Inspired by functional equations, dynamicals systems and sieve theory I came to the following questions.
Let $x>0$
I was thinking about these individual equations
$$f( g(x) f(x) ) = g(x)$$
$$h(\frac{g(x)}{h(x)}) = g(x)$$
$$k(\frac{k(x)}{g(x)}) = g(x)$$
How to solve such equations ?
I assume solving them is done similar to one another.
To be clear, I am interested in the general case ;
For instance :
$$f( g(x) f(x) ) = g(x)$$
I wonder what if $g(x)$ is given and we want to find some $g(x)$ if it exists ? Or what if $f(x)$ is given and we want to find some $g(x)$ if it exists ?
Or what if we look for such pairs.
edit
I am aware that
$$f( g(x) f(x) ) = g(x)$$
is solved by $f(x) =g(x) = \sqrt x$.
Also $f,g,h $ being constant or moebius transformations is not really what I am looking for.