Fixed points and functional equations connected with derivations on Banach algebras
type: Item
instance of: scholarly article
title: Fixed points and functional equations connected with derivations on Banach algebras
zbMATH Open document ID: 1274.39051
published in: Acta Mathematica Hungarica
publication date: 2012-02-13T00:00:00Z
review text: Let \(\mathcal{A}\) and \(\mathcal{B}\) be two algebras and \(1\leq m\leq 4\) be a fixed positive integer. A function \(f\colon \mathcal{A}\to \mathcal{B}\) is called \(m\)-derivation, if it satisfies both functional equations \[ \begin{aligned} f(ax+y)+f(ax-y)&=a^{m-2}\left[f(x+y)+f(x-y)\right]\\ &\quad+2(a^{2}-1)\left[a^{m-2}f(x)+\frac{1}{6}(m-2)(1-(m-2)^{2})f(y)\right]\end{aligned} \] and \[ f(xy)=x^{m}f(y)+f(x)y^{m} \] for all \(x,y\in \mathcal{A}\) and for some \(a\in \mathbb{R}\setminus \{-1,0,1\}\). In this paper, the authors prove the stability and hyperstability of \(m\)-derivations on Banach algebras using a fixed point method.
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46L57
zbMATH DE Number: 6006178
zbMATH Keywords: Banach algebra
zbMATH Keywords: approximate derivation
zbMATH Keywords: additive, quadratic, cubic, and quartic functional equation
zbMATH Keywords: fixed point approach
zbMATH Keywords: stability
zbMATH Keywords: hyperstability
author: Zahra Alizadeh
author: Madjid Eshaghi Gordji
author: H. Khodaei
MaRDI profile type: Publication
full work available at URL: s10474-011-0133-z
OpenAlex ID: W2030780801
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DOI: 10.1007/S10474-011-0133-Z
description: scientific article; zbMATH DE number 6006178