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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}
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참고문헌 (25)
참고문헌 신청
J. Aczel / 1966 / Lectures on Functional Equations and Their Applications / Academic Press
J. Aczel / 1989 / Functional Equations in Several Variables / Cambridge University Press
O. Ajebbar / 2017 / The cosine-sine functional equation on a semigroup with an involutive automorphism / Aequationes Math 91(6):1115~1146
T. Aoki / 1950 / On the stability of the linear transformation in Banach spaces / J. Math. Soc. Japan 2:64~66
J. A. Baker / 1980 / The stability of the cosine equation / Proc. Amer. Math. Soc 80(3):411~416
J. Aczel / 1989 / Functional Equations in Several Variables / Cambridge University Press
O. Ajebbar / 2017 / The cosine-sine functional equation on a semigroup with an involutive automorphism / Aequationes Math 91(6):1115~1146
T. Aoki / 1950 / On the stability of the linear transformation in Banach spaces / J. Math. Soc. Japan 2:64~66
J. A. Baker / 1980 / The stability of the cosine equation / Proc. Amer. Math. Soc 80(3):411~416
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