonlinear Analysis and Optimization (S. Akashi, W. Takahashi and T. Tanaka Eds.), Yokohama Publishers, Yokohama 75–85 (2008)

  13. Itoh, S., Takahashi, W.: The common fixed point theory of singlevalued mappings and multivalued mappings. Pacific J. Math. 79, 493–508 (1978)

    Article  MathSciNet  Google Scholar 

  14. Kimura, Y., Takahashi, W., Toyoda, M.: Convergence to common fixed points of a finite family of nonexpansive mappings. Arch. Math. 84, 350–363 (2005)

    Article  MathSciNet  Google Scholar 

  15. Kocourek, P., Takahashi, W., Yao, J.C.: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert Spaces. Taiwanese J. Math. 14, 2497–2511 (2010)

    Article  MathSciNet  Google Scholar 

  16. Kohsaka, F.: Existence and approximation of common fixed points of two hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 16, 2193–2205 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)

    Article  MathSciNet  Google Scholar 

  18. Kondo, A.: Strong approximation using hybrid methods to find common fixed points of noncommutative nonlinear mappings in Hilbert spaces, unpublished manuscript

  19. Kondo, A., Takahashi, W.: Attractive point and weak convergence theorems for normally \(N\)-generalized hybrid mappings in Hilbert spaces. Linear Nonlinear Anal. 3, 297–310 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Kondo, A., Takahashi, W.: Strong convergence theorems of Halpern’s type for normally \(2\)-generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 19, 617–631 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Kondo, A., Takahashi, W.: Approximation of a common attractive point of noncommutative normally \(2\)-generalized hybrid mappings in Hilbert spaces. Linear Nonlinear Anal. 5, 279–297 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Kondo, A., Takahashi, W.: Weak convergence theorems to common attractive points of normally \(2\)-generalized hybrid mappings with errors. J. Nonlinear Convex Anal. 21, 2549–2570 (2020)

    MathSciNet  MATH  Google Scholar 

  23. Kondo, A., Takahashi, W.: Strong convergence theorems for finding common attractive points of normally \(2\)-generalized hybrid mappings and applications. Linear Nonlinear Anal. 6, 421–438 (2020)

    MathSciNet  MATH  Google Scholar 

  24. Ma, L., GAO, X., Zhou, H.: Three kinds of hybrid algorithms and their numerical realizations for finite family of quasi-nonexpansive mappings, J. Nonlinear Funct. Anal. 2019(8), (2019)

  25. Maruyama, T., Takahashi, W., Yao, M.: Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal. 12, 185–197 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–378 (2003)

    Article  MathSciNet  Google Scholar 

  27. Shimizu, T., Takahashi, W.: Strong convergence to common fixed points of families of nonexpansive mappings. J. Math. Anal. Appl. 211, 71–83 (1997)

    Article  MathSciNet  Google Scholar 

  28. Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5, 387–404 (2001)

    Article  MathSciNet  Google Scholar 

  29. Takahashi, W.: Nonlinear Functional Analysis. Fixed Point Theory and its Applications. Yokohama Publishes, Yokohama (2000)

    MATH  Google Scholar 

  30. Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishes, Yokohama (2009)

    MATH  Google Scholar 

  31. 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 

  32. Takahashi, W.: Weak and strong convergence theorems for noncommutative two generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 19, 867–880 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)

    Article  MathSciNet  Google Scholar 

  34. Takahashi, W., Wen, C.F., Yao, J.C.: Strong convergence theorems by hybrid methods for noncommutative normally 2-generalized hybrid mappings in Hilbert spaces. Appl. Anal. Optim. 3, 43–56 (2019)

    MathSciNet  Google Scholar 

  35. Takahashi, W., Wong, N.C., Yao, J.C.: Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 13, 745–757 (2012)

    MathSciNet  MATH  Google Scholar 

  36. Tan, B., Xu, S.: Strong convergence of two inertial projection algorithms in Hilbert spaces. J. Appl. Numer. Optim 2(2), 171–186 (2020)

    Google Scholar 

  37. Zegeye, H., Shahzad, N.: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62(11), 4007–4014 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This study is partially supported by the Ryousui Gakujutsu Foundation of Shiga University. The author would like to thank two anonymous referees for their helpful comments and advice.

Author information

Authors and Affiliations

  1. Department of Economics, Shiga University, Banba 1-1-1, Hikone, Shiga, 522-0069, Japan

    Atsumasa Kondo

Authors
  1. Atsumasa Kondo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atsumasa Kondo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kondo, A. Mean convergence theorems using hybrid methods to find common fixed points for noncommutative nonlinear mappings in Hilbert spaces. J. Appl. Math. Comput. 68, 217–248 (2022). https://doi.org/10.1007/s12190-021-01527-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01527-8

Keywords

Mathematics Subject Classification

Search

Navigation