On the Asymptoticity Aspect of Hyers-Ulam Stability of Quadratic Mappings

The question concerning the stability of group homomorphisms was posed by Ulam 1 . Hyers 2 solved the case of approximately additive mappings on Banach spaces. Aoki 3 provided a generalization of the Hyers’ theorem for additive mappings. In 4 , Rassias generalized the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded see also 5 . The result of Rassias has been generalized by Găvruţa 6 who permitted the norm of the Cauchy difference f x y − f x − f y to be bounded by a general control function under some conditions. This stability concept is also applied to the case of various functional equations by a number of authors. For more results on the stability of functional equations, see 7–32 . We also refer the readers to the books 33–37 . It is easy to see that the function f : → defined by f x cx2 with c an arbitrary constant is a solution of the functional equation