On a Variant of μ-Wilson’s Functional Equation with an Endomorphism

  • K. H. Sabour
  • A. Charifi
  • S. Kabbaj


The main goal of this chapter is to find the solutions (f, g) of the generalized variant of μ-d’Alembert’s functional equation
$$\displaystyle f(xy)+\mu (y)f(\varphi (y)x)=2f(x)f(y), $$
and μ-Wilson’s functional equation
$$\displaystyle f(xy)+\mu (y)f(\varphi (y)x)=2f(x)g(y), $$
in the setting of semigroups, monoids, and groups, where φ is an endomorphism not necessarily involutive and μ is a multiplicative function. We prove that their solutions can be expressed in terms of multiplicative and additive functions. Many consequences of these results are presented.


  1. 1.
    A.L. Cauchy, in Cours d’Analyse de l’ecole Polytechnique. Analyse Algebrique, V., Paris, vol. 1 (1821)Google Scholar
  2. 2.
    J. d’Alembert, Addition au Mémoire sur la courbe que forme une corde tendue mise en vibration. Hist. Acad. Berlin 1750, 355–360 (1750)Google Scholar
  3. 3.
    B.R. Ebanks, H. Stetkær, d’Alembert’s other functional equation on monoids with an involution. Aequationes Math. 89, 187–206 (2015)Google Scholar
  4. 4.
    B.R. Ebanks, H. Stetkær, On Wilson’s functional equations. Aequationes Math. 89(2), 339–354 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    E. Elqorachi, A. Redouani, Solutions and Stability of Variant of Wilson’s Functional Equation (2015). arXiv:1505.06512v1 [math.CA]Google Scholar
  6. 6.
    B. Fadli, S. Kabbaj, K.H. Sabour, D. Zeglami, Functional equation on semigroups with an endomorphism. Acta Math. Hung. 150(2) (2016).
  7. 7.
    P.L. Kannappan, A functional equation for the cosine. Can. Math. Bull. 2, 495–498 (1968)CrossRefGoogle Scholar
  8. 8.
    P.L. Kannappan, Functional Equations and Inequalities with Applications (Springer, New York, 2009)CrossRefGoogle Scholar
  9. 9.
    H. Stetkær, Functional Equations on Groups (World Scientific Publishing Co, Singapore, 2013)CrossRefGoogle Scholar
  10. 10.
    W.H. Wilson, On certain related functional equations. Bull. Am. Math. Soc. 26, 300–312 (1919–1920). Fortschr.: 47, 320 (1919–20)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • K. H. Sabour
    • 1
  • A. Charifi
    • 1
  • S. Kabbaj
    • 1
  1. 1.Department of Mathematics, Faculty of SciencesUniversity of Ibn TofailKenitraMorocco

Personalised recommendations