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Afrika Matematika

, Volume 29, Issue 1–2, pp 1–22 | Cite as

Solution of several functional equations on abelian groups with involution

  • B. Fadli
  • D. Zeglami
  • S. Kabbaj
Article

Abstract

Let G be a locally compact abelian Hausdorff group, let \(\sigma \) be a continuous involution on G, and let \(\mu ,\nu \) be regular, compactly supported, complex-valued Borel measures on G. We determine the continuous solutions \(f,g:G\rightarrow {\mathbb {C}}\) of each of the two functional equations
$$\begin{aligned}&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=f(x)g(y),\quad x,y\in G,\\&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=g(x)f(y),\quad x,y\in G, \end{aligned}$$
in terms of characters and additive functions. These equations provides a common generalization of many functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Van Vleck’s, or Wilson’s equations. So, several functional equations will be solved.

Keywords

Functional equation Involution d’Alembert Gajda Wilson 

Mathematics Subject Classification

Primary 39B32 39B52 

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Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesIbn Tofail UniversityKenitraMorocco
  2. 2.Department of Mathematics, E.N.S.A.MMoulay Ismail UniversityMeknesMorocco

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