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 equationwhere 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 [9] by Janusz Matkowski concerning a very special case of that equation.Mathematics Subject Classification (2010). Primary 39B12; Secondary 26A30, 26A46, 28A80.

On a problem of Janusz Matkowski and Jacek Wesołowski, II

“…, f −1 N (A) have Lebesgue measure zero for every set A ⊂ [0, 1] of Lebesgue measure zero 1 ), then the class C is determined by two of its subclasses C a and C s of all absolutely continuous and all singular functions, respectively. Repeating directly the proof of Remark 2.2 from [11] with the use of Lemma 2.2 we get the following result. It is clear that 0 is the unique fixed point of f 0 and 1 is the unique fixed point of f N , i.e.…”

“…Repeating the proof of Theorem 3.3 from [11] we get the following result. We finish this section by giving a precise formula for ϕ.…”

“…. , N − 1} there exists c n1,...,n k ∈ [0, 1] such that (11) ϕ| [fn 1 ,...,n k (1),fn 1 ,...,n k +1(0)] = c n1,...,n k .…”

“…We continue our study of the existence of solutions ϕ of the functional equation In this paper we assume, in contrast to [11], that…”

Attractor of Cantor Type with Positive Measure

“…[9,7,1]). Studying a functional equation connected with the problem posed in [6,8], we come to the following question. Is the attractor of this IFS necessary of Lebesgue measure zero?…”