Hyers-ulam Stability of Butler-rassias Functional Equation

In 1940, Ulam [9] gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms. Let G1 be a group and let G2 be a metric group with a metric d(·,·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy),h(x)h(y)) < δ for all x, y ∈ G1, then a homomorphism H : G1 → G2 exists with d(h(x),H(x)) < ε for all x ∈ G1? The case of approximately additive functions was solved by Hyers [5] under the assumption that G1 and G2 are Banach spaces. Taking this fact into account, the additive Cauchy functional equation f (x + y) = f (x) + f (y) is said to have the Hyers-Ulam stability. This terminology is also applied to the case of other functional equations. For a more detailed definition of such terminology, one can refer to [4, 6, 7]. In 2003, Butler [3] posed the following problem.