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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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