Approximately Quadratic Mappings on Restricted Domains

Under what conditions does there exist a group homomorphism near an approximate group homomorphism? This question concerning the stability of group homomorphisms was posed by Ulam 1 . The case of approximately additive mappings was solved by Hyers 2 on Banach spaces. In 1950 Aoki 3 provided a generalization of the Hyers’ theorem for additive mappings and in 1978 Th. M. Rassias 4 generalized the Hyers’ theorem for linear mappings by allowing the Cauchy difference to be unbounded see also 5 . The result of Rassias’ theorem has been generalized by Găvruţa 6 who permitted the Cauchy difference to be bounded by a general control function. This stability concept is also applied to the case of other functional equations. For more results on the stability of functional equations, see 7– 24 . We also refer the readers to the books in 25–29 . It is easy to see that the quadratic function f x x2 is a solution of each of the following functional equations: