How to solve for the nonlinear functional equation? [closed]

Question

I got a nonlinear functional equation like:

$f(x) = g(x) + h(f(Ax))$,

where $A$ is a constant, $x$ is a scalar, $g()$ and $h()$ are given. The objective is to solve for the expression of $f(x)$.

Note that $g(x)$ has a very simple form: $g(x) = x^2$, but $h(x)$ is very complicated and highly non-linear.

I have checked some methods related to the functional analysis (iterative solution), it is hard to obtain any viable solution in the iterative process $(f_0, f_1, ...)$. Specifically in the second step of the iterative process, due to the non-linear nature of $h()$, there is no closed form for $f_1$, not even for $f_2$, $f_3$...

I guess maybe there exist some iterative method like: for example I wanna calculate $f(x=x_0)$, and we construct an iterative method (this is for obtaining $f(x)$ for all $x$, just for obtaining $f(x_0)$) very much alike newton method, so that we can obtain $f(x_0)$.

Any method or textbook solution suggested, thanks!

Answer 1

If $A$ were equal to 1, and supposing f has an inverse function $f^{[-1]}(x)$, you could compose the left and right hand side with it so to get

$$f^{[-1]}(x)^2=x-h(x)$$

$$f^{[-1]}(x)=\pm\sqrt{x-h(x)}$$

$$f(x)=\left(\pm\sqrt{x-h(x)}\right)^{[-1]}$$

There are multiple numerical methods for finding an inverse function, so given $h(x)$ you can easily find $f(x)$ numerically.