Research Communication | Open Access
Volume 2019 | Communication ID 120
Volume 2019 | Communication ID 120
| New Idea of Generalized of Variant of d’Alembert Functional Equation |
Hajira Dimou, Samir Kabbaj
Received |
Accepted |
Published |
Jan 27, 2019 |
Feb 26, 2019 |
Mar 01, 2019 |
Abstract: Let $(S, .)$ be a semigroup and let $\sigma\in Hom(S,S)$ satisfies $\sigma\circ\sigma=id.$ We show that any solution $ f:S \to \mathbb{C}$ of the functional equation $$ \chi_{1}(y)f(xy)+\chi_{2}(y)f(\sigma(y)x)=2f(x)f(y),\ \ \ x,y\in S, $$ has the form $f=\frac{\mu+\chi_{2}\chi_{1}\mu\circ\sigma}{2},$ where $\mu$ is a multiplicative function on $S$ and $\chi_{1},\chi_{2}:S \to (\mathbb{C} \backslash\lbrace{0}\rbrace,.)$ be two characters on $S$ (i.e, $\chi_{1}(xy)=\chi_{1}(x)\chi_{1}(y)$ and $\chi_{2}(xy)=\chi_{2}(x)\chi_{2}(y)\\ \ for\ all \ x,y\in S$) such that ...