Let , be two unital -algebras. We prove that every almost unital almost linear mapping : which satisfies for all , all , and all , is a Jordan homomorphism. Also, for a unital -algebra of real rank zero, every almost unital almost linear continuous mapping is a Jordan homomorphism when holds for all (), all , and all . Furthermore, we investigate the Hyers- Ulam-Aoki-Rassias stability of Jordan -homomorphisms between unital -algebras by using the fixed points methods.