On the stability of the generalized sine functional equations
type: Item
instance of: scholarly article
title: On the stability of the generalized sine functional equations
zbMATH Open document ID: 1181.39021
author: Gwang Hui Kim
publication date: 2009-11-11T00:00:00Z
review text: The superstability bounded by a constant for the sine functional equation \[ f(x)f(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] was improved by \textit{G. H. Kim} [J. Math. Anal. Appl. 331, No. 2, 886--894 (2007; Zbl 1119.39024)] and \textit{R. Badora} and \textit{R. Ger} [Adv. Math., Dordr. 3, 3--15 (2002; Zbl 1010.39012)]. In this paper the author investigates the superstability for the generalized sine functional equations \[ g(x)f(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] \[ f(x)g(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] \[ g(x)g(y)=f\big(\frac{x+y}{2}\big)^2-f\big(\frac{x-y}{2}\big)^2 \] where \((G,+)\) is a uniquely 2-divisible Abelian group and \(f, g:G \to \mathbb C\) are nonzero functions. The author also obtains the superstability bounded by a constant for each equation and extends his results when \(G\) is a Banach algebra.
reviewed by: Maryam Amyari
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 5629480
zbMATH Keywords: stability
zbMATH Keywords: superstability
zbMATH Keywords: sine functional equation
zbMATH Keywords: cosine functional equation
zbMATH Keywords: Abelian group
zbMATH Keywords: Banach algebra
MaRDI profile type: Publication
full work available at URL: s10114-008-6621-6
OpenAlex ID: W1569568031
cites work: Q5511350
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DOI: 10.1007/S10114-008-6621-6
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 5629480