On a Problem of Janusz Matkowski and Jacek Wesołowski, II

Abstract

We continue our study started in "On a problem of Janusz Matkowski and Jacek Wesołowski" (see arXiv:1703.08459) of the functional equation \begin{equation*} φ(x)=\sum_{n=0}^{N}φ(f_n(x))-\sum_{n=0}^{N}φ(f_n(0)) \end{equation*} and its increasing and continuous solutions $\phi :[0,1]\to [0,1]$ such that $\phi (0)=0$ and $\phi (1)=1$. In this paper we assume that $f_0,\dots ,f_N:[0,1]\to [0,1]$ are strictly increasing contractions such that \begin{equation*} 0\leq f_0(0)<f_0(1)\leq f_1(0)<\cdots <f_{N-1}(1)\leq f_N(0)<f_N(1)\leq 1 \end{equation*} and at least one of the weak inequalities is strong.