On the stability of Cauchy additive mappings
type: Item
instance of: scholarly article
title: On the stability of Cauchy additive mappings
zbMATH Open document ID: 1156.39018
author: Kil-Woung Jun
author: Jaiok Roh
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2008-11-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1222783087
review text: The following inequality and the two other of similar type is considered: \[ \| f(x)+f(y)+f(z)\| \leq \left\| 2f\left(\frac{x+y+z}{2}\right)\right\| ,\qquad x,y,z\in X\tag{1} \] where \(f\colon X\to Y\), \(X,Y\) are Banach spaces. It is easy to see that a solution of the above inequality has to be an additive mapping. The aim of the paper is to show the stability of inequality (1). It is proved that if \(f\) satisfies \[ \| f(x)+f(y)+f(z)\| \leq \left\| 2f\left(\frac{x+y+z}{2}\right)\right\| +\varepsilon\left(\| x\| ^r+\| y\| ^r+\| z\| ^r\right),\qquad x,y,z\in X\tag{2} \] with \(r\neq 1\) or \[ \| f(x)+f(y)+f(z)\| \leq \left\| 2f\left(\frac{x+y+z}{2}\right)\right\| +\varepsilon\left(\| x\| ^r\cdot\| y\| ^r\cdot\| z\| ^r\right),\qquad x,y,z\in X\tag{3} \] with \(r\neq \frac{1}{3}\), then \(f\) can be approximated by an additive mapping. Namely, there exists a unique additive mapping \(a\colon X\to Y\) such that \(\| f(x)-a(x)\| \) is bounded by \(M\| x\| ^r\) or \(N\| x\| ^{3r}\) in the case of (2) or (3), respectively (with some constants \(M,N\) depending on \(\varepsilon\) and \(r\)). In the proofs, the authors use the standard technique involving the Hyers' sequence. However, in the reviewers opinion, it would be much simpler to reduce the considered inequalities so that the classical results might be applied directly. For example, it is easy to see that (2) yields (with some \(k\)) \[ \| f(x+y)-f(x)-f(y)\| \leq k\varepsilon (\| x\| ^r+\| y\| ^r),\qquad x,y\in X \] and the assertion follows immediately from the result of \textit{T. Aoki} [J. Math. Soc. Japan 2, 64--66 (1950; Zbl 0040.35501)] (with further generalizations by \textit{Th. M. Rassias}, \textit{Z. Gajda} and others).
reviewed by: Jacek Chmieliński
Mathematics Subject Classification ID: 39B82
zbMATH DE Number: 5363263
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: Cauchy additive mapping
zbMATH Keywords: Jordan-von Neumann type
zbMATH Keywords: Cauchy-Jensen functional equation
zbMATH Keywords: Banach spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5363263