Numerical solution of integral equation involving unknown random variable transformation.

Question

Let Y be a random variable normally distributed, and let $X=g(Y)$ be a transformation of Y such that g solves a certain integral equation

$$I[g,F_X]=0$$

where $F_X$ is the distribution function of X.

Solve for g.

The only algorithm I naively came out is:

1. guess an initial probability distribution for X
2. solve I[.,.] for g, given the guessed $F_X$
3. estimate the new $F_X$ given g
4. goto step 2 or exit

Is there a more sophisticated strategy I can adopt for solving this equation? Any reference to articles or books can be helpful.