One-parameter system of functional equations

Summary

In this paper we reduce a system of functional equations to the functional equation

$$f(x) = \frac{1}{2}\left\{ {f\left( {\frac{x}{{1 - a}}} \right) + f\left( {\frac{{x - a}}{{1 - a}}} \right)} \right\}$$

wherea is a parameter witha ∈ (0, 1) and the unknown functionf:R → R is a bounded function withf _|(−∞,0)=0 andf _|(1,+∞)=1.

We prove that, for anya ∈ (0, 1), this functional equation admits exactly one solution. Moreover this solution is continuous and nondecreasing.