Solutions and stability of trigonometric functional equations on an amenable group with an involutive automorphism
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Given $\sigma:G\rightarrow G$ an involutive automorphism of a semigroup $G$, we study the solutions and stability of the following functional equations \begin{equation*}f(x\sigma(y))=f(x)g(y)+g(x)f(y),\quad x,y\in G,\end{equation*} \begin{equation*}f(x\sigma(y))=f(x)f(y)-g(x)g(y),\quad x,y\in G\end{equation*} and \begin{equation*}f(x\sigma(y))=f(x)g(y)-g(x)f(y),\quad x,y\in G,\end{equation*} from the theory of trigonometric functional equations. \begin{itemize} \item[(1)] We determine the solutions when $G$ is a semigroup generated by its squares. \item[(2)] We obtain the stability results for these equations, when $G$ is an amenable group. \end{itemize}
Abstract
Given $\sigma:G\rightarrow G$ an involutive automorphism of a semigroup $G$, we study the solutions and stability of the following functional equations \begin{equation*}f(x\sigma(y))=f(x)g(y)+g(x)f(y),\quad x,y\in G,\end{equation*} \begin{equation*}f(x\sigma(y))=f(x)f(y)-g(x)g(y),\quad x,y\in G\end{equation*} and \begin{equation*}f(x\sigma(y))=f(x)g(y)-g(x)f(y),\quad x,y\in G,\end{equation*} from the theory of trigonometric functional equations. \begin{itemize} \item[(1)] We determine the solutions when $G$ is a semigroup generated by its squares. \item[(2)] We obtain the stability results for these equations, when $G$ is an amenable group. \end{itemize}
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