Skip to main content

A composite functional equation from algebraic aspect

Aequationes mathematicae volume 86, pages 57–64 (2013)Cite this article

Abstract

In this paper we discuss the composite functional equation

$$f(x+2f(y)) = f(x)+y+f(y)$$

on an Abelian group. This equation originates from Problem 10854 of the American Mathematical Monthly. We give an algebraic description of the solutions on uniquely 3-divisible Abelian groups, and then we construct all solutions f of this equation on finite Abelian groups without elements of order 3 and on divisible Abelian groups without elements of order 3 including the additive group of real numbers.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Aczél, J., Dhombres, J.: Functional equations in several variables. Cambridge University Press, Cambridge (1989)

  2. Boros Z., Daróczy Z.: A composite functional equation with additive solutions. Publ. Math. Debrecen 69(1-2), 245–253 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Daróczy Z., Páles Zs.: Gauss-composition of means and the solution of the Matkowski–Sutô problem. Publ. Math. Debrecen 61(1-2), 157–218 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Chao, W.W.: Problem 10854. Amer. Math. Monthly 108(2), 171 (2001)

  5. Fechner W.: On a composite functional equation on Abelian groups. Aequationes Math. 78(1-2), 185–193 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fuchs, L.: Abelian groups. Publishing House of the Hungarian Acedemy of Sciences, Budapest (1958)

  7. Gilányi A., Páles Zs.: A regularity theorem for composite functional equations. Arch. Math. (Basel) 77, 317–322 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Henderson D.: Continuous additive functions; solution to Problem 10854. Amer. Math. Monthly 111(8), 725–726 (2004)

    Google Scholar 

  9. Kochanek T.: On a composite functional equation fulfilled by modulus of an additive function. Aequationes Math. 80(1–2), 155–172 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kuczma, M.: In: Gilányi, A. (ed.) An introduction to the theory of functional equations and inequalities, 2nd edn. Birkhäuser, Basel (2009)

  11. Járai, A.: Regularity properties of functional equations in several variables. Advances in Mathematics, vol. 8. Springer, New York (2005)

  12. Maksa Gy., Páles Zs.: On a composite functional equation arising in utility theory. Publ. Math. Debrecen 65(1-2), 215–221 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Matkowski J., Okrzesik J.: On a composite functional equation. Demonstratio Math. 36(3), 653–658 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Okrzesik J.: On the existence of Lipschitzian solution of a composite functional equation. Demonstratio Math. 39(1), 97–100 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Páles Zs.: A regularity theorem for composite functional equations. Acta Sci. Math. (Szeged) 69(3-4), 591–604 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Páles, Zs.: Regularity problems and results concerning composite functional equations. Tatra Mt. Math. Publ. 34, part II, 289–306 (2006)

  17. Rätz J.: On a functional equation related to projections of Abelian groups. Aequationes Math. 70(3), 279–297 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Probability Theory, University of Debrecen, Pf. 12, 4010, Debrecen, Hungary

    Pál Burai

  2. TU Berlin Institut für Mathematik, Str. des 17.Juni 136, 10623, Berlin, Germany

    Pál Burai

  3. Department of Applied Mathematics, University of Miskolc, 3515, Miskolc-Egyetemváros, Hungary

    Attila Házy

  4. Institute of Mathematics and Informatics, Eszterházy Károly College, Leányka út 4, 3300, Eger, Hungary

    Tibor Juhász

Authors
  1. Pál Burai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Attila Házy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tibor Juhász
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tibor Juhász.

Additional information

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK-81402 and OTKA “Mobility” call Human-MB08A-84581. The second-named author’s research was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Burai, P., Házy, A. & Juhász, T. A composite functional equation from algebraic aspect. Aequat. Math. 86, 57–64 (2013). https://doi.org/10.1007/s00010-013-0211-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-013-0211-0

Mathematics Subject Classification (2010)

  • 39B12
  • 39B52

Keywords

  • Composite functional equation
  • iterative functional equation
  • additive mapping
  • automorphism
  • Hamel basis
  • Abelian group