On the Ulam stability of Jensen and Jensen type mappings on restricted domains
type: Item
instance of: scholarly article
title: On the Ulam stability of Jensen and Jensen type mappings on restricted domains
zbMATH Open document ID: 1028.39011
DOI: 10.1016/S0022-247X(03)00136-7
author: Matina J. Rassias
author: J. M. Rassias
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: The functional equations \[ \begin{aligned} f(x+y)+f(x-y)&=2f(x),\tag{1}\\ f(x+y)-f(x-y)&=2f(y)\tag{2}\end{aligned} \] are considered for functions mapping a real normed space \(X\) into a real Banach space \(Y\). The stability of the equation (1) was proved by \textit{J. M. Rassias} [Geometry, Analysis and Mechanics, World Scientific, 365-376 (1994; Zbl 0842.39014)]. For a fixed \(d>0\) the stability of equations (1), (2) on a restricted domain \(\|x\|+\|y\|\geq d\) is proved. As a consequence the following results are obtained. Corollary 1. A mapping \(f:X\to Y\) fulfils (1) iff the asymptotic condition \[ \bigl\|f(x+y)+f(x-y)-2f(x)\bigr\|\to 0\quad\text{as } \|x\|+\|y\|\to\infty \] holds. Corollary 2. A mapping \(f:X\to Y\) fulfils (2) iff the asymptotic conditions \[ \bigl\|f(-x)+f(x)\bigr\|\to 0\quad\text{ as }\|x\|\to\infty \] and \[ \bigl\|f(x+y)-f(x-y)-2f(y)\bigr\|\to 0\quad\text{ as } \|x\|+\|y\|\to\infty \] hold.
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 1956878
zbMATH Keywords: Ulam stability problem
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: approximately odd mapping
zbMATH Keywords: Jensen equation
zbMATH Keywords: Jensen type equation
zbMATH Keywords: restricted domain
zbMATH Keywords: functional equations
zbMATH Keywords: normed space
zbMATH Keywords: Banach space
reviewed by: Szymon Wąsowicz
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00136-7
OpenAlex ID: W1963497765
description: scientific article; zbMATH DE number 1956878