On the stability of Hosszú's functional equation
type: Item
instance of: scholarly article
title: On the stability of Hosszú's functional equation
zbMATH Open document ID: 0872.39017
DOI: 10.1007/BF03322226
author: László Losonczi
published in: Results in Mathematics
publication date: 1997-10-19T00:00:00Z
review text: Let \(X\) be a normed space and \(Y\) be a Banach space. The following two theorems are proved. 1. If \(B\) is a subset of \(X^2\) such that the first (or second) coordinates of the points of \(B\) form a bounded set and \(f:X\to Y\) satisfies \(|f(x+y)-f(x)-f(y)|\leq\varepsilon\) for all \((x,y)\in X^2\setminus B\) with an \(\varepsilon\geq 0\), then there exists exactly one additive function \(A:X\to Y\) such that \(|f(x)-A(x)|\leq 5\varepsilon\) for every \(x\in X\). 2. If \(f:\mathbb{R}\to Y\) satisfies \(|f(x+y-xy)+f(xy)-f(x)-f(y)|\leq\varepsilon\) for all \(x,y\in\mathbb{R}\) with an \(\varepsilon\geq 0\), then there exists exactly one additive function \(A:\mathbb{R}\to Y\) and exactly one constant \(b\in Y\) such that \(|f(x)-A(x)-b|\leq 20\varepsilon\) for every \(x\in X\). The first theorem generalizes a result of \textit{F. Skof} [Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 117, No. 4-6, 377-389 (1983; Zbl 0794.39008)] whereas the second one contains that of \textit{C. Borelli} [Result. Math. 26, No. 3-4, 221-224 (1994; Zbl 0828.39019)].
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 920813
zbMATH Keywords: Hosszú functional equation
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: Banach space
zbMATH Keywords: additive function
reviewed by: Karol Baron
MaRDI profile type: Publication
cites work: The functional equation \(f(x + y - xy) + f(xy) = f(x) + f(y)\)
cites work: On Hyers-Ulam stability of Hosszú's functional equation
cites work: Q5599368
cites work: On the general solution of the functional equation \(f(x + y - xy) + f(xy) = f(x) + f(y)\)
cites work: The complete solution of Hosszu's functional equation over a field
cites work: On the general solution of a functional equation in the domain of distributions
cites work: On the Stability of the Linear Functional Equation
cites work: Applications of extensions of additive functions
cites work: Proprieta’ locali e approssimazione di operatori
cites work: Q5551856
cites work: Q4349914
description: scientific article; zbMATH DE number 920813