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Remarks on Stability of the Linear Functional Equation in Single Variable

In: Nonlinear Analysis

Author

Listed:
  • Janusz Brzdȩk

    (Pedagogical University)

  • Dorian Popa

    (Technical University)

  • Bing Xu

    (Sichuan University)

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).

Suggested Citation

  • Janusz Brzdȩk & Dorian Popa & Bing Xu, 2012. "Remarks on Stability of the Linear Functional Equation in Single Variable," Springer Optimization and Its Applications, in: Panos M. Pardalos & Pando G. Georgiev & Hari M. Srivastava (ed.), Nonlinear Analysis, edition 127, chapter 0, pages 91-119, Springer.
    • Handle: RePEc:spr:spochp:978-1-4614-3498-6_7
    DOI: 10.1007/978-1-4614-3498-6_7
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