Communications of the
Korean Mathematical Society

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024



Commun. Korean Math. Soc. 2019; 34(1): 55-82

Online first article April 11, 2018      Printed January 1, 2019

Copyright © The Korean Mathematical Society.

Solutions and stability of trigonometric functional equations on an amenable group with an involutive automorphism

Omar Ajebbar, Elhoucien Elqorachi

Faculty of Sciences; Faculty of Sciences


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