Variants of Wilson's functional equation on semigroups
Commun. Korean Math. Soc.
Published online March 10, 2020
omar AJEBBAR and Elhoucien Elqorachi
Departement of Mathematics, Ibn Zohr University, Faculty of Sciences, Agadir, Morocco
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 equation; Wilson equation.
MSC numbers : Primary 39B52; Secondary 39B32.
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