Skip to main content
SpringerLink
Log in

Convergence theorems using Ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaces

  • Original Paper
  • Published:
Advances in Operator Theory Aims and scope Submit manuscript

Abstract

This paper presents weak and strong convergence theorems for finding common fixed points of two nonlinear mappings, where one mapping is demiclosed, and the other is 2-demiclosed. For this purpose, we use Ishikawa type iteration and obtain weak convergence theorems. Nakajo and Takahashi’s hybrid method and Takahashi, Takeuchi, and Kubota’s shrinking projection method are also employed alongside Ishikawa iteration to derive strong convergence. Our proofs do not require the mappings to be commutative or continuous, and the results obtained in this paper extend many theorems in the literature.

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. Alizadeh, S., Moradlou, F.: A weak convergence theorem for 2-generalized hybrid mappings. ROMAI J. 11(1) 131-138 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Alizadeh, S., Moradlou, F.: Weak and strong convergence theorems for m-generalized hybrid mappings in Hilbert spaces. Topol. Methods Nonlinear Anal. 46(1), 315–328 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atsushiba, S.: Nonlinear ergodic theorems for normally 2-generalized hybrid sequences. J. Nonlinear Convex Anal. 20, 251–264 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. 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  MATH  Google Scholar 

  6. Berinde, V.: On the convergence of the Ishikawa iteration in the class of quasi-contractive operators. Acta Math. Univ. Comen. 73(1), 119–126 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Berinde, V.: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics, vol. 1912, 2nd edn. Springer, Berlin (2007)

    MATH  Google Scholar 

  8. 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  MATH  Google Scholar 

  9. 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. Appl. 10(2), 838–844 (2018)

    MathSciNet  Google Scholar 

  10. Hojo, M.: Attractive point and mean convergence theorems for normally generalized hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 18, 2209–2120 (2017)

    MathSciNet  MATH  Google Scholar 

  11. 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 

  12. 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 

  13. 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  MATH  Google Scholar 

  14. 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  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. 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  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  20. 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 

  21. 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  MATH  Google Scholar 

  22. Kohsaka, F., Takahashi, W.: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. 91, 166–177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kondo, A.: Strong approximation using hybrid methods to find common fixed points of noncommutative nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal. 23, 33–58 (2022)

    MathSciNet  MATH  Google Scholar 

  24. Kondo, A.: Mean convergence theorems using hybrid methods to find common fixed points of noncommutative nonlinear mappings in Hilbert spaces. J. Appl. Math. Comput. 68, 217–248 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kondo, A.: Ishikawa’s type mean convergence theorems for finding common fixed points of nonlinear mappings in Hilbert spaces. Rendiconti del Circolo Mat. di Palermo Series 2 (to appear). https://doi.org/10.1007/s12215-022-00742-x

  26. 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 

  27. 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 

  28. 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 

  29. 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 

  30. 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 

  31. 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, Article ID 8, 1–11 (2019)

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

    Article  MathSciNet  MATH  Google Scholar 

  33. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Panyanak, B.: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54(6), 872–877 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rouhani, B.D.: Ergodic theorems for hybrid sequences in a Hilbert space with applications. J. Math. Anal. Appl. 409(1), 205–211 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rouhani, B.D.: Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space. Demonstr. Math. 51(1), 27–36 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sadeewong, P., Saleewong, T., Kumam, P., Cho, Y.J.: The modified viscosity iteration with \(m\)-generalized hybrid mappings and \((a, b)\)-monotone mappings for equilibrium problems. Thai J. Math. 16(1), 243–265 (2018)

    MathSciNet  MATH  Google Scholar 

  39. 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  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  43. 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 

  44. 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 

  45. 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  MATH  Google Scholar 

  46. Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. 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  MATH  Google Scholar 

  48. 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 

  49. Takahashi, W., Yao, J.C.: Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces. Taiwan. J. Math. 15, 457–472 (2011)

    MathSciNet  MATH  Google Scholar 

  50. Tan, K.K., Xu, H.K.: Approximating fixed points of non-expansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wongyai, K., Thianwan, T.: Projection type Ishikawa iteration with perturbations for common fixed points of two nonself generalized asymptotically quasi-nonexpansive mappings. Thai J. Math. 17(3), 843–859 (2019)

    MathSciNet  MATH  Google Scholar 

  52. Yan, F., Su, Y., Feng, Q.: Convergence of Ishikawa’method for generalized hybrid mappings. Commun. Korean Math. Soc. 28(1), 135–141 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. 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  MATH  Google Scholar 

Download references

Acknowledgements

This work is financially supported by the Ryousui Gakujutsu Foundation of Shiga University. The author would appreciate the anonymous reviewers for their useful comments and encouragement.

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

Communicated by Timur Oikhberg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kondo, A. Convergence theorems using Ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaces. Adv. Oper. Theory 7, 26 (2022). https://doi.org/10.1007/s43036-022-00190-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43036-022-00190-5

Keywords

Mathematics Subject Classification

Search

Navigation