On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings on the Gaussian plane
type: Item
instance of: scholarly article
title: On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings on the Gaussian plane
zbMATH Open document ID: 1025.39018
published in: Southeast Asian Bulletin of Mathematics
publication date: 2003-07-08T00:00:00Z
review text: Let \(Z\), \(W\) be complex linear spaces and \(Q:Z\to W\). The functional equation \[ Q\Biggl(\sum_{i=1}^na_iz_i\Biggr)+\sum_{1\leq i<j\leq n} (a_j\bar{z}_i-a_i\bar{z}_j) =m\sum_{i=1}^nQ(z_i) \] for every complex \(z_i\in Z\) \((i=1,2,\dots,n)\) and fixed \({\mathbb C}^n\ni a=(a_1,a_2,\dots,a_n) \neq (0,0,\dots,0)\) with \[ 0<m=\sum_{i=1}^n|a_i|^2\neq\frac{1}{n}\Biggl[1+\binom{n}{2}\Biggr] \] is introduced. Its solutions are called \textit{quadratic mappings with respect to~\(a\)}. The Hyers-Ulam stability problem for such mappings is solved.
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B32
zbMATH DE Number: 1947334
zbMATH Keywords: Ulam problem
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: quadratic weighted means
zbMATH Keywords: Gaussian plane
zbMATH Keywords: quadratic mapping
zbMATH Keywords: mean equation
zbMATH Keywords: fundamental equation
zbMATH Keywords: complex linear spaces
author: J. M. Rassias
reviewed by: Szymon Wąsowicz
MaRDI profile type: Publication
full work available at URL: s10012-002-0483-0
OpenAlex ID: W2078363847
DOI: 10.1007/S10012-002-0483-0
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description: scientific article; zbMATH DE number 1947334