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In this paper, we proved the generalized Hyers-Ulam stability ofhomomorphisms in $C^*$- ternary algebras and of derivations on $C^*$-ternaryalgebras for the following Cauchy- Jensen functional equation$$3f\bigg(\frac{x+y+z}{3}\bigg)=2f\bigg(\frac{x+y}{2}\bigg)+f(z).$$ These wereapplied to investigate isomorphisms between $C^*$-ternary algebras.

Jordan derivations on $C^*$-ternary algebras for a Cauchy-Jensen functional equation