On a Zoltán Boros' problem connected with polynomial-like iterative equations

Abstract

We determine all continuous solutions $g:I\to I$ 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 Zoltán Boros (during the Fiftieth International Symposium on Functional Equations, 2012) of determining all continuous functions $f:(0,+\infty )\to (0,+\infty )$ satisfying $f^3(x)=\frac{[f(x){]}^3}{x^2}$.