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.
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Dedicated to Professor Karol Baron on the occasion of his 70th birthday.
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Sudzik, M. On a functional equation related to a problem of G. Derfel. Aequat. Math. 93, 137–148 (2019). https://doi.org/10.1007/s00010-018-0600-5
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DOI: https://doi.org/10.1007/s00010-018-0600-5