# On stability of the Cauchy equation on semigroups

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## 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 any*f:S* → ℂ such that the set {*f(x + y) − f(x) − f(y): x, y ∈ S*} is bounded, there exists an additive function*a:S* → ℂ for which the function*f − a* is bounded.

Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada*8* (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions on*S* implies its validity for functions mapping*S* into a semi-reflexive locally convex linear topological space*X.* We improve this result by assuming sequential completeness of the space*X* instead of its semi-reflexiveness. Our assumption on*X* 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.*

### AMS (1980) subject classification

Primary 39B40 Secondary 46A05, 46A25## Preview

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### References

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*Remark 17.*In:*The 22nd International Symposium on Functional Equations, December 16–December 22, 1984, Oberwolfach*, Aequationes Math.*29*(1985), 95–96.Google Scholar - [6]Székelyhidi, L.,
*Note on Hyers's theorem*. C.R. Math. Rep. Sci. Canada*8*(1986), 127–129.Google Scholar