We apply a fixed point theorem to investigate some new stability results of $(\alpha,\beta,\gamma)$-derivations on Lie $C^*$-algebras associated to the generalized Cauchy–-Jensen type additive functional equation \begin{equation*} \sum^{n}_{i=1}f\left(x_i+ \frac{1}{n-1}\sum^{n}_{j=1,j\neq i}x_j\right)=2\sum^{n}_{i=1}f(x_i). \end{equation*}}
