Communications of the
Korean Mathematical Society CKMS

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. (1) We determine the solutions when $G$ is a semigroup generated by its squares. (2) We obtain the stability results for these equations, when $G$ is an amenable group.

Keywords: Hyers-Ulam stability, semigroup, group, cosine equation, sine equation, involutive automorphism, multiplicative function, additive function

MSC numbers: 39B32, 39B72, 39B82