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

Janusz Brzdȩk () and Magdalena Piszczek ()
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Janusz Brzdȩk: Pedagogical University
Magdalena Piszczek: Pedagogical University

A chapter in Handbook of Functional Equations, 2014, pp 59-81 from Springer

Abstract: 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 $$\varphi(x)=\sum_{i=1}^m a_i(x)\varphi(\xi_i(x))+F(x).$$

Keywords: Hyers–Ulam stability; Polynomial functional equation; Linear functional equation; Single variable; Banach space; Characteristic root (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-1-4939-1286-5_3

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