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

Abstract

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 $\ddot{I}†:[0,1]\to [0,1]$, distinct from the identity on $[0,1]$, such that $\ddot{I}†(0)=0$, $\ddot{I}†(1)=1$ and $\ddot{I}†(x)=\ddot{I}†(\frac{x}{2})+\ddot{I}†(\frac{x+1}{2})-\ddot{I}†(\frac{1}{2})$ 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 $\hat{I}¼$ concentrated on $(0,1)$ the formula $\ddot{I}{•}_{\hat{I}}¼(x)={\int }_{(0,1)}{R}_p(x)d\hat{I}¼(p)$ defines a solution $\ddot{I}{•}_{\hat{I}}¼:[0,1]\to [0,1]$ 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 $\hat{I}¼$.