Aequationes mathematicae

, Volume 89, Issue 1, pp 49–56 | Cite as

Stability of generalized Cauchy equations

  • Roman Badora
  • Barbara Przebieracz
  • Peter Volkmann
Open Access
Article

Abstract

We investigate the stability of the functional equation
$$f(xy) = g(x)h(y) + k(y)$$
on amenable semigroups. This equation is a common generalization of two Pexider equations stemming from Cauchy’s additive and multiplicative functional equations, and it is a simple case of the Levi-Civita equation.

Keywords

Cauchy equations stability in the sense of Hyers–Ulam Pexider equations Levi-Civita functional equation 

Mathematics Subject Classification

Primary 39B82 Secondary 39B22 

References

  1. 1.
    Aczél, J.: Vorlesungen über Funktionalgleichungen und ihre Anwendungen. Birkhäuser Verlag Basel, 1961. English edition: Lectures on Functional Equations and their Applications, Academic Press, New York (1966)Google Scholar
  2. 2.
    Aczél J.: On Applications and Theory of Functional Equations. Birkhäuser, Basel (1969)MATHGoogle Scholar
  3. 3.
    Aczél J., Dhombres J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)CrossRefMATHGoogle Scholar
  4. 4.
    Ebanks B.: General solution of a simple Levi–Civitá functional equation on non-abelian groups. Aequationes Math. 85(3), 359–378 (2013)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Járai A., Székelyhidi L.: Regularization and general methods in the theory of functional equations. Aequationes Math. 52(1), 10–29 (1996)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    McKiernan M.A.: Equations of the form \({H(x \circ y) = \sum_i f_i(x)g_i(y)}\). Aequationes Math. 16(1–2), 51–58 (1977)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Shulman, E.: Functional equations of homological type, Ph.D. Thesis, Moscow Institute of Electronics and Mathematics, Moscow 1994 (Russian)Google Scholar
  8. 8.
    Shulman E.: Group representations and stability of functional equations. J. London Math. Soc. 52(2), 111–120 (1996)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Stetkaer H.: Functional equations and matrix-valued spherical functions. Aequationes Math. 69(3), 271–292 (2005)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Székelyhidi, L.: On the Levi-Civita functional equation. Ber. Math.-Statist. Sekt. Forsch. Graz, vol. 301, pp. 23 (1988)Google Scholar
  11. 11.
    Székelyhidi L.: Stability properties of functional equations describing the scientific laws. J. Math. Anal. Appl. 150, 151–158 (1990)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Vincze E.: Eine allgemeinere Methode in der Theorie der Funktionalgleichungen I. Publ. Math. Debrecen 9, 149–163 (1963)MathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Roman Badora
    • 1
  • Barbara Przebieracz
    • 1
  • Peter Volkmann
    • 2
  1. 1.Instytut MatematykiUniwersytet Śla̧skiKatowicePoland
  2. 2.Institut für AnalysisKITKarlsruheGermany

Personalised recommendations