Approximate functional inequalities by additive mappings
type: Item
instance of: scholarly article
title: Approximate functional inequalities by additive mappings
zbMATH Open document ID: 1256.39020
author: Hark-Mahn Kim
author: Juri Lee
author: Eunyoung Son
published in: Journal of Mathematical Inequalities
publication date: 2012-09-21T00:00:00Z
full work available at URL: jmi-06-43.pdf
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6084712
zbMATH Keywords: Ulam-Hyers stability
zbMATH Keywords: additive function
zbMATH Keywords: \(n\)-divisible abelian group
zbMATH Keywords: Banach space
zbMATH Keywords: Cauchy Jensen inequality
MaRDI profile type: Publication
OpenAlex ID: W2313274239
DOI: 10.7153/JMI-06-43
review text: Let \(f:G \to Y\), where \(G\) is an \(n\)-divisible abelian group and \(Y\) is a Banach space. After showing that \(f\) is additive if and only if it satisfies the inequality NEWLINE\[NEWLINE \left\| f(x)+f(y)+nf(z) \right\| \leq \left\| nf\Big(\frac{x+y}{n}+z\Big) \right\|, \quad x,y,z \in G, NEWLINE\]NEWLINE the authors prove the following stability result:NEWLINENEWLINETheorem: Suppose that \(f:G \to Y\) satisfies the inequality NEWLINE\[NEWLINE \left\| f(x)+f(y)+nf(z) \right\| \leq \left\| nf\Big(\frac{x+y}{n}+z\Big) \right\|+\phi(x,y,z) NEWLINE\]NEWLINE where \(\phi: G^3 \to [0,\infty)\) is a given function such that NEWLINE\[NEWLINE \lim_{k \to \infty} \frac{1}{n^k}\phi(n^kx,n^ky,n^kz)=0 NEWLINE\]NEWLINE for all \(x, y, z \in G\) and NEWLINE\[NEWLINE \check{\phi}(x,z):=\sum_{i=0}^{\infty} \frac{1}{2n^{i+1}} \Big(\phi(n^{i+1}x,0,-n^iz)+\phi(-n^{i+1}x,0,n^iz)\Big)<\infty NEWLINE\]NEWLINE for all \(x, z \in G\). Then there exists a unique additive mapping \(h:G \to Y\) such that NEWLINE\[NEWLINE \left\| f(x)-h(x)\right\| \leq \check{\phi}(x,x)+\frac{\phi(x,-x,0)}{2}+\frac{n^2}{n-1}\left\| f(0)\right\| NEWLINE\]NEWLINE for all \(x \in G\).NEWLINENEWLINEMoreover, the paper contains corollaries and other similar results.
reviewed by: Gian Luigi Forti
description: scientific article; zbMATH DE number 6084712