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

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Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 96))

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

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

2010 Mathematics Subject Classification: Primary 39B82; Secondary 39B62

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Authors and Affiliations

  1. Department of Mathematics, Pedagogical University, Podchorążych 2, 30-084, Kraków, Poland

    Janusz Brzdȩk & Magdalena Piszczek

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  1. Janusz Brzdȩk
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  2. Magdalena Piszczek
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Correspondence to Janusz Brzdȩk .

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Brzdȩk, J., Piszczek, M. (2014). On Stability of the Linear and Polynomial Functional Equations in Single Variable. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 96. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1286-5_3

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