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A composite functional equation from algebraic aspect

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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.

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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
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  2. Attila Házy
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  3. Tibor Juhász
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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.

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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

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

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