arXiv subjects
Publications and source records attributed to Omar Ajebbar.
Given a semigroup $S$ equipped with an involutive automorphism $σ$, we determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(xσ(y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\,\,x,y\in S,\end{equation*} in terms of multiplicative functions and solutions of the special cases of sine and cosine-sine functional equations.
Let $S$ be a semigroup, $z_0$ a fixed element in $S$ and $σ:S \longrightarrow S$ an involutive automorphism. We determine the complex-valued solutions of Kannappan-sine subtraction law $f(xσ(y)z_0)=f(x)g(y)-f(y)g(x),\; x,y \in S$. As an application we solve the following variant of Kannappan-sine subtraction law viz. $f(xσ(y)z_0)=f(x)g(y)-f(y)g(x)+λg(xσ(y)z_0) ,\; x,y \in S,$ where $λ\in \mathbb{C}^{*}$. The continuous solutions on topological semigroups are given and an example to illustrate the main results is also given.
Let $S$ be a semigroup. We determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y), x,y\in S,\end{equation*} in terms of multiplicative functions, solutions of the special case $$φ(xy)=φ(x)χ(y)+χ(x)φ(y), x,y\in S$$ of the sine addition law, where $χ:S\to\mathbb{C}$ is a multiplicative function, and also in terms of solutions of the particular case $$ψ(xy)=ψ(x)χ(y)+χ(x)ψ(y)+φ(x)φ(y), x,y\in S$$ of the cosine-sine functional equation where $χ:S\to\mathbb{C}$ is a multiplicative function and $φ:S\to\mathbb{C}$ such that the pair $(φ,χ)$ satisfies the sine addition law.
We determine the complex-valued solutions of the Kannappan cosine functional law $g(xyz_{0})=g(x)g(y)-f(x)f(y)$, $x,y\in S$, where $S$ is a semigroup and $z_{0}$ is a fixed element in $S.$
Let $S$ be a semigroup and $z_{0}$ a fixed element in $S.$ We determine the complex-valued solutions of the following Kannappan-sine addition law $f(xyz_{0})=f(x)g(y)+f(y)g(x),x,y\in S.$
Given a semigroup $S$ generated by its squares, we determine the complex-valued solutions of the following system of cosine-sine functional equations \begin{align*} f(xy)=f(x)g_{1}(y)+g_{1}(x)f(y)+λ_{1}^{2}\,h(x)h(y),\; x,y\in S,\\ h(xy)=h(x)g_{2}(y)+g_{2}(x)h(y)+λ_{2}^{2}\,f(x)f(y),\; x,y\in S, \end{align*} where $λ_{1},λ_{2}\in\mathbb{C}$ are given constants and $f,g_{1},g_{2},h:S\to\mathbb{C}$ are unknown functions.
Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $σ$ and a multiplicative function $μ:S\to\mathbb{C}$ such that $μ(xσ(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equations \begin{equation*}f(xy)+μ(y)f(σ(y)x)=2f(x)g(y),\, x,y\in S\end{equation*} and \begin{equation*}f(xy)+μ(y)f(σ(y)x)=2f(y)g(x),\, x,y\in S\end{equation*}
Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $σ$ and a multiplicative function $μ:S\to\mathbb{C}$ such that $μ(xσ(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equation.
The aim of the present paper is to give extensions of the cosine-sine functional equation.