On a problem of Janusz Matkowski and Jacek Weso{l}owski

We study the problem of the existence of increasing and continuous solutions $\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$ of the functional equation \begin{equation*} \varphi(x)=\sum_{n=0}^{N}\varphi(f_n(x))-\sum_{n=1}^{N}\varphi(f_n(0)), \end{equation*} where $N\in\mathbb N$ and $f_0,\ldots,f_N\colon[0,1]\to[0,1]$ 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 the article Remark on BV-solutions of a functional equation connected with invariant measures by Janusz Matkowski concerning a very special case of that equation.