**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.