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Erschienen in:
On Some Functional Equations Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

On Stability of the Linear and Polynomial Functional Equations in Single Variable

verfasst von : Janusz Brzdȩk, Magdalena Piszczek

Erschienen in: Handbook of Functional Equations

Verlag: Springer New York

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Abstract

We present a survey of selected recent results of several authors concerning stability of the following polynomial functional equation (in single variable)
$$\varphi(x)=\sum_{i=1}^m a_i(x)\varphi(\xi_i(x))^{p(i)}+F(x),$$
in the class of functions ϕ mapping a nonempty set S into a Banach algebra X over a field \(\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}\), where m is a fixed positive integer, \(p(i)\in \mathbb{N}\) for \(i=1,\ldots,m\), and the functions \(\xi_i:S\to S\), \(F:S\to X\) and \(a_i:S\to X\) for \(i=1,\ldots,m\), are given. A particular case of the equation, with \(p(i)=1\) for \(i=1,\ldots,m\), is the very well-known linear equation
$$\varphi(x)=\sum_{i=1}^m a_i(x)\varphi(\xi_i(x))+F(x).$$

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Vorheriges Kapitel Remarks on Stability of the Equation of Homomorphism for Square Symmetric Groupoids
Nächstes Kapitel Selections of Set-valued Maps Satisfying Some Inclusions and the Hyers–Ulam Stability
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Metadaten
Titel
On Stability of the Linear and Polynomial Functional Equations in Single Variable
verfasst von
Janusz Brzdȩk
Magdalena Piszczek
Copyright-Jahr
2014
Verlag
Springer New York
Buch
Handbook of Functional Equations

Print ISBN: 978-1-4939-1285-8

Electronic ISBN: 978-1-4939-1286-5

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-1-4939-1286-5_3