Remarks on Stability of the Linear Functional Equation in Single Variable

Janusz Brzdȩk (), Dorian Popa () and Bing Xu ()
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Janusz Brzdȩk: Pedagogical University
Dorian Popa: Technical University
Bing Xu: Sichuan University

Chapter Chapter 7 in Nonlinear Analysis, 2012, pp 91-119 from Springer

Abstract: Abstract We present some observations concerning stability of the following linear functional equation (in single variable) $$\varphi\bigl(f^m(x) \bigr)=\sum_{i=1}^m a_i(x)\varphi\bigl(f^{m-i}(x) \bigr)+F(x), $$ in the class of functions φ mapping a nonempty set S into a Banach space X over a field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ , where m is a fixed positive integer and the functions f:S→S, F:S→X and $a_{i}:S\to\mathbb{K}$ , i=1,…,m, are given. Those observations complement the results in our earlier paper (Brzdȩk et al. in J. Math. Anal. Appl. 373:680–689, 2011).

Keywords: Hyers-Ulam stability; Linear functional equation; Single variable; Banach space; Characteristic root; 39B82; 39B62 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-1-4614-3498-6_7

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