On stability of the Cauchy equation on semigroups. (English) Zbl 0658.39006

D. H. Hyers [Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.264)] proved that if \(\delta\) is a positive real number and f a mapping from a vector space over the rationals X into a Banach space Y such that \(\| f(x_ 1)+f(x_ 2)-f(x_ 1+x_ 2)\| \leq \delta\) for all \(x_ 1,x_ 2\in X\), then there is a unique additive mapping \(\ell:X\to Y\) such that \(\| f(x)-\ell (x)\| \leq \delta\) for all \(x\in X\). J. Rätz [General Inequalities II, Proc. 2nd int. Conf., Oberwolfach 1978, ISNM 47, 233-251 (1980; Zbl 0433.39014)] generalized Hyer’s results and analyzed such theorems in rather general situations. Stimulated by two notes of L. Székelyhidl [C. R. Math. Acad. Sci., Soc. R. Can. 8, 127-129 (1986; Zbl 0604.39007)], the author utilizes some of Rätz’ techniques to prove the theorem:
Suppose \((S,+)\) is a (not necessarily commutative) semigroup such that Hyers’ theorem holds for all complex-valued functions defined on S. Let X be a sequentially complete locally convex linear Hausdorff space. Then, if F:S\(\to X\) is a function for which the transformation \((x,y)\to F(x+y)-F(x)-F(y)\) is bounded, there exists an additive function A:S\(\to X\) such that F-A is bounded.
Reviewer: S.L.Segal


39B52 Functional equations for functions with more general domains and/or ranges
46A03 General theory of locally convex spaces
Full Text: DOI EuDML


[1] Greenleaf, F. P.,Invariant means on topological groups. (Van Nostrand Mathematical Studies, vol. 16). Van Nostrand New York–Toronto–London–Melbourne, 1969. · Zbl 0174.19001
[2] Hewitt, E. andRoos, K. A.,Abstract harmonic analysis. Vol. I, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1963.
[3] Jarchow, H.,Locally convex spaces, Teubner, Stuttgart, 1981. · Zbl 0466.46001
[4] Rätz, J.,On approximately additive mappings. In:General Inequalities 2, ISNM Vol. 47 (edited by E. F. Beckenbach), Birkhäuser Verlag, Basel–Boston–Stuttgart, 1980, pp. 233–251. · Zbl 0433.39014
[5] Székelyhidi, L.,Remark 17. In:The 22nd International Symposium on Functional Equations, December 16–December 22, 1984, Oberwolfach, Aequationes Math.29 (1985), 95–96.
[6] Székelyhidi, L.,Note on Hyers’s theorem. C.R. Math. Rep. Sci. Canada8 (1986), 127–129. · Zbl 0604.39007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.