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

Gajda, Zbigniew

Aequationes Math. 36, No. 1, 76-79 (1988).

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

MSC:

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

Keywords:

stability; Cauchy equation; semigroup; Hyers’ theorem; complex-valued functions; locally convex linear Hausdorff space

Citations:

Zbl 0061.264; Zbl 0433.39014; Zbl 0604.39007

References:

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