Superstability of a Van Vleck’s type functional equation for the sine



Let G be a group and let a be a fixed element of G, not necessarily belongs to the center of G. In this paper we study the superstability of the following functional equations
$$\begin{aligned} f(ax\sigma (y))-f(axy)=2f(x)g(y),\quad x,\ y\in G, \end{aligned}$$
$$\begin{aligned} f(ax\sigma (y))-f(axy)=2g(x)h(y),\quad x,\ y\in G, \end{aligned}$$
where \(\sigma : G \rightarrow G\) is an involution and fgh are unknown complex valued functions.


Functional equation Superstability Stability Involution 

Mathematics Subject Classification

Primary 39B32 Secondary 39B72 



We wish to express our thanks to the referee for valuable comments and useful suggestions.


  1. 1.
    Badora, R.: On the stability of the cosine functional equation. Rocznik NaukowoDydaktyczny. Prace Matematyczne 15, 5–14 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Badora, R., Ger, R.: On some trigonometric functional inequalities, Functional Equations—Results and Advances, pp. 3–15 (2002)Google Scholar
  3. 3.
    Baker, J.A.: The stability of the cosine equation. Proc. Am. Math. Soc. 80, 411–416 (1980)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baker, J., Lawrence, J., Zorzitto, F.: The stability of the equation \(f(x+y) = f(x)f(y)\). Proc. Am. Math. Soc. 74(2), 242–246 (1979)MATHGoogle Scholar
  5. 5.
    Hyers, D.H.: On the stability of the linear functional equatinon. Proc. Natl. Acad. Sci. USA 27, 93–101 (1941)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hyers, D.H., Isac, G.I., Rassias, ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications, vol. 34. Birkhauser, Boston (1998)CrossRefMATHGoogle Scholar
  7. 7.
    Kim, G.H.: On the stability of trigonometric functional equations. Advances in Difference Equations, Article ID 90405 (2007)Google Scholar
  8. 8.
    Kim, G.H.: On the stability of the Pexiderized trigonometric functional equation. Appl. Math. Comput. 203(1), 99–105 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Lehlou, F., Moussa, M., Roukbi, A., Kabbaj, S.: On the superstability of the cosine and sine type functional equations. Ann. Univ. Paedagog. Crac. Stud. Math. 15, 113–121 (2016)MathSciNetMATHGoogle Scholar
  10. 10.
    Lehlou, F., Moussa, M., Roukbi, A., Kabbaj, S.: On the stability of vector valued Pexiderized functional equations. Res. Commun. Math. Math. Sci. 8(1), 57–67 (2017)Google Scholar
  11. 11.
    Rassias, J.M., Zeglami, D., Charifi, A.: On the stability of a class of cosine type functional equations. Bol. Soc. Paran. Mat. 37(2), 35–49 (2019)CrossRefGoogle Scholar
  12. 12.
    Roukbi, A., Zeglami, D., Kabbaj, S.: Hyers–Ulam stability of Wilson’s functional equation. Math. Sci. Adv. Appl. 22, 19–26 (2013)MATHGoogle Scholar
  13. 13.
    Stetkaer, H.: Van Vlek’s functional equation for the sine. Aeq. Math. 90(1), 25–34 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ulam, S.M.: Problems in Modern Mathematics, chap.VI, Science edition. Wiley, New York (1964)Google Scholar
  15. 15.
    Zeglami, D., Kabbaj, S.: On the superstability of trigonometric type functional equations. Br. J. Math. Comput. Sci. 4(8), 1146–1155 (2014)CrossRefMATHGoogle Scholar
  16. 16.
    Zeglami, D., Roukbi, A., Kabbaj, S.: Hyers–Ulam stability of generalized Wilson’s and d’Alembert’s functional equations. Afr. Mat. 26, 215–223 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesIbn Tofail universityKenitraMorocco

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