Remark on hyperstability of the general linear equation
type: Item
instance of: scholarly article
title: Remark on hyperstability of the general linear equation
zbMATH Open document ID: 1304.39033
author: Magdalena Piszczek
published in: Aequationes Mathematicae
publication date: 2014-10-02T00:00:00Z
review text: Let \(\mathbb{F}\) and \(\mathbb{K}\) be fields of real or complex numbers. Using a fixed point theorem of \textit{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 \[ g(ax + by) = Ag(x) + Bg(y), \;\;x, y \in X \setminus \{0\}. \] .
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6351395
zbMATH Keywords: hyperstability
zbMATH Keywords: general linear equation
zbMATH Keywords: normed space
zbMATH Keywords: Banach space
zbMATH Keywords: functional equation
reviewed by: Andrzej Smajdor
MaRDI profile type: Publication
full work available at URL: s00010-013-0214-x
OpenAlex ID: W2017171006
Wikidata QID: Q59399179
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DOI: 10.1007/S00010-013-0214-X
description: scientific article; zbMATH DE number 6351395