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
Open Access


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.


Archetypal equation Quasi-arithmetic mean Functional equations Equations with rescaling 

Mathematics Subject Classification

Primary 39B22 Secondary 39B05 


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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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