Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A VARIANT OF D'ALEMBERT'S AND WILSON'S FUNCTIONAL EQUATIONS FOR MATRIX VALUED FUNCTIONS

  • Abdellatif Chahbi (Equipe d'Equations Fonctionnelles et Applications Department of Mathematics Faculty of Sciences Ibn Zohr University) ;
  • Mohamed Chakiri (Equipe d'Equations Fonctionnelles et Applications Department of Mathematics Faculty of Sciences Ibn Zohr University) ;
  • Elhoucien Elqorachi (Equipe d'Equations Fonctionnelles et Applications Department of Mathematics Faculty of Sciences Ibn Zohr University)
  • Received : 2022.03.13
  • Accepted : 2024.04.05
  • Published : 2024.07.31
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Abstract

Given M a monoid with a neutral element e. We show that the solutions of d'Alembert's functional equation for n × n matrices Φ(pr, qs) + Φ(sp, rq) = 2Φ(r, s)Φ(p, q), p, q, r, s ∈ M are abelian. Furthermore, we prove under additional assumption that the solutions of the n-dimensional mixed vector-matrix Wilson's functional equation $$\begin{cases}f(pr, qs) + f(sp, rq) = 2\phi(r, s)f(p, q),\\Φ(p, q) = \phi(q, p),{\quad}p, q, r, s {\in} M\end{cases}$$ are abelian. As an application we solve the first functional equation on groups for the particular case of n = 3.

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References

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