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Remarks on Stability of the Linear Functional Equation in Single Variable

  • Chapter
Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 68))

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).

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Authors and Affiliations

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

    Janusz Brzdȩk

  2. Department of Mathematics, Technical University, Str. Memorandumului 28, Cluj-Napoca, 400114, Romania

    Dorian Popa

  3. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, P.R. China

    Bing Xu

Authors
  1. Janusz Brzdȩk
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  2. Dorian Popa
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  3. Bing Xu
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Correspondence to Bing Xu .

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Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. Center for Applied Optimization, Dept. Industrial & Systems, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Pando G. Georgiev

  3. , Department of Mathematics & Statistics, University of Victoria, Finnerty Road 3800, Victoria, V8W 3R4, British Columbia, Canada

    Hari M. Srivastava

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Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.

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Brzdȩk, J., Popa, D., Xu, B. (2012). Remarks on Stability of the Linear Functional Equation in Single Variable. In: Pardalos, P., Georgiev, P., Srivastava, H. (eds) Nonlinear Analysis. Springer Optimization and Its Applications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3498-6_7

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