On a problem of Janusz Matkowski and Jacek Wesołowski, II.

We continue our study started in Morawiec and Zürcher (Aequ Math 92(4):601-615, 2018) of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=0Nφ(fn(0))and its increasing and continuous solutions φ:[0,1]→[0,1] such that φ(0)=0 and φ(1)=1. In this paper we assume that f0,...,fN:[0,1]→[0,1] are strictly increasing contractions such that 0≤f0(0)<f0(1)≤f1(0)<⋯<fN-1(1)≤fN(0)<fN(1)≤1and at least one of the weak inequalities is strong. [ABSTRACT FROM AUTHOR]