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Elhoucien Elqorachi

Publications and source records attributed to Elhoucien Elqorachi.

14 recordsLinked to original sources
arXiv2024

Kannappan-Wilson and Van Vleck-Wilson functional equations on semigroups

Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $σ:S\rightarrow S$ is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) +\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] and the Van Vleck-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) -\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] where $μ$ is a measure that is a linear combination of Dirac measures $(δ_{z_i})_{i\in I}$, such that $z_i\in Z(S)$ for all $i\in I$. Interesting consequences of these results are presented.

math.FA↗
arXiv2023

A Kannappan-sine subtraction law on semigroups

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.

math.GM↗
arXiv2023

Cosine-sine functional equation on semigroups

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.

math.GM↗
arXiv2023

A Kannappan-sine addition law on semigroups

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 $f(xyz_{0})=f(x)g(y)+f(y)g(x),x,y\in S.$

math.GM↗
arXiv2023

Cosine and Sine addition and subtraction law with an automorphism

Let $S$ be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional equations \[g(xσ(y)) = g(x)g(y)+f(x)f(y),\ x,y\in S,\] \[f(xσ(y)) = f(x)g(y)+f(y)g(x),\ x,y\in S,\] and \[f(xσ(y)) = f(x)g(y)-f(y)g(x),\ x,y\in S,\] where $σ:S\rightarrow S$ is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications.

math.FA↗
arXiv2022

A variant of Wilson's functional equation on semigroups

We determine the complex-valued solutions of the following functional equation \[f(xy)+μ(y)f(σ(y)x) = 2f(x)g(y),\quad x,y\in S,\] where $S$ is a semigroup and $σ$ an automorphism, $μ:S\rightarrow \mathbb{C}$ is a multiplicative function such that $μ(xσ(x))=1$ for all $x\in S$.

math.FA↗
arXiv2022

A d'Alembert type functional equation on semigroups

We treat two related trigonometric functional equations on semigroups. First we solve the $μ$-sine subtraction law \[μ(y) k(x σ(y))=k(x) l(y)-k(y) l(x), \quad x, y \in S,\] for $k, l : S\rightarrow \mathbb{C}$, where $S$ is a semigroup and $σ$ an involutive automorphism, $μ:S\rightarrow \mathbb{C}$ is a multiplicative function such that $μ(xσ(x))=1$ for all $x\in S$, then we determine the complex-valued solutions of the following functional equation \[f(xy) - μ(y)f(σ(y)x) = g(x)h(y),\quad x,y\in S,\] on a larger class of semigroups.

math.FA↗
arXiv2022

A generalization of the cosine addition law on semigroups

Our main result is that we describe the solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation \[g(xσ(y))=g(x)g(y)-f(x)f(y)+αf(xσ(y)),\quad x,y\in S,\] where $S$ is a semigroup, $α\in \mathbb{C}$ is a fixed constant and $σ:S\rightarrow S$ an involutive automorphism.

math.FA↗
arXiv2022

Five Trigonometric addition laws on semigroups

In this paper, we determine the complex-valued solutions of the following functional equations \[g(xσ(y)) = g(x)g(y)+f(x)f(y),\quad x,y\in S,\]\[f(xσ(y)) = f(x)g(y)+f(y)g(x),\quad x,y\in S,\]\[f(xσ(y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\quad x,y\in S,\]\[f(xσ(y))=f(x)g(y)+f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\]\[f(xσ(y))=f(x)g(y)-f(y)g(x)+αg(xσ(y)),\quad x,y\in S,\] where $S$ is a semigroup, $α\in \mathbb{C}\backslash \lbrace 0\rbrace$ is a fixed constant and $σ:S\rightarrow S$ an involutive automorphism.

math.FA↗
arXiv2022

A system of cosine-sine functional equations on a semigroup generated by its squares

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.

math.FA↗
arXiv2019

Variants of Wilson's functional equation on semigroups

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*}

math.GM↗
arXiv2019

A generalization of d'Alembert's functional equation on semigroups

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.

math.GM↗