Asymptotic aspect of Drygas, quadratic and Jensen functional equations in metric abelian groups
type: Item
instance of: scholarly article
title: Asymptotic aspect of Drygas, quadratic and Jensen functional equations in metric abelian groups
zbMATH Open document ID: 1413.39056
author: Peng Zhang
published in: Acta Mathematica Hungarica
publication date: 2018-09-03T00:00:00Z
review text: Let \((X, +)\) and \((Y, +)\) be abelian groups. A function \(f \colon X \to Y \) is said to be -- additive, if \(f (x +y) = f (x) + f (y)\) holds for all \(x, y \in X\); -- Jensen, if \(f (x + y) + f (x-y) = 2f (x)\) is satisfied for all \(x, y \in X\); -- quadratic, if \(f (x +y) + f (x-y) = 2f (x) + 2f (y)\) is fulfilled for all \(x, y \in X\). Furthermore, the functional equation \[ f (x +y) + f (x-y) = 2f (x) + f (y) + f (-y) \;\;\; \left( x, y \in X\right) \] is known as Drygas equation. The main purpose of this paper is to investigate the asymptotic stability behaviour of the Drygas equation and quadratic and Jensen functional equations. In their main result the authors show that if these equations hold approximately for large arguments with an upper bound \(\varepsilon\), then they are also valid approximately everywhere with a new upper bound which is a constant multiple of \(\varepsilon\). These results are applied to the study of asymptotic properties of Drygas, quadratic and Jensen functional equations. Some results of hyperstability character for these functional equations are also obtained.
reviewed by: Eszter Gselmann
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 6931182
zbMATH Keywords: Drygas equation
zbMATH Keywords: quadratic equation
zbMATH Keywords: Jensen equation
zbMATH Keywords: stability
zbMATH Keywords: asymptotic stability
zbMATH Keywords: metric group
MaRDI profile type: Publication
full work available at URL: s10474-018-0807-x
OpenAlex ID: W2793452882
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DOI: 10.1007/S10474-018-0807-X
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description: scientific article; zbMATH DE number 6931182