## 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).

### MSC:

 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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