The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces
type: Item
instance of: scholarly article
title: The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces
zbMATH Open document ID: 1221.39036
DOI: 10.1155/2010/140767
published in: Discrete Dynamics in Nature and Society
publication date: 2010-09-29T00:00:00Z
full work available at URL: https://eudml.org/doc/232111
review text: Suppose that \(X\) is a linear space, \((Z,N')\) a fuzzy normed space, \((Y,N)\) a fuzzy Banach space, \(f: X\to Y\), \(n\geq 2\) a fixed integer. The authors consider the stability of the functional equation \[ \Delta f(x_1,\dots, x_n)=0 \] where \[ \Delta f(x_1,\dots, x_n)=\sum_{i=1}^{n}f\left(x_i-\frac{1}{n}\sum_{j=1}^{n}x_j\right)-\sum_{i=1}^{n}f(x_i)+nf\left(\frac{1}{n}\sum_{i=1}^{n}x_i\right). \] The main result reads, roughly, as follows. Suppose that \(f: X\to Y\) satisfy \(f(0)=0\) and \[ N(\Delta f(x_1,\dots,x_n),t_1+\cdots+t_n)\geq \min\{N'(\phi(x_1),t_1),\dots,N'(\phi(x_n),t_n)\} \] for all \(x_1,\dots,x_n\in X\), \(t_1,\dots,t_n>0\) where \(\phi: X\to (Z,N')\) is a control mapping satisfying \(\phi(2x)=\alpha \phi(x)\), \(x\in X\) with some \(|\alpha|<2\). Then, there exists a unique quadratic function \(Q: X\to Y\) and a unique additive function \(A: X\to Y\) such that the mapping \(Q+A\) approximates \(f\) (in terms of the fuzzy norm \(N\)). Moreover, if \(f\) is odd, it can be approximated by an additive function \(A\) and if \(f\) is even, a quadratic mapping \(Q\) approximates \(f\).
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 46S40
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 5793477
zbMATH Keywords: stability of functional equations
zbMATH Keywords: fuzzy norm
zbMATH Keywords: fixed point theorem
zbMATH Keywords: fixed point method
zbMATH Keywords: fuzzy approximation
zbMATH Keywords: inner product spaces
zbMATH Keywords: fuzzy normed space
zbMATH Keywords: fuzzy Banach space
zbMATH Keywords: quadratic function
zbMATH Keywords: additive function
author: Madjid Eshaghi Gordji
author: H. Khodaei
reviewed by: Jacek Chmieliński
MaRDI profile type: Publication
OpenAlex ID: W2056278043
Wikidata QID: Q58650736
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description: scientific article; zbMATH DE number 5793477