On a problem of Janusz Matkowski and Jacek Wesołowski

Abstract We study the problem of the existence of increasing and continuous solutions %$\varphi :[0,1]\rightarrow [0,1]%$ such that %$\varphi (0)=0%$ and %$\varphi (1)=1%$ of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=1Nφ(fn(0)),%$\begin{aligned} \varphi (x)=\sum _{n=0}^{N}\varphi (f_n(x))-\sum _...
Ausführliche Beschreibung

Abstract We study the problem of the existence of increasing and continuous solutions %$\varphi :[0,1]\rightarrow [0,1]%$ such that %$\varphi (0)=0%$ and %$\varphi (1)=1%$ of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=1Nφ(fn(0)),%$\begin{aligned} \varphi (x)=\sum _{n=0}^{N}\varphi (f_n(x))-\sum _{n=1}^{N}\varphi (f_n(0)), \end{aligned}%$where %$N\in {\mathbb {N}}%$ and %$f_0,\ldots ,f_N:[0,1]\rightarrow [0,1]%$ are strictly increasing contractions satisfying the following condition %$0=f_0(0)<f_0(1)=f_1(0)<\cdots<f_{N-1}(1)=f_N(0)<f_N(1)=1%$. 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. Ausführliche Beschreibung