Functional inequalities associated with additive mappings
type: Item
instance of: scholarly article
title: Functional inequalities associated with additive mappings
zbMATH Open document ID: 1161.26012
DOI: 10.1155/2008/136592
author: Jaiok Roh
author: Ick-Soon Chang
published in: Abstract and Applied Analysis
publication date: 2008-10-16T00:00:00Z
full work available at URL: https://eudml.org/doc/55367
review text: Consider a Banach space \(X,\) an additive group \(G\) divisible by 2 and mappings \(\varphi:G\times G\times G\rightarrow[0,\infty)\) such that \(\varphi(2x,-x,0)=\varphi(0,x,-x)\) and \(\lim_{n\rightarrow\infty} (1/2^{n})\varphi(2^{n+1}x,2^{n}y,2^{n}z)=0\) (or \(\lim_{n\rightarrow\infty }2^{n}\varphi(x/2^{n-1},y/2^{n},z/2^{n})=0)\) for all \(x,y,z\in G.\) Under these hypotheses it is proved that any mapping \(f:G\rightarrow X\) that verifies the functional inequality \(\left\| f(x)+2f(y)+2f(z)\right\| \leq\left\| 2f(x/2+y+z)\right\| +\varphi(x,y,z)\) for all \(x,y,z\in G\) is additive. This result is related to the paper of \textit{Gy. Maksa} and \textit{P. Volkmann} [Publ. Math. 56, No. 1--2, 197--200 (2000; Zbl 0991.39015)].
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 47H14
zbMATH DE Number: 5352107
zbMATH Keywords: Functional inequality
zbMATH Keywords: Hyers-Ulam theorem
zbMATH Keywords: additive function
Wikidata QID: Q58644276
MaRDI profile type: Publication
OpenAlex ID: W2005269432
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description: scientific article; zbMATH DE number 5352107