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On a problem of Janusz Matkowski and Jacek Wesołowski, II

  • Janusz MorawiecEmail author
  • Thomas Zürcher
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Abstract

We continue our study started in Morawiec and Zürcher (Aequ Math 92(4):601–615, 2018) of the functional equation
$$\begin{aligned} \varphi (x)=\sum _{n=0}^{N}\varphi (f_n(x))-\sum _{n=0}^{N}\varphi (f_n(0)) \end{aligned}$$
and its increasing and continuous solutions \(\varphi :[0,1]\rightarrow [0,1]\) such that \(\varphi (0)=0\) and \(\varphi (1)=1\). In this paper we assume that \(f_0,\ldots ,f_N:[0,1]\rightarrow [0,1]\) are strictly increasing contractions such that
$$\begin{aligned} 0\le f_0(0)<f_0(1)\le f_1(0)<\cdots<f_{N-1}(1)\le f_N(0)<f_N(1)\le 1 \end{aligned}$$
and at least one of the weak inequalities is strong.

Keywords

Functional equations Iterated function systems Singular functions Absolutely continuous functions 

Mathematics Subject Classification

Primary 39B12 Secondary 37A05 

Notes

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Copyright information

© The Author(s) 2019

OpenAccessThis 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.Instytut MatematykiUniwersytet ŚląskiKatowicePoland

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