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Hyers-Ulam stability of functional equations with a square-symmetric operation. (English) Zbl 0930.39020
Let $$S$$ be a nonempty set and $$\circ$$ be a square-symmetric operation on the set $$S$$. Let $$G$$ be a multiplicative subsemigroup of $$\mathbb{C}$$ and let $$H$$: $$G\times G\to G$$ be a continuous and $$G$$-homogeneous function i.e. $$H(uv,uw)=uH(u,w),u,v$$, $$w\in G$$. Assuming that $$|H(1,1) |>1$$ the authors obtained the stability in the Hyers-Ulam sense of the functional equation $$f(x\circ y)= H(f(x),f(y))$$, $$x,y\in S$$. In the case $$|H(1,1) |<1$$ under divisibility assumption on the operation $$\circ$$ another stability result is established. The results presented include and generalize the classical theorem of D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] on the stability of the Cauchy functional equation.
Reviewer: M.C.Zdun (Kraków)

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations
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