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Hyperstability of the Cauchy equation on restricted domains. (English) Zbl 1313.39037
The author extends classical results about Hyers-Ulam stability of the additive Cauchy equation from one normed space $$E_1$$ to another $$E_2$$. His maps are not required to be defined on all of $$E_1$$, but just on non-empty subsets $$X$$ of $$E_1\setminus \{0\}$$ with the following property: There exists a positive integer $$m_0$$ such that $$x \in X$$ $$\Rightarrow$$ $$-x \in X$$ and $$nx \in X$$ for all integers $$n \geq m_0$$.
Under these conditions his main result is: Let $$c \geq 0$$ and $$p < 0$$. Any map $$g:X \to E_2$$ satisfying $\| g(x+y) - g(x) - g(y)\| \leq c(\| x \|^p + \| y \|^p) \text{ whenever } x,y,x+y \in X,$ is additive on $$X$$.
The corresponding result for $$p \geq 0$$ is not true, so $$p<0$$ is essential.
The proof is based on the work by the author et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6728–6732 (2011; Zbl 1236.39022)].

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
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