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 ∑ n =0 φ ( f n ( x )) − N ∑ n =1 φ ( f n (0)) , where N ∈ N and f 0 ,...,f N :[0 , 1] → [0 , 1] are strictly increasing contractions satisfying the following condition 0 = f 0 (0) <f 0 (1) = f 1 (0) < ··· <f N − 1 (1) = f N (0) <f N (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