Approximation of a generalized Euler-Lagrange type additive mapping on Lie $C^{ast}$-algebras

Using fixed point method, we prove some new stability results for Lie $(alpha,beta,gamma)$-derivations and Lie $C^{ast}$-algebra homomorphisms on Lie $C^{ast}$-algebras associated with the Euler-Lagrange type additive functional equation begin{align*} sum^{n}_{j=1}f{bigg(-r_{j}x_{j}+sum_{1leq i leq n, ineq j}r_{i}x_{i}bigg)}+2sum^{n}_{i=1}r_{i}f(x_{i})=nf{bigg(sum^{n}_{i=1}r_{i}x_{i}bigg)} end{align*} where $r_{1},ldots,r_{n}in {mathbb{R}}$ are given and $r_{i},r_{j}neq 0$ for some $1leq i< jleq n$.