Stability of the multi-Jensen equation
type: Item
instance of: scholarly article
title: Stability of the multi-Jensen equation
zbMATH Open document ID: 1211.39017
author: Krzysztof Ciepliński
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-12-10T00:00:00Z
review text: Let \(V\) and \(W\) denote normed spaces. A function \(f:V^m\to W\) is called multi-Jensen, if it satisfies the Jensen functional equation in each of its variables. It is well-known [see \textit{W. Prager} and \textit{J. Schwaiger}, Bull. Korean Math. Soc. 45, No.~1, 133--142 (2008; Zbl 1151.39023)] that multi-Jensen functions are characterized by the functional equation \[ J_f(x,y):=f\left(\frac{x+y}{2}\right) -\frac{1}{2^m}\sum_{S\subset\mathbf{m}}f(x_S+y_{\mathbf{m}\setminus S})=0 \quad (x,y\in V^m), \] where \(\mathbf{m}:=\{1,\dots,m\}\) and, for \(S\subset\mathbf{m}\) and an \(m\)-tuple \(z=(z_1,\dots,z_m)\in V^m\), the \(i\)th coordinate of \(z_S\in V^m\) is defined to be \(z_i\) if \(i\in\mathbf{m}\) and to be zero if \(i\not\in\mathbf{m}\). The stability problems of this functional equation were considered in several recent papers. Prager and Schwaiger [loc. cit.] proved the stability in the Hyers-Ulam sense, while the author [ibid. 45, No.~4, 729--737 (2008; Zbl 1172.39033)] investigated the stability in Gavruta's sense. The main result of this paper concerns the stability of the above functional equation in the Rassias sense, i.e., when the stability estimate for \(J_f\) is expressed in terms of the sum of \(p\)th power of the coordinates of \(x\) and \(y\).
reviewed by: Zsolt Páles
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 5644924
zbMATH Keywords: Rassias stability
zbMATH Keywords: multi-Jensen functional equation
zbMATH Keywords: normed spaces
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.08.021
OpenAlex ID: W2043937759
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DOI: 10.1016/J.JMAA.2009.08.021
description: scientific article; zbMATH DE number 5644924