On a problem of Janusz Matkowski and Jacek Wesołowski

Abstract

We study the problem of the existence of increasing and continuous solutions φ: [0 , 1] → [0 , 1] such that φ(0) = 0 and φ(1) = 1 of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=1Nφ(fn(0)),where N∈ N and f0, … , fN: [0 , 1] → [0 , 1] are strictly increasing contractions satisfying the following condition 0 = f0(0) < f0(1) = f1(0) < ⋯ < fN - 1(1) = fN(0) < fN(1) = 1. In particular, we give an answer to the problem posed in Matkowski (Aequationes Math. 29:210–213, 1985) by Janusz Matkowski concerning a very special case of that equation.