Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ subject to the conditions:

Question

Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ subject to the conditions:

$f(f(f(x)))+2x=f(3x)$ for all $x>0$ and $\displaystyle\lim_{x\to\infty}(f(x)-x)=0$

I tried as follows: Suppose $x_0=x,x_1=f(x_0)=f(x),...,x_n=f(x_{n-1})$. But I am facing problem due to $f(3x)$ term.

Answer 1

Note that $f(x)=x$ satisfies all three of your conditions, by inspection.