Variants of Wilson's functional equation on semigroups
Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 711-722
Published online March 10, 2020
Printed July 31, 2020
Omar Ajebbar, Elhoucien Elqorachi
Ibn Zohr University; Ibn Zohr University
Abstract : Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equations \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(x)g(y),\, x,y\in S\end{equation*} and \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(y)g(x),\, x,y\in S.\end{equation*}
Keywords : Semigroup, involutive automorphism, multiplicative function, d'Alembert's equation, Wilson's equation
MSC numbers : Primary 39B52; Secondary 39B32
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