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Elqorachi Elhoucien

Publications and source records attributed to Elqorachi Elhoucien.

17 recordsLinked to original sources
arXiv2025

Integral Kannappan-Sine subtraction and addition law on semigroups

Let $S$ be a semigroup, $μ$ a discrete measure on $S$ and $σ:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)-f(y)g(x),\; x,y \in S,$$ and the integral Kannappan-Sine addition law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)+f(y)g(x),\; x,y \in S.$$ We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)χ(y)+f(y)χ(x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law $\int_{S}f(xσ(y)t)dμ(t)=[f(x)χ(y)+f(y)χ(x)]\int_{S}χ(t)dμ(t), $ $x,y\in S$, and where $χ$: $S\longrightarrow \mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given.

math.GM↗
arXiv2020

A variant of d'Alembert functional equation on monoids

In this paper, we determine the complex-valued solutions of the functional equation $$ f(xσ(y))+f(τ(y)x)=2f(x)f(y)$$ for all $x,y \in M$, where $M$ is a monoid, $σ$: $M\longrightarrow M$ is an involutive automorphism and $τ$: $M\longrightarrow M$ is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of $2$-dimensional irreducible representations of $M$.

math.GM↗
arXiv2019

A note on Levi-Civita functional equation

In this paper we find the solutions of the functional equation $$f(xy) = g(x)h(y) + \sum_{j=1}^n g_j(x)h_j(y), \;x,y \in M,$$ where $M$ is a monoid, $n\geq 2$, and $g_j$ (for $j=1,...,n$) are linear combinations of at least $2$ distinct nonzero multiplicative functions.

math.CA↗
arXiv2017

On the Operator-valued $μ$-cosine functions

Let $(G,+)$ be a topological abelian group with a neutral element $e$ and let $μ: G\longrightarrow\mathbb{C}$ be a continuous character of $G$. Let $(\mathcal{H}, \langle \cdot,\cdot \rangle)$ be a complex Hilbert space and let $\mathbf{B}(\mathcal{H})$ be the algebra of all linear continuous operators of $\mathcal{H}$ into itself. A continuous mapping $ Φ: G\longrightarrow \mathbf{B}(\mathcal{H})$ will be called an operator-valued $μ$-cosine function if it satisfies both the $μ$-cosine equation $$Φ(x+y)+μ(y)Φ(x-y)=2Φ(x)Φ(y),\; x,y\in G$$ and the condition $Φ(e)=I,$ where $I$ is the identity of $\mathbf{B}(\mathcal{H})$. We show that any hermitian operator-valued $μ$-cosine functions has the form $$Φ(x)=\frac{Γ(x)+μ(x)Γ(-x)}{2}$$ where $ Γ: G\longrightarrow \mathbf{B}(\mathcal{H})$ is a continuous multiplicative operator. As an application, positive definite kernel theory and W. Chojnacki's results on the uniformly bounded normal cosine operator are used to give explicit formula of solutions of the cosine equation.

math.RT↗
arXiv2016

Solutions and stability of a variant of Van Vleck's and d'Alembert's functional equations

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$\int_{S}f(σ(y)xt)dμ(t)-\int_{S}f(xyt)dμ(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $σ$ is an involutive morphism of $S$, and $μ$ is a complex measure that is linear combinations of Dirac measures $(δ_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation $$\int_{S}f(xty)d\upsilon(t)+\int_{S}f(σ(y)tx)d\upsilon(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a topological semigroup, $σ$ is a continuous involutive automorphism of $S$, and $\upsilon$ is a complex measure with compact support and which is $σ$-invariant. (3) We prove the superstability theorems of the first functional equation.

math.CA↗
arXiv2016

An extensions of Kannappan's and Van Vleck's functional equations on semigroups

This paper treats two functional equations, the Kannppan-Van Vleck functional equation $$μ(y)f(xτ(y)z_0)\pm f(xyz_0) =2f(x)f(y), \;x,y\in S$$ and the following variant of it $$μ(y)f(τ(y)xz_0)\pm f(xyz_0) = 2f(x)f(y), \;x,y\in S,$$ in the setting of semigroups $S$ that need not be abelian or unital, $τ$ is an involutive morphism of $S$, $μ$ : $S\longrightarrow \mathbb{C}$ is a multiplicative function such that $μ(xτ(x))=1$ for all $x\in S$ and $z_0$ is a fixed element in the center of $S$. We find the complex-valued solutions of these equations in terms of multiplicative functions and solutions of d'Alembert's functional equation.

math.CA↗
arXiv2016

Solutions and stability of generalized Kannappan's and Van Vleck's functional equations

We study the solutions of the integral Kannappan's and Van Vleck's functional equations $$\int_{S}f(xyt)dμ(t)+\int_{S}f(xσ(y)t)dμ(t) = 2f(x)f(y), \;x,y\in S;$$ $$\int_{S}f( xσ(y)t)dμ(t)-\int_{S}f(xyt)dμ(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $σ$ is an involutive automorphism of $S$ and $μ$ is a linear combination of Dirac measures $(δ_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. We show that the solutions of these equations are closely related to the solutions of the d'Alembert's classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems that these functional equations are superstable in the general case, where $σ$ is an involutive morphism.

math.CA↗
arXiv2016

Integral Van Vleck's and Kannappan's functional equations on semigroups

In this paper we study the solutions of the integral Van Vleck's functional equation for the sine $$\int_{S}f(xτ(y)t)dμ(t)-\int_{S}f(xyt)dμ(t) =2f(x)f(y),\; x,y\in S$$ and the integral Kannappan's functional equation $$\int_{S}f(xyt)dμ(t)+\int_{S}f(xτ(y)t)dμ(t) =2f(x)f(y),\; x,y\in S,$$ where $S$ is a semigroup, $τ$ is an involution of $S$ and $μ$ is a measure that is linear combinations of point measures $(δ_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. \\ We express the solutions of the first equation by means of multiplicative functions on $S$, and we prove that the solutions of the second equation are closely related to the solutions of the classic d'Alembert's functional equation with involution.

math.CA↗
arXiv2016

On the integral functional equations: On the integral d'Alembert's and Wilson's functional equations

Let $G$ be a locally compact group, and let $K$ be a compact subgroup of $G$. Let $μ: G\longrightarrow\mathbb{C}\backslash\{0\}$ be a character of $G$. In this paper, we deal with the integral equations $$W_μ(K):\; \;\int_{K}f(xkyk^{-1})dk+μ(y)\int_{K}f(xky^{-1}k^{-1})dk=2f(x)g(y),$$ and $$D_μ(K):\; \;\int_{K}f(xkyk^{-1})dk+μ(y)\int_{K}f(xky^{-1}k^{-1})dk=2f(x)f(y)$$ for all $x, y\in G$ where $f, g: G\longrightarrow \mathbb{C}$, to be determined, are complex continuous functions on $G$. When $K\subset Z(G)$, the center of $G$, $D_μ(K)$ reduces to the new version of d'Almbert's functional equation $f(xy)+μ(y)f(xy^{-1})=2f(x)f(y)$, recently studied by Davison [18] and Stetkær [35]. We derive the following link between the solutions of $W_μ(K)$ and $D_μ(K)$ in the following way : If $(f,g)$ is a solution of equation $W_μ(K)$ such that $C_{K}f=\int_{K}f(kxk^{-1})dω_{K}(k)\neq 0$ then $g$ is a solution of $D_μ(K)$. This result is used to establish the superstability problem of $W_μ(K)$. In the case where $(G,K)$ is a central pair, we show that the solutions are expressed by means of $K$-spherical functions and related functions. Also we give explicit formulas of solutions of $D_μ(K)$ in terms of irreducible representations of $G$. These formulas generalize Euler's formula $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ on $G=\mathbb{R}$.

math.CA↗
arXiv2016

A class of functional equations on monoids

In \cite{05} B. Ebanks and H. Stetkær obtained the solutions of the functional equation $f(xy)-f(σ(y)x)=g(x)h(y)$ where $σ$ is an involutive automorphism and $f,g,h$ are complex-valued functions, in the setting of a group $G$ and a monoid $M$. Our main goal is to determine the complex-valued solutions of the following more general version of this equation, viz $f(xy)-μ(y)f(σ(y)x)=g(x)h(y)$ where $μ: G\longrightarrow \mathbb{C}$ is a multiplicative function such that $μ(xσ(x))=1$ for all $x\in G$. As an application we find the complex-valued solutions $(f,g,h)$ on groups of the equation $f(xy)+μ(y)g(σ(y)x)=h(x)h(y)$.

math.CA↗
arXiv2015

An extension of Van Vleck's functional equation for the sine

In \cite{St3} H. Stetkær obtained the solutions of Van Vleck's functional equation for the sine $$f(xτ(y)z_0)-f(xyz_0) =2f(x)f(y),\; x,y\in G,$$ where $G$ is a semigroup, $τ$ is an involution of $G$ and $z_0$ is a fixed element in the center of $G$. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck's functional equation for the sine $$μ(y)f(xτ(y)z_0)-f(xyz_0) =2f(x)f(y), \;x,y\in G,$$ where $μ$ : $G\longrightarrow \mathbb{C}$ is a multiplicative function such that $μ(xτ(x))=1$ for all $x\in G$. Furthermore, we obtain the solutions of a variant of Van Vleck's functional equation for the sine $$μ(y)f(σ(y)xz_0)-f(xyz_0) = 2f(x)f(y), \;x,y\in G$$ on monoids, and where $σ$ is an automorphism involutive of $G$.

math.CA↗
arXiv2015

Solutions and stability of variant of Wilson's functional equation

In this paper we will investigate the solutions and stability of the generalized variant of Wilson's functional equation $$ (E):\;\;\;\; f(xy)+χ(y)f(σ(y)x)=2f(x)g(y),\; x,y\in G,$$ where $G$ is a group, $σ$ is an involutive morphism of $G$ and $χ$ is a character of $G$. (a) We solve $(E)$ when $σ$ is an involutive automorphism, and we obtain some properties about solutions of $(E)$ when $σ$ is an involutive anti-automorphism. (b) We obtain the Hyers Ulam stability of equation $(E)$. As an application, we prove the superstability of the functional equation $f(xy)+χ(y)f(σ(y)x)=2f(x)f(y),\; x,y\in G.$

math.CA↗
arXiv2015

Solutions and stability of a generalization of Wilson's equation

In this paper we study the solutions and stability of the generalized Wilson's functional equation $\int_{G}f(xty)dμ(t)+\int_{G}f(xtσ(y))dμ(t)=2f(x)g(y),\; x,y\in G$, where $G$ is a locally compact group, $σ$ is a continuous involution of $G$ and $μ$ is an idempotent complex measure with compact support and which is $σ$-invariant. We show that $\int_{G}g(xty)dμ(t)+\int_{G}g(xtσ(y))dμ(t)=2g(x)g(y),\; x,y\in G$ if $f\neq 0$ and $\int_{G}f(t.)dμ(t)\neq 0$. We also study some stability theorems of that equation and we establish the stability on noncommutaive groups of the classical Wilson's functional equation $f(xy)+χ(y)f(xσ(y))=2f(x)g(y)\; x,y\in G$, where $χ$ is a unitary character of $G$.

math.CA↗
arXiv2014

On functional inequalities associated with Drygas functional equation

In the paper, the equivalence of the functional inequality $$\|2f(x)+f(y)+f(-y)-f(x-y)\|\leq\|f(x+y)\|\;\;\;(x,y\in{G})$$ and the Drygas functional equation $$f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)\;\;\;(x,y\in{G})$$ is proved for functions $f:G\rightarrow E$ where $(G, +)$ is an abelian group, $(E, <\cdot, \cdot>)$ is an inner product space, and the norm is derived from the inner product in the usual way.

math.FA↗