On stability of the Cauchy equation on semigroups

Summary

We say that Hyers's theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S → ℂ such that the set {f(x + y) − f(x) − f(y): x, y ∈ S} is bounded, there exists an additive functiona:S → ℂ for which the functionf − a is bounded.

Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi.

Theorem.Suppose that Hyers's theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S → X is a function for which the mapping (x, y) → F(x + y) − F(x) − F(y) is bounded, then there exists an additive function A : S → X such that F — A is bounded.