On a Problem of Janusz Matkowski and Jacek Weso{\l}owski, II

TL;DR

This paper investigates the solutions of a specific functional equation involving increasing contractions on [0,1], extending previous work by analyzing continuous solutions with boundary conditions under certain strict inequalities.

Contribution

It extends prior research by characterizing continuous increasing solutions of the functional equation with strict contraction conditions and boundary constraints.

Findings

Identifies conditions for existence of solutions

Provides explicit descriptions of solutions under strict inequalities

Extends previous results to more general contraction mappings

Abstract

We continue our study started in "On a problem of Janusz Matkowski and Jacek Weso{\l}owski" (see arXiv:1703.08459) of the functional equation \begin{equation*} \varphi(x)=\sum_{n=0}^{N}\varphi(f_n(x))-\sum_{n=0}^{N}\varphi(f_n(0)) \end{equation*} and its increasing and continuous solutions $\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$. In this paper we assume that $f_0,\ldots,f_N\colon[0,1]\to[0,1]$ are strictly increasing contractions such that \begin{equation*} 0\leq f_0(0)<f_0(1)\leq f_1(0)<\cdots <f_{N-1}(1)\leq f_N(0)<f_N(1)\leq 1 \end{equation*} and at least one of the weak inequalities is strong.