Ulam stability problem for quadratic mappings of Euler--Lagrange
type: Item
instance of: scholarly article
title: Ulam stability problem for quadratic mappings of Euler--Lagrange
zbMATH Open document ID: 1074.39027
author: Kil-Woung Jun
author: Hark-Mahn Kim
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-06-01T00:00:00Z
review text: The authors consider a generalization of the \textit{quadratic functional equation}. Let \({\mathcal A}\) and \({\mathcal B}\) be linear spaces and \(a,b\in\mathbb{N}\). A mapping \(Q:{\mathcal A}\to {\mathcal B}\) satisfying the equation \[ (a+b)aQ(x)+(a+b)bQ(y)=Q(ax+by)+abQ(x-y),\;\;\;x,y\in {\mathcal A} \] is called the {quadratic mapping of Euler--Lagrange}. The aim of the paper is to prove the stability of the above equation. It is shown that, with \({\mathcal B}\) being a normed space, for \(f:{\mathcal A}\to {\mathcal B}\) satisfying the inequality \[ \| (a+b)aQ(x)+(a+b)bQ(y)-Q(ax+by)-abQ(x-y)\| \leq\varphi(x,y),\;\;\;x,y\in {\mathcal A} \] with the \textit{approximate reminder} \(\varphi\) satisfying some assumptions, there exists a unique quadratic mapping of Euler-Lagrange \(Q:{\mathcal A}\to{\mathcal B}\) which is, in some sense, close to \(f\). Additionally, similar results are given for mappings between Banach modules.
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 2173620
zbMATH Keywords: quadratic functional equation
zbMATH Keywords: Euler-Lagrange mapping
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: normed space
zbMATH Keywords: Banach modules
zbMATH Keywords: linear spaces
reviewed by: Jacek Chmieliński
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.058
OpenAlex ID: W2082154351
cites work: Q3992445
cites work: Remarks on the stability of functional equations
cites work: On the stability of the quadratic mapping in normed spaces
cites work: Q4879432
cites work: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings
cites work: Stability of Isometries
cites work: On the Stability of the Linear Functional Equation
cites work: On the asymptoticity aspect of Hyers-Ulam stability of mappings
cites work: Approximate homomorphisms
cites work: On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality
cites work: Q4012607
cites work: Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings
cites work: On the Stability of the Linear Mapping in Banach Spaces
cites work: On the stability of functional equations in Banach spaces
cites work: Q3259049
cites work: On a conjecture of Z. Ditzian
DOI: 10.1016/J.NA.2004.07.058
description: scientific article; zbMATH DE number 2173620