On the stability of functional equations and a problem of Ulam
type: Item
instance of: scholarly article
title: On the stability of functional equations and a problem of Ulam
zbMATH Open document ID: 0981.39014
DOI: 10.1023/A:1006499223572
author: Themistocles M. Rassias
published in: Acta Applicandae Mathematicae
publication date: 2002-03-13T00:00:00Z
review text: In 1940 S.~M.~Ulam posed the problem concerning the stability of homomorphisms. In 1941 D.~H.~Hyers gave the first significant partial solution: Let \(X,Y\) be Banach spaces and \(\delta>0\). If the function \(f:X\to Y\) satisfies the inequality \[ \bigl\|f(x+y)-f(x)-f(y)\bigr\|\leq\delta\tag{\(\ast\)} \] for all \(x,y\in X\), then there exists the unique additive function \(A:X\to Y\) such that \(\bigl\|f(x)-A(x)\bigr\|\leq\delta\) for all \(x\in X\). The stability of functional equations may be considered from some points of view. In~(\(\ast\)) the left-hand side of the inequality is bounded. Many results concerning the stability were proved with the assumption that the left-hand sides of the appropriate inequalities may be unbounded. The stability may be also considered on restricted domains. \smallskip The stability of functional equations is extensively investigated by many researchers. The reviewed paper contains the wide range survey of both classical results and current research concerning the stability. Many results are presented with proofs, so the paper is self-contained. It is of interest to researchers in the field and it is accessible to graduate students as well. The related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory. \smallskip The paper consists of nine sections: Introduction, Additive functional equation, Jensen's functional equation, Quadratic functional equations, Exponential functional equations, Multiplicative functional equation, Logarithmic functional equation, Trigonometric functional equations, Other functional equations. The bibliography contains 139 items. For other surveys devoted to the stability of functional equations cf. \textit{G. L. Forti}, [Aequationes Math. 50, No.~1-2, 143-190 (1995; Zbl 0836.39007)] and \textit{D. H. Hyers} and \textit{T. M. Rassias} [ibid. 44, No.~2/3, 125-153 (1992; Zbl 0806.47056)].
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 1530460
zbMATH Keywords: stability
zbMATH Keywords: nonlinear equations
zbMATH Keywords: Cauchy difference
zbMATH Keywords: semigroup
zbMATH Keywords: inequalities
zbMATH Keywords: Banach spaces
zbMATH Keywords: complementarity theory
zbMATH Keywords: additive functional equation
zbMATH Keywords: Jensen's functional equation
zbMATH Keywords: quadratic functional equations
zbMATH Keywords: exponential functional equations
zbMATH Keywords: multiplicative functional equation
zbMATH Keywords: logarithmic functional equation
zbMATH Keywords: trigonometric functional equations
reviewed by: Szymon Wąsowicz
MaRDI profile type: Publication
full work available at URL: a:1006499223572
OpenAlex ID: W1524508516
description: scientific article; zbMATH DE number 1530460