On a Zolt\'an Boros' problem connected with polynomial-like iterative equations

TL;DR

This paper characterizes all continuous solutions to a specific polynomial-like iterative functional equation, addressing a problem posed by Zoltán Boros, and provides a complete classification of solutions on real intervals.

Contribution

It fully determines continuous solutions to the polynomial-like iterative equation, solving an open problem posed by Boros in 2012.

Findings

All continuous solutions are explicitly characterized.

The solutions include specific polynomial and functional forms.

The results answer Boros's problem on a related iterative equation.

Abstract

We determine all continuous solutions $g\colon I\to I$ of the polynomial-like iterative equation $g^3(x)=3g(x)-2x$, where $I\subset\mathbb R$ is an interval. In particular, we obtain an answer to a problem posed by Zolt\'{a}n Boros (during the Fiftieth International Symposium on Functional Equations, 2012) of determining all continuous functions $f\colon (0,+\infty)\to (0,+\infty)$ satisfying $f^3(x)=\frac{[f(x)]^3}{x^2}$.