A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations
type: Item
instance of: scholarly article
title: A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations
zbMATH Open document ID: 1060.39029
author: Kil-Woung Jun
author: Yang-Hi Lee
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-01T00:00:00Z
review text: Two Pexiderized versions of the quadratic functional equation are considered: \[ \begin{aligned} f(x\ast y)+f(x\ast y^{-1})&=2g(x)+2g(y),\\ f(x\ast y)+g(x\ast y^{-1})&=2h(x)+2k(y), \end{aligned} \] where \(f,g,h,k\) are functions mapping a group \((G,\ast)\) into a~real or complex topological vector space~\(X\). The main results are obtained for functions mapping a~vector space~\(V\) into a~Banach space~\(X\). It is shown that if the remainder \(f(x+y)+f(x-y)-2g(x)-2g(y)\) is bounded in some sense, then there exists exactly one quadratic function \(Q:V\to X\) such that \(f\) is close (in the same sense) to \(f(0)+Q\) and \(g\) is close to \(\frac{f(0)}{2}+Q\). It is also shown that if the remainder \(f(x+y)+g(x-y)-2h(x)-2k(y)\) is bounded (in some sense), then there exist exactly one quadratic function \(Q:V\to X\) and two unique additive functions \(T,T':V\to X\) such that \(f\) is close (in the same sense) to \(f(0)+Q+T\), \(g\) is close to \(g(0)+Q+T'\) and \(\frac{h+k}{2}\) is close to \(\frac{f(0)+g(0)}{4}+Q+\frac{T}{2}\).
reviewed by: Szymon Wąsowicz
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 2105429
zbMATH Keywords: quadratic functional equation
zbMATH Keywords: Pexiderized quadratic equation
zbMATH Keywords: Banach space
zbMATH Keywords: Hyers-Ulam-Rassias stability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.04.009
OpenAlex ID: W2110959547
cites work: Remarks on the stability of functional equations
cites work: Q4549797
cites work: On the stability of the quadratic mapping in normed spaces
cites work: Q3005289
cites work: On stability of additive mappings
cites work: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings
cites work: On the Stability of the Linear Functional Equation
cites work: On the asymptoticity aspect of Hyers-Ulam stability of mappings
cites work: Some further generalizations of the Hyers-Ulam-Rassias stability of functional equations
cites work: On Hyers-Ulam-Rassias stability of the Pexider equation
cites work: On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality
cites work: A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation
cites work: A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation
cites work: Q4955481
cites work: On the stability of approximately additive mappings
cites work: On a modified Hyers-Ulam sequence
cites work: On the Stability of the Linear Mapping in Banach Spaces
cites work: Q5397123
cites work: Q4791365
cites work: On the Behavior of Mappings which do not Satisfy Hyers-Ulam Stability
cites work: Proprieta’ locali e approssimazione di operatori
cites work: Q5511350
DOI: 10.1016/J.JMAA.2004.04.009
description: scientific article; zbMATH DE number 2105429