Stability of an additive functional inequality with the fixed point alternative
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.39022
publication date: 2012-09-21T00:00:00Z
full work available at URL: index.html
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6085204
zbMATH Keywords: additive functional inequality
zbMATH Keywords: fixed point alternative
zbMATH Keywords: Ulam-Hyers stability
zbMATH Keywords: Banach space
MaRDI profile type: Publication
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author: W.-G. Park
author: M. H. Han
title: Stability of an additive functional inequality with the fixed point alternative
published in: International Journal of Pure and Applied Mathematics
review text: Let \(f:\mathcal X \to \mathcal Y\), \(\mathcal X\) be a normed space and \(\mathcal Y\) be a Banach space. After showing that \(f\) is additive if and only if it satisfies the inequality NEWLINE\[NEWLINE \| f(x)+2f(y)+2f(z) \| \leq \| f(x)+f(2y+2z) \|, NEWLINE\]NEWLINE the authors prove the following stability result:NEWLINENEWLINETheorem: Suppose that an odd mapping \(f:\mathcal X \to \mathcal Y\), \(\mathcal X\) satisfies the inequality NEWLINE\[NEWLINE \| f(x)+2f(y)+2f(z) \| \leq \| f(x)+f(2y+2z) \|+\phi(x,y,z) NEWLINE\]NEWLINE where \(\phi: \mathcal X^3 \to [0,\infty)\) is a given function. If there exists \(L <1\) such that NEWLINE\[NEWLINE \phi(x,y,z) \leq \frac{1}{2} L\phi(2x,2y,2z) NEWLINE\]NEWLINE for all \(x, y, z \in \mathcal X\), then there exists a unique additive mapping \(A:\mathcal X \to \mathcal Y\) such that NEWLINE\[NEWLINE \| f(x)-A(x)\| \leq \frac{L}{2-2L}\phi(2x,-x,0) NEWLINE\]NEWLINE for all \(x \in \mathcal X\).NEWLINENEWLINEThe proof is based on the fixed point method.NEWLINENEWLINEReviewer's remark: The quoted generalization of Rassias' theorem due to Gǎvruta, is a special case of a result published in 1980 by the reviewer [\textit{G. L. Forti}, Stochastica 4, No. 1, 23--30 (1980; Zbl 0442.39005)].
reviewed by: Gian Luigi Forti
description: scientific article; zbMATH DE number 6085204