We introduce the following additive functional equation 1 \(\displaystyle \begin{aligned} {} g(\lambda u +v+ 2y)= \lambda g(u) + g(v) + 2 g(y) \end{aligned} \) for all \(\lambda \in \mathbf {C}\) , all unitary elements \(u, v\) in a unital \(C^*\) -algebra-ternary algebra P and all \(y\in P\) . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the additive functional equation ( 1 ) in unital \(C^*\) -algebra-ternary algebras. Furthermore, we apply to study \(C^*\) -bi-ternary homomorphisms and \(C^*\) -bi-ternary derivations in unital \(C^*\) -algebra-ternary algebras.