Skip to main content
Log in

Equivalence of certain iteration processes via averaged mappings

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

In this paper, we show that Picard, Mann, Ishikawa and Picard-Mann hybrid iterative processes associated with average mapping converge strongly to the fixed point of the mapping satisfies enriched Zamfirescu condition and all these iterative processes are equivalent to each others. An application of the main results to variational inequality problem are also given.

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

Access this article

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

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

References

  1. Abbas, M., R. Anjum, and V. Berinde. 2021. Equivalence of Certain Iteration Processes Obtained by Two New Classes of Operators. Mathematics. 9 (18): 2292.

    Article  Google Scholar 

  2. Abbas, M., R. Anjum, and V. Berinde. 2021. Enriched multivalued contractions with applications to differential inclusions and dynamic programming. Symmetry. 13 (8): 1350.

    Article  ADS  Google Scholar 

  3. Abbas, M., R. Anjum, and H. Iqbal, 2022. Generalized enriched cyclic contractions with application to generalized iterated function system. Chaos, Solitons and Fractals. 154.

  4. Abbas, M., R. Anjum, and N. Ismail. 2022. Approximation of fixed points of enriched asymptotically nonexpansive mappings in CAT(0) spaces. II, Ser: Rend. Circ. Mat. Palermo.

    Google Scholar 

  5. Anjum, R., and M. Abbas. 2021. Common Fixed point theorem for modified Kannan enriched contraction pair in Banach spaces and its Applications. J. Filomat. 35 (8): 2485–2495.

    Article  MathSciNet  Google Scholar 

  6. Anjum, R., N. Ismail, and A. Bartwal. 2023. Implication between certain iterative processes via some enriched mappings. The Journal of Analysis. https://doi.org/10.1007/s41478-023-00558-7.

    Article  MathSciNet  Google Scholar 

  7. Abbas, M., R. Anjum, and S. Riasat. 2022. Fixed point results of enriched interpolative Kannan type operators with applications. Appl. Gen. Topol. 23 (2): 391–404.

    Article  MathSciNet  Google Scholar 

  8. Berinde, V. 2002. Iterative approximation of fixed points. Baia Mare, Romania: Efemeride.

    Google Scholar 

  9. Berinde, V., and M. Păcurar. 2020. Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 22: 1–10.

    Article  MathSciNet  Google Scholar 

  10. Berinde, V., and M. Păcurar. 2021. Approximating fixed points of enriched Chatterjea contractions by Krasnoselskij iterative algorithm in Banach spaces. J. Fixed Point Theory Appl. 23: 66.

    Article  MathSciNet  Google Scholar 

  11. Berinde, V., and M. Păcurar. 2020. Kannan’s fixed point approximation for solving split feasibility and variational inequality problems. J. Comput. Appl. Math. 386: 113217.

    Article  MathSciNet  Google Scholar 

  12. Berinde, V. 2019. Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces. Carpathian J. Math. 2019 (35): 293–304.

    Article  MathSciNet  Google Scholar 

  13. Baillon, J.B., R.E. Bruck, and S. Reich. 1978. On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4: 1–9.

    MathSciNet  Google Scholar 

  14. Byrne, C. 2004. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20: 103–120.

    Article  ADS  MathSciNet  Google Scholar 

  15. Ishikawa, S. 1974. Fixed points by a new iteration method, Proc.Amer.Math.Soc., 44:147-150.

  16. Mann, WR. 1953. Mean value methods in iterations, Proc.Amer.Math.Soc., 4:506-510.

  17. Picard, E. 1890. Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pures et Appl. 6: 145–210.

    Google Scholar 

  18. Psupathi, R., A.K.B. Chand, and M.A. Navascués. 2020. Cyclic iterated function systems. Journal of Fixed Point Theory and Applications 22 (3): 1–17.

    MathSciNet  Google Scholar 

  19. Rhoades, B.E., and ŞM. Şoltuz. 2003. On the equivalence of Mann and Ishikawa iteration methods. Int. J. Math. Math. Sci. 7: 451–459.

    Article  MathSciNet  Google Scholar 

  20. Rhoades, B.E., and Ş Şoltuz. 2003. The equivalence of the Mann and Ishikawa iteration for non-Lipschitzian operators. Int. J. Math. Math. Sci. 42: 2645–2652.

    Article  MathSciNet  Google Scholar 

  21. Rhoades, B.E., and ŞM. Şoltuz. 2003. The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically pseudocontractive map. J. Math. Anal. Appl. 283: 681–688.

    Article  MathSciNet  Google Scholar 

  22. Rhoades, B.E., and ŞM. Şoltuz. 2004. The equivalence of Mann and Ishikawa iteration for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly accretive maps. Tamkang J. Math. 35: 235–245.

    Article  MathSciNet  Google Scholar 

  23. Rhoades, B.E., and ŞM. Şoltuz. 2004. The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps. J. Math. Anal. Appl. 289: 266–278.

    Article  MathSciNet  Google Scholar 

  24. Khan, S.H. 2013. A Picard-Mann hybrid iterative process. Fixed Point Theory and Applications. 1: 1–10.

    MathSciNet  CAS  Google Scholar 

  25. Şoltuz, ŞM. 2003. An equivalence between the convergences of Ishikawa. Mann and Picard iterations. Math. Commun. 8: 15–22.

    MathSciNet  Google Scholar 

  26. Weng, X. 1991. Fixed point iteration for local strictly psedocontractive mapping. Proc. Amer. Math. Soc. 113: 727–731.

    Article  MathSciNet  Google Scholar 

  27. Vasilev, F.P. 1988. Numerical Methods for Solving Extremal Problems, 2nd ed. Moscow, Russia: Nauka.

    Google Scholar 

  28. Zamfirescu, T. 1972. Fix point theorems in metric spaces. Archiv der Mathematik 23 (1): 292–298.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rizwan Anjum.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by S Ponnusamy.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anjum, R., Khan, S.H. Equivalence of certain iteration processes via averaged mappings. J Anal 32, 1181–1198 (2024). https://doi.org/10.1007/s41478-023-00679-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-023-00679-z

Keywords

Mathematics Subject Classification

Navigation