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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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