Let $X, Y$ be Banach modules over a $C^*$-algebra and let $r_1, \cdots, r_n \in (0, \infty)$ be given.
We prove the Hyers-Ulam-Rassias stability of the following functional
equation in Banach modules over a unital
$C^*$-algebra:
\begin{eqnarray}
\sum_{i=1}^{n} r_i f \left( \sum_{j=1}^{n}r_j(x_i-x_j) \right)
+ \left(\sum_{i=1}^{n} r_i\right) f \left(\sum_{i=1}^{n} r_ix_i \right)
=\left(\sum_{i=1}^{n}r_i\right) \sum_{i=1}^{n} r_i f(x_i) .
\end{eqnarray}
We show that if $r_1=\cdots=r_n = r$ and an odd mapping $f : X
\rightarrow Y$ satisfies the functional equation {\rm (0.1)} then
the odd mapping $f : X \rightarrow Y$ is Cauchy additive.
As an application, we show that every almost
linear bijection $h : A \rightarrow B$ of a unital $C^*$-algebra
$A$ onto a unital $C^*$-algebra $B$ is a $C^*$-algebra isomorphism
when $h((nr)^d u y) = h((nr)^d u) h(y)$ for
all unitaries $u \in A$, all $y \in A$, and all $d \in {\bf Z}$.