Currently, I am facing this problem:

Given two real functions $A( \vec x )$ and $B( \vec x ):\Bbb R^N\to \Bbb R$, I want to find a third real, monotonic function $f(x):\Bbb R\to\Bbb R$ such that:

$$A\big(f( \vec x )\big)=f\big(B( \vec x )\big)$$

where the simplified the notation writing $f( \vec x )$ means $$ f(x_1,x_2,\ldots,x_n) = \big(f(x_1), f(x_2),\ldots, f(x_n)\big). $$

I am interested in either having a formula/method for finding $f$, or even just having a proof that $f$ exists (or doesn't) under some specific conditions. Eventually, I am interested also in the solution in the case $N=1$.

Also, does this type of problem have a specific name?

Thank you very much!