Let A be a unital semisimple Banach \(*\) -algebra whose centre is denoted by Z(A), and let m be a fixed element in Z(A). Let f, g be two continuous linear mappings on A satisfying the identity \(f(u)u^{*}+u^{*}f(u)+u{g(u)}^{*}+{g(u)}^{*}u=m\) for each unitary element u in A. Then we prove that there exist two derivations \(d_{1},d_{2}\) such that \(f(a)=d_{1}(a)+f(1)a, g(a)=d_{2}(a)+g(1)a\) and \(d_{1}(a^{*})={d_{2}(a)}^{*}\) for each \(a\in A\) . Let T be a conjugate linear mapping which is ternary derivable at the unit element from a unital \(C^*\) -algebra into its dual space. We establish the automatic continuity of any such a mapping T.
