E. Elqorachi, M. Akkouchi

On Hyers-Ulam stability of Cauchy and Wilson Equations

Abstract:
We study the Hyers-Ulam stability problem for the Cauchy and Wilson integral equations
\begin{gather*}
\int_{G}f(xty)d\mu(t)=f(x)g(y),\;\;x,y\in G,\\
\int_{G}f(xty)d\mu(t)+\int_{G}f(xt\sigma(y))d\mu(t)=2f(x)g(y),
\;\;x,y\in G,
\end{gather*}
where $G$ is a topological group, $f$, $g$ : $G\rightarrow \mathbb{C}$ are continuous functions, $\mu$ is a complex measure with compact support and $\sigma$ is a continuous involution of $G$. The result obtained in this paper are natural extensions of the previous works concerning the Hyers-Ulam stability of the Cauchy and Wilson functional equations done in the particular case of $\mu$=$\delta_{e}$: The Dirac measure concentrated at the identity element of $G$.

Keywords:
Topological group, Hyers--Ulam stability, Superstability, Cauchy equation, D'Alembert equation, Wilson equation.

MSC 2000: 39B72.