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Title: On a problem of Janusz Matkowski and Jacek Wesołowski
Authors: Morawiec, Janusz
Zürcher, Thomas
Keywords: Absolutely continuous functions; Continuously singular functions; Functional equations; Probabilistic iterated function systems
Issue Date: 2018
Citation: Aequationes mathematicae, Vol. 92, iss. 4 (2018), s. 601-615
Abstract: We study the problem of the existence of increasing and continuous solutions φ: [0 , 1] → [0 , 1] such that φ(0) = 0 and φ(1) = 1 of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=1Nφ(fn(0)),where N∈ N and f0, … , fN: [0 , 1] → [0 , 1] are strictly increasing contractions satisfying the following condition 0 = f0(0) < f0(1) = f1(0) < ⋯ < fN - 1(1) = fN(0) < fN(1) = 1. In particular, we give an answer to the problem posed in Matkowski (Aequationes Math. 29:210–213, 1985) by Janusz Matkowski concerning a very special case of that equation.
Description: The research of the first author was supported by the Silesian University Mathematics Department (Iterative Functional Equations and Real Analysis program). Furthermore, the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 291497 while the second author was a postdoctoral researcher at the University of Warwick.
DOI: 10.1007/s00010-018-0556-5
ISSN: 0001-9054
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