The Ulam Type Stability of a Generalized Additive Mapping and Concrete Examples

In 1940, Ulam [1] proposed the following stability problem: “When is it true that a function which satisfies some functional equation approximatelymust be close to one satisfying the equation exactly?” Next year, Hyers [2] gave an answer to this problem for additive mappings between Banach spaces. Furthermore, Aoki [3] and Rassias [4] obtained independently generalized results of Hyers’ theorem which allow the Cauchy difference to be unbounded. Let X and Y be normed spaces over K, which denotes either the real field R or the complex field C. Throughout the paper, we fix scalars a, b, c, d ∈ K \ {0} and vectors