Let be a monotonically increasing symmetric homogeneous function of degree , where . Let be a mapping from a real normed space into a real Banach space . Assume that
The authors proved that
exists for every when (resp. ).
(ii) is the unique additive mapping satisfying
Furthermore, is linear if for every fixed in there exists a real number such that the function is bounded on . A counterexample has been given for the cae . These results partially answer a question of Ulam about the existence of additive mapping near an approximately additive mapping.