Nearly Jordan ∗‐Homomorphisms between UnitalC∗‐Algebras

LetA,Bbe two unitalC∗‐algebras. We prove that every almost unital almost linear mappingh:A→Bwhich satisfiesh(3nuy+ 3nyu) =h(3nu)h(y) +h(y)h(3nu) for allu∈U(A), ally∈A, and alln= 0,1, 2, …, is a Jordan homomorphism. Also, for a unitalC∗‐algebraAof real rank zero, every almost unital almost linear continuous mappingh:A→Bis a Jordan homomorphism whenh(3nuy+ 3nyu) =h(3nu)h(y) +h(y)h(3nu) holds for allu∈I1(Asa), ally∈A, and alln= 0,1, 2, …. Furthermore, we investigate the Hyers‐ Ulam‐Aoki‐Rassias stability of Jordan ∗‐homomorphisms between unitalC∗‐algebras by using the fixed points methods.