Skip to main content
SpringerLink
Log in

On the functional equation \(G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) \) and means

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We consider the functional equation \(G\left( x,G\left( y,x\right) \right) =G\left( y,G\left( x,y\right) \right) \), posed in Jarczyk and Jarczyk (Aequ Math 72:198–200, 2006). We show that every continuous and reducible solution generates a mean resembling the weighted quasi-arithmetic mean, but no weighted quasi-arithmetic mean is a solution of this equation. This fact particularly implies that the equation is not a direct consequence of the bisymmetry equation and the reflexivity condition. The closedness of the family of solutions with respect to conjugacy is noted. Finally, the translative solutions, homogeneous solutions, and suitable iterative composite functional equation for single variable functions are discussed.

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. J. Aczél, Lectures on Functional Equations and Their Applications (Academic Press, New York, 1966)

    MATH  Google Scholar 

  2. M. Balcerowski, On the functional equation \( x+f(y+f\left( x\right) )=y+f(x+f\left( y\right) )\). Aequ. Math. 75, 297–303 (2008)

    Article  MathSciNet  Google Scholar 

  3. J. Jarczyk, W. Jarczyk, On a problem of N Brillouët–Belluot. Aequ. Math. 72, 198–200 (2006)

    Article  Google Scholar 

  4. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities-Cauchy’s Equations and Inequalities (Polish Scientific Publishers, Warsaw, 1985)

    MATH  Google Scholar 

  5. J. Rätz, On the functional equation \( x+f(y+f\left( x\right) )=y+f(x+f\left( y\right) )\). Aequ. Math. 86, 187–200 (2013)

    Article  Google Scholar 

  6. J. Rätz, On the functional equation \( x+f(y+f\left( x\right) )=y+f(x+f\left( y\right) )\) II. Aequ. Math. 89, 169–186 (2015)

    Article  Google Scholar 

  7. M. Sablik, Remark 4, Report of meeting, 10th International Conference on Functional Equations and Inequalities (Bȩ dlewo, 2005). Ann. Acad. Paed. Crac. Stud. Math. 5, 127–165 (2006)

    Google Scholar 

Download references

Acknowledgements

This work was partially supported by Zhejiang Provincial Natural Science Foundation of China under Grant No.LY18A010017 and the National Science Foundation of China Grant #11701400.

Author information

Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang, 314001, People’s Republic of China

    Lin Li

  2. Computer Science and Econometrics, Faculty of Mathematics, University of Zielona Góra, Szafrana 4a, 65-516, Zielona Gora, Poland

    Janusz Matkowski

Authors
  1. Lin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Janusz Matkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lin Li.

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

Li, L., Matkowski, J. On the functional equation \(G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) \) and means. Period Math Hung 80, 28–37 (2020). https://doi.org/10.1007/s10998-019-00301-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-019-00301-5

Keywords

Mathematics Subject Classification

Search

Navigation