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

We determine all continuous solutions g:I→Ig:I→I of the polynomial-like iterative equation g3(x)=3g(x)−2xg3(x)=3g(x)−2x, where I⊂RI⊂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,+∞)→(0,+∞)f:(0,+∞)→(0,+∞) satisfying f3(x)=[f(x)]3x2.