Skip to main content
Log in

Comments on “Fractal functions associated with Reich contractions: an approximation of chaotic attractors”

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

A Correction to this article was published on 04 March 2024

This article has been updated

Abstract

It is the aim of this paper to demonstrate the error in Lemma 1 and Theorem 3 of the contribution from Priyanka et al. (Numer. Algo., 2023). Their novel attempt to study fractal interpolation remains open.

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.

Similar content being viewed by others

Data availability

No datasets were generated or analysed during the current study.

Change history

References

  1. Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)

    Article  MathSciNet  Google Scholar 

  2. Barnsley, M. F. : Fractals everywhere. Academic Press, Dublin (1988). 2nd Edition, Morgan Kaufmann 1993; 3rd Edition, Dover Publications, (2012)

  3. Secelean, N.A.: The existence of the attractor of countable iterated function systems. Mediterr. J. Math. 9, 61–79 (2012)

    Article  MathSciNet  Google Scholar 

  4. Secelean, N.A.: Iterated function systems consisting of F-contractions. Fixed Point Theory Appl. 277(1), 1–13 (2013)

    MathSciNet  Google Scholar 

  5. Imdad, M., Alfapih, W.M., Khan, I.A.: Weak \(\theta \)-contractions and some fixed point results with applications to fractal theory. Adv Differ Equ. 439, 1–18 (2018)

    MathSciNet  Google Scholar 

  6. Pasupathi, R., Chand, A.K.B., Navascueś, M.A.: Cyclic iterated function systems. J. Fixed Point Theory Appl. 22(58), 1–17 (2020)

    MathSciNet  Google Scholar 

  7. Navascueś, M. A., Pacurar, C., Drakopoulus, V. : Scale-free fractal interpolation. Fractal and Fractional 6(10) (2022)

  8. Ri, S.: A new nonlinear fractal interpolation function. Fractals 25(6), 1750063 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  9. Pacurar, C. M.: A countable fractal interpolation scheme. Fixed point theory and Appl. 76 (161) (2021)

  10. Dung, N.V., Petruşel, A.: On iterated function systems consisting of Kannan maps, Reich maps, Chatterjea type maps, and related results. J. Fixed Point Theory Appl. 19, 2271–2285 (2017)

    Article  MathSciNet  Google Scholar 

  11. Kifayat, U., Katiyar, S.K.: Generalized G-Hausdorff space and applications in fractals. Chaos Solit. Fractals 174, 113819 (2023)

    Article  MathSciNet  Google Scholar 

  12. Prithvi, B.V., Katiyar, S.K.: Comments on “Fractal set of generalized countable partial iterated function system with generalized contractions via partial Hausdorff metric’’. Topol & Appl. 341, 108687 (2024)

    Article  MathSciNet  Google Scholar 

  13. Prithvi, B. V., Katiyar, S. K. : Generalized Kannan maps with application to iterated function system, Advanced Mathematical Analysis and Applications, Taylor & Francis group (2023). (CRC Press)

  14. Prithvi, B. V., Katiyar, S. K. : A revisit of the Kannan map. J. Anal. (2024) . https://doi.org/10.1007/s41478-023-00715-y

  15. Shouchuan Hu, Papageorgiou, N. S. : Handbook of multivalued analysis 1. Kluwer Academic Publishers, New York, (1997)

  16. Reich, S.: Some remarks concerning contraction mappings. Canad. Math. Bull. 14, 121–124 (1971)

    Article  MathSciNet  Google Scholar 

  17. Rus, I.A.: Picard operators and applications. Sci. Math. Jpn. 58, 191–219 (2003)

    MathSciNet  Google Scholar 

  18. Prithvi, B.V., Katiyar, S.K.: Revisiting fractal through nonconventional iterated function systems. Chaos Solitons & Fractals 170, 113337 (2023)

    Article  MathSciNet  Google Scholar 

  19. Prithvi, B.V., Katiyar, S.K.: Interpolative operators: Fractal to Multivalued fractal. Chaos Solitons & Fractals 164, 112449 (2022)

    Article  MathSciNet  Google Scholar 

  20. Kifayat, U., Katiyar, S.K.: Cyclic weak \(\phi \) iterated function system. Topol Algebra and its Appl. 10, 161–166 (2022)

    MathSciNet  Google Scholar 

  21. Prithvi, B. V., Katiyar, S. K.: Revisiting Ćirić-Reich-Rus type iterated function systems. Rend. Circ. Mat. Palermo, II. Ser (2024) (To appear)

  22. Chandra, S., Verma, S., Abbas, S.: Construction of fractal functions using Kannan mappings and smoothness analysis, (2023). arXiv:2301.03075

  23. Priyanka, T.M.C., Gowrishankar, A., Cao, J.: Fractal functions associated with Reich contractions: an approximation of chaotic attractors. Numer Algor (2023). https://doi.org/10.1007/s11075-023-01687-7

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the editor and reviewer(s) for their useful comments and constructive remarks that helped to improve the presentation of the paper. S. K. Katiyar acknowledges the financial support received from Dr B R Ambedkar National Institute of Technology (NIT), Jalandhar, Punjab, 144011, India (Institute seed grant).

Author information

Authors and Affiliations

Authors

Contributions

Both the authors have equally contributed.

Corresponding author

Correspondence to S. K. Katiyar.

Ethics declarations

Ethics approval

Not applicable

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

The original online version of this article was revised: In the Abstract the year in “(Numer. Algo., 2003)” is incorrect. It should be presented as “(Numer. Algo., 2023)”.

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

Prithvi, B.V., Katiyar, S.K. Comments on “Fractal functions associated with Reich contractions: an approximation of chaotic attractors”. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01780-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11075-024-01780-5

Keywords

Mathematics Subject Classification (2010)

Navigation