arXiv subjects

Redouani Ahmed

Publications and source records attributed to Redouani Ahmed.

4 recordsLinked to original sources

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↗

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↗