  In this paper, we investigate homomorphisms between $C^*$-ternary algebras
and derivations on $C^*$-ternary algebras, associated with the following
functional equation
$$f(\frac{x_2-x_1}{3})+f(\frac{x_1-3x_3}{3})+f(\frac{3x_1+3x_3-x_2}{3})=f(x_1).$$
Moreover, we prove the generalized Hyers-Ulam -Rassias stability of
homomorphisms in $C^*$-ternary algebras and of derivations on $C^*$-ternary
algebras.
