A characterization of Hyers-Ulam stability of first order linear differential operators.
type: Item
instance of: scholarly article
title: A characterization of Hyers-Ulam stability of first order linear differential operators.
zbMATH Open document ID: 1045.47037
DOI: 10.1016/S0022-247X(03)00458-X
author: T. Miura
author: Shizuo Miyajima
author: S.-E. Takahasi
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-11-16T00:00:00Z
review text: Let \(E_1\), \(E_2\) be two real Banach spaces and \(f: E_1\to E_2\) is a mapping such that \(f(tx)\) is continuous in \(t\in\mathbb{R}\) (the set of real numbers), for each fixed \(x\in E_1\). \textit{Th. M. Rassias} [Proc. Am. Math. Soc. 72, 297--300 (1978; Zbl 0398.47040)] introduced the following inequality: Assume that there exist \(\theta\geq 0\) and \(p\in [0,1)\) such that \[ \| f(x+ y)- f(x)- f(y)\|\leq \theta(\| x\|^p+\| y\|^p) \] for every \(x,y\in E_1\). Then there exists a unique linear mapping \(T: E_1\to E_2\) such that \(\| f(x)- T(x)\|\leq 2\theta\| x\|^p/(2-2^p)\) for every \(x\in E_1\). \textit{D. H. Hyers} [Proc. Natl. Acad. Sci. USA 27, 222--224 (1941; Zbl 0061.26403)] had obtained the result for \(p= 0\). Rassias' proof also works for \(p< 0\). In 1990, the reviewer, during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for \(p\geq 1\). In 1991, \textit{Z. Gajda} [Int. J. Math. Math. Sci. 14, 431--434 (1991; Zbl 0739.39013)], following the reviewer's approach, gave an affirmative solution to this question for \(p> 1\). The authors of the present paper consider the following problem: Let \(X\) be a complex Banach space and \(h: \mathbb{R}\to\mathbb{C}\) a continuous function. Assume that \(T_h: C^1(\mathbb{R}, X)\to C(\mathbb{R}, X)\) is the linear differential operator defined by \(T_hu= u'+ hu\). Then a very essential and interesting necessary and sufficient condition is obtained in order for the operator \(T_h\) to be stable in the sense of Hyers-Ulam.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 39B42
zbMATH DE Number: 2003894
zbMATH Keywords: Ulam's problem
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: complex Banach space
zbMATH Keywords: minimum
zbMATH Keywords: strongly differentiable
zbMATH Keywords: linear differential operator
zbMATH Keywords: additive mappings
reviewed by: Themistocles M. Rassias
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00458-x
OpenAlex ID: W1974688961
description: scientific article; zbMATH DE number 2003894