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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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