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Aequationes mathematicae

, Volume 93, Issue 1, pp 137–148 | Cite as

On a functional equation related to a problem of G. Derfel

  • Mariusz SudzikEmail author
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Abstract

Two years ago, during the 21st European Conference on Iteration Theory, Gregory Derfel asked: “Does there exist a non-trivial bounded continuous solution of the equation \(2f(x) = f(x-1) + f(-2x)\)?” He repeated the question during the 55th International Symposium on Functional Equations. In this paper we present a partial solution of a more general problem, connected to the functional equation \(f(x) = M \big ( f(x+t_{1}), f(x + t_{2} ),\ldots , f(x + t_{n-1} ), \,f(ax) \big ),\) where \(n \in \mathbb {N}, \,t_{1},t_{2},\ldots ,t_{n-1} \in \mathbb {R} {\setminus } \{ 0\}, \,a \in (-\infty , 0)\) and M is a given function in n variables satisfying some additional properties. In particular, M can be a weighted quasi-arithmetic mean in n variables.

Keywords

Archetypal equation Quasi-arithmetic mean Functional equations Equations with rescaling 

Mathematics Subject Classification

Primary 39B22 Secondary 39B05 

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GóraPoland

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