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Hyperstability of a class of linear functional equations. (English) Zbl 1004.39022
First the following result is offered: Suppose that \(M: ]0,1]\to \mathbb{R}\) is multiplicative and assumes a value greater than 1, and that \(f:]0,1]\to \mathbb{R}\) satisfies \(|f(xy)-M(x)f(y)-M(y)f(x)|\leq \varepsilon\) for some \(\varepsilon\geq 0\). Then \(f(xy)-M(x)f(y)-M(y)f(x)=0\) \((x,y\in ]0,1])\). The rest of the paper offers similar and more general results for equations of the form \(f(x)+f(y)=\sum_{k=1}^n f[sg_k (t)]/n\) on a semigroup (\(f\) maps the semigroup into a real normed space, \(g_1,\dots,g_n\) are pairwise distinct automorphisms of the semigroup, forming a group under composition).

39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
20M20 Semigroups of transformations, relations, partitions, etc.
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