On a problem of Janusz Matkowski and Jacek Wesołowski. II (Q1731430)

The authors continue their previous research work [Aequationes Math. 92, No. 4, 601--615 (2018; Zbl 1404.39024)] about the existence of solutions $\varphi$ of the functional equation \[ \varphi (x)=\sum_{n=0}^{N}\varphi (f_{n}(x))-\sum_{n=0}^{N}\varphi (f_{n}(0)) \] in the class $\mathcal{C}$ consisting of all increasing and continuous functions $\varphi : [0, 1]\rightarrow [0, 1]$ such that $\varphi (0)=0$ and $\varphi (1)=1$. Here they assume that $f_{0}, \dots, f_{N} : [0, 1]\rightarrow [0, 1]$ are strictly increasing contractions such that $0 \leq f_{0}(0) < f_{0}(1)\leq f_{1}(0) < \dots < f_{N-1}(1) \leq f_{N}(0) < f_{N}(1)\leq 1$ and at least one of the weak inequalities is strong. To prove their results, they construct several lemmas. In the beginning, they give preliminaries and basic property of solutions. For this purpose, the authors use the following notations. The class $\mathcal{C}$ is determined by two of its subclasses $\mathcal{C}_{\alpha}$ and ${\mathcal{C}}_{s}$ of all absolutely continuous and all singular functions respectively. They suppose \[ \mathbf{0}=\lim_{k\rightarrow \infty} f_{\underbrace{0, \dots, 0}_k}(0)\text{ and }\mathbf{1}=\lim_{k\rightarrow \infty} f_{\underbrace{N, \dots, N}_k}(1). \] Then, $\varphi (\mathbf{0})=0$ and $\varphi (\mathbf{1})=1$ under the assumption $\varphi\in\mathcal{C}$. The authors define $A_{*}=\bigcap_{k\in N} A_{k}$, then the set $A_{*}$ is exactly the set of points in $[0, 1]$ that have an address. The authors prove that if the set $A_{*}$ has Lebesgue measure zero, then ${\mathcal{C}}={\mathcal{C}}_{s}$. They also provide an example to verify this result. Further, they prove that the set $A_{*}$ is perfect. For the existence of solutions of the functional equation, the authors show that $\mathcal{C}\neq\emptyset$. To achieve this result, they prove that either $\varphi\in \mathcal{C}_{\alpha}$ or $\varphi\in {\mathcal{C}}_{s}$. Under certain assumptions, they give a precise formula for $\varphi$. The authors denote by $\mathcal{W}$ the set of all solutions. As a main result, using several lemmas, they prove that the set $\mathcal{W}$ is linearly independent and its convex hull is contained in $\mathcal{C}$. Finally, the authors give the application of this main result.