On the generalized Hyers-Ulam stability of module left \((m,n)\)-derivations
type: Item
instance of: scholarly article
title: On the generalized Hyers-Ulam stability of module left \((m,n)\)-derivations
zbMATH Open document ID: 1267.39022
published in: Aequationes Mathematicae
publication date: 2012-11-21T00:00:00Z
review text: The stability problem of functional equations originates from a question of Ulam in 1940, concerning the stability of group homomorphisms. In 1941, D. H. Hyers gave a first affirmative answer to the question for Banach spaces. Let \(A\) be an algebra over the real or complex field \(\mathbb{F}\) and \(M\) be a left \(A\)-module. An additive mapping \(d:A\to M\) is called a module left \((m,n)\)-derivation if \[ (m+n)d(xy)=2mx.d(y)+2ny.d(x) \] holds for all \(x,y\in A\), where \(m\geq 0\), \(n\geq 0\) with \(m+n\neq 0\) are some fixed integers. The main purpose of the present paper is to prove the following theorem: Let \(A\) be a normed algebra, \(M\) a Banach left \(A\)-module, and \(F:A^2\to [0,\infty)\) a function such that \(F(2x,y)=\xi F(x,Y)\) and \(F(x,2y)=\chi F(x,y)\) for some scalars \(\xi,\chi\geq 0\), \(\xi\chi<1\). Suppose that \(f:A\to M\) is a map with the properties \[ \|f(x+y)-f(x)f(y)\|\leq F(x,y) \] for all \(x,y\in A\). Then there exists a unique module left \((m,n)\)-derivation \(d:A\to M\) such that \[ \|f(x)-d(x)\|\leq \frac{F(x,x)}{1-\xi\chi} \] for all \(x\in A\).
reviewed by: Nasrin Eghbali
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46L57
Mathematics Subject Classification ID: 16W25
zbMATH DE Number: 6109267
zbMATH Keywords: generalized Hyers-Ulam stability
zbMATH Keywords: normed algebra
zbMATH Keywords: Banach left \(A\)-module
zbMATH Keywords: module left \((m,n)\)-derivation
author: Ajda Fošner
MaRDI profile type: Publication
full work available at URL: s00010-012-0124-3
OpenAlex ID: W1970659317
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DOI: 10.1007/S00010-012-0124-3
description: scientific article; zbMATH DE number 6109267