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Remark on hyperstability of the general linear equation. (English) Zbl 1304.39033
Let $$\mathbb{F}$$ and $$\mathbb{K}$$ be fields of real or complex numbers. Using a fixed point theorem of J. Brzdȩk et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6728–6732 (2011; Zbl 1236.39022)], the authors prove the following theorem:
Let $$X$$ be a normed space over the field $$\mathbb{F}$$, $$Y$$ be a Banach space over $$\mathbb{K}$$, $$a, b \in \mathbb{F} \setminus \{0\}$$, $$A, B \in \mathbb{K}$$, $$c \geq 0$$, $$p < 0$$ and $$g: X \to Y$$ satisfy $\|g(ax + by) - Ag(x) - Bg(y)\| \leq c( \|x\|^{p} + \|y\|^{p}), \;x, y \in X \setminus \{0\}.$ Then $$g$$ satisfies the functional equation $g(ax + by) = Ag(x) + Bg(y), \;\;x, y \in X \setminus \{0\}.$ .

##### 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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##### References:
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