Jordan $*-$homomorphisms between unital $C^*-$algebras

Let $A,B$ be two unital $C^*-$algebras. We prove that every almost unital almost linear mapping $h:A\longrightarrow B$ which satisfies $h(3^nuy+3^nyu) = h(3^nu)h(y)+h(y)h(3^nu)$ for all $u\in U(A)$, all $y\in A$, and all $n = 0, 1, 2,...$, is a Jordan homomorphism. Also, for a unital $C^*-$algebra $A$ of real rank zero, every almost unital almost linear continuous mapping $h:A\longrightarrow B$ is a Jordan homomorphism when $h(3^nuy+3^nyu) = h(3^nu)h(y)+h(y)h(3^nu)$ holds for all $u\in I_1(A_{sa})$, all $y\in A,$ and all $n = 0, 1, 2,... ~$. Furthermore, we investigate the Hyers--Ulam--Rassias stability of Jordan $*-$homomorphisms between unital $C^*-$algebras by using the fixed points methods.