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Let 𝐴, 𝐵 be two unital 𝐶∗-algebras. We prove that every almost unital almost linear mapping ℎ : 𝐴→𝐵 which satisfies ℎ(3𝑛𝑢𝑦+3𝑛𝑦𝑢)=ℎ(3𝑛𝑢)ℎ(𝑦)+ℎ(𝑦)ℎ(3𝑛𝑢) for all 𝑢∈𝑈(𝐴), all 𝑦∈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→𝐵 is a Jordan homomorphism when ℎ(3𝑛𝑢𝑦+3𝑛𝑦𝑢)=ℎ(3𝑛𝑢)ℎ(𝑦)+ℎ(𝑦)ℎ(3𝑛𝑢) holds for all 𝑢∈𝐼1 (𝐴sa), all 𝑦∈𝐴, and all 𝑛=0,1,2,…. Furthermore, we investigate the Hyers- Ulam-Aoki-Rassias stability of Jordan ∗-homomorphisms between unital 𝐶∗-algebras by using the fixed points methods.