On approximate $sigma$-homomorphisms and derivations on $C^*$-ternary

In this paper, we prove the generalized Hyers-Ulam stability of $sigma$-homomorphisms and derivations on $C^*$-ternary algebras associated with the generalized Cauchy-Jensen type additive functional equation begin{equation*} sum_{i=1}^n fleft(x_i +{1 over n-1}sum_{j=1, j ne i}^n x_jright) =2 sum_{i=1}^n f(x_i) end{equation*} where $n in Z^+$ is a fixed integers with $n ge 2$.