Another look at the Matkowski and Wesołowski problem yielding a new class of solutions

The following MW-problem was posed independently by Janusz Matkowski and Jacek Wesołowski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions \(\varphi :[0,1]\rightarrow [0,1]\), distinct from the identity on [0, 1], such that \(\varphi (0)=0\), \(\varphi (1)=1\) and \(\varphi (x)=\varphi (\frac{x}{2})+\varphi (\frac{x+1}{2})-\varphi (\frac{1}{2})\) for every \(x\in [0,1]\)? By now, it is known that each of the de Rham functions \(R_p\), where \(p\in (0,1)\), is a solution of the MW-problem, and for any Borel probability measure \(\mu \) concentrated on (0, 1) the formula \(\phi _\mu (x)=\int _{(0,1)}R_p(x)\, d\mu (p)\) defines a solution \(\phi _\mu :[0,1]\rightarrow [0,1]\) of this problem as well. In this paper, we give a new family of solutions of the MW-problem consisting of Cantor-type functions. We also prove that there are strictly increasing solutions of the MW-problem that are not of the above integral form with any Borel probability measure \(\mu \).