On Hyers-Ulam-Rassias stability of the Pexider equation
type: Item
instance of: scholarly article
title: On Hyers-Ulam-Rassias stability of the Pexider equation
zbMATH Open document ID: 0940.39021
author: Kil-Woung Jun
author: Dong-Soo Shin
author: Byung-Do Kim
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-07-17T00:00:00Z
review text: Let \((G,+)\) be an abelian group, \((X,\|\cdot\|)\) be a Banach space and \(f,g,h:G\rightarrow X\) be mappings. An equation \(f(x+y)=g(x)+h(y)\) is called a Pexider functional equation. In the paper the stability of that equation in the spirit of Hyers-Ulam-Rassias is considered. The main theorem is the following: Let \(\varphi:G\times G\rightarrow[0,\infty)\) be such that \(\varepsilon(x):=\sum_{j=1}^{\infty}2^{-j}(\varphi(2^{j-1}x,0)+\varphi(0,2^{j-1}x)+\varphi(2^{j-1}x,2^{j-1}x))<\infty\) and \(\lim_{n\rightarrow\infty}2^{-n} \varphi(2^nx,2^ny)=0\) for all \(x,y\in G\). Assume that \(\|f(x+y)-g(x)-h(y)\|\leq\varphi(x,y)\) for all \(x,y\in G\). Then there exists a unique additive mapping \(T:G\rightarrow X\) such that \(\|f(x)-T(x)\|\leq\|g(0)\|+\|h(0)\|+\varepsilon(x)\), \(\|g(x)-T(x)\|\leq\|g(0)\|+2\|h(0)\|+\varphi(x,0)+\varepsilon(x)\) and \(\|h(x)-T(x)\|\leq 2\|g(0)\|+\|h(0)\|+\varphi(0,x)+\varepsilon(x)\) for all \(x,y\in G\). Some possible applications of that theorem are presented.
reviewed by: Szymon Wąsowicz
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 1389232
zbMATH Keywords: Hyers-Ulam-Rassias stability
zbMATH Keywords: Pexider functional equation
zbMATH Keywords: abelian group
zbMATH Keywords: Banach space
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6521
OpenAlex ID: W1982403343
cites work: On approximate solutions of the Pexider equation
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: Approximate homomorphisms
cites work: On the Hyers-Ulam stability of \(\psi\)-additive mappings
cites work: Stability of \(\Psi\)-additive mappings: Applications to nonlinear analysis
cites work: On the Stability of the Linear Mapping in Banach Spaces
cites work: Q5511350
DOI: 10.1006/JMAA.1999.6521
description: scientific article; zbMATH DE number 1389232