How to find function G(x), and make for every x, G(x) always returns fixed point for another function F(G(x))

Question

I have a problem such that $F(G(x))=G(x)$, where $x$ is real numbers and I know that function $F$ can be computed if value $G(x)$ is given. $F$ is a probability function.

I want to obtain a function form $G$ (or an algorithm G).

What is the problem? Is there any knowledge about "finding a function that gives fixed points of another given function"?