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 $φ\colon [0,1]\to [0,1]$, distinct from the identity on $[0,1]$, such that $φ(0)=0$, $φ(1)=1$ and $φ(x)=φ(\frac{x}{2})+φ(\frac{x+1}{2})-φ(\frac{1}{2})$ 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 $μ$ concentrated on $(0,1)$ the formula $ϕ_μ(x)=\int_{(0,1)}R_p(x) dμ(p)$ defines a solution $ϕ_μ\colon[0,1]\to[0,1]$ 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 $μ$.