Analytical solution of a non-linear equation with a 'min' function

Question

I am building a mathematical model of a non-linear dynamical system and I have an expression of this form:

$$x=\min\left(\frac{y}{a+y},\frac{y}{c+y(d+ex)}\right)$$

or let's consider any form like:

$$x=\min\big(f(x),g(x)\big)$$

How to solve this equation, considering that the individual equations $x=f(x)$ and $x=g(x)$ can be solved?

Should I solve $x=f(x)$ and $x=g(x)$ individually and choose the solution with minimum value?

Answer 1

Obtain all solutions $F_i$ of $x=f(x)$ and $G_j$ of $x=g(x)$.

Then the solutions of $x=\min(f(x),g(x))$ are all the $F_i$ such that $F_i\le g(F_i)$ and all the $G_j$ such that $G_j\le f(G_j)$, if any.