Possibly, this is a bad (stupid) question, but sometimes some discussion helps.
I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed point on some parameter $x$. Denote $z(x)$ a fixed point for the parameter $x$ and let the function be $y(z(x))$.
For sure for hyperbolic tangent there exists a region where there are three fixed points with corresponding values of $y$ e.g. $y_1, y_2, y_3$. If I just draw this dependency of $y$ on $x$ (i.e. $y(z(x))$, I will get something that looks similar to what is depicted on the picture.
Now I would love to know the value of parameter $x$ where $y$ is maximized. How do I deal with this case when there is multiplicity of fixed points (i.e. three branches of the function)? What would be your approach to tackle this problem?
Thank you in advance.
I need to recapitulate to understand this:
You have a family of functions $y_x(z)$ for $x \in X$, for example $$ y_x(z) = \tanh(x z - 5) $$
Here is a plot, note that the $x$ from the label is the variable $z$ from above and thus $f(x,7)$ rather means $y_7(z)$ etc.
The family of functions $y_x$ has an associated family of sets of fixed points $$ \mbox{F}_x = \{ z^* \,\left\vert\, y_x(z^*) = z^* \right. \} $$
In the plot $F_7$ features no fixed point in the plotted range, $F_8$ contains $\{ 0.7.., 1 \}$ and $F_{20}$ contains $\{ 0.2.., 1 \}$.
For each set $F_x$ we can define a function, which picks the maximum fixed point or returns a special value $\bot$ if $F_x$ is empty (I drop the case where $F_x$ is not empty and where no maximum element exists) $$ \mu(x) = \left \{ \begin{array}{cr} \max F_x & \mbox { if } F_x \ne \emptyset \\ \bot & \mbox { if } F_x = \emptyset \end{array} \right. $$
Here we would have $\mu(7) \approx -1$, $\mu(8) = \mu(20) = 1$.
You are interested in those $x \in X$ which result in the largest fixed point $z^*$ selected over all sets $F_x$.
It seems you want to calculate $$ \operatorname*{arg\,max}_{x \in X} \mu(x) \quad (*) $$
In our example with $X = \{ 7, 8, 20 \}$ it would be $$ \operatorname*{arg\,max}_{x \in X} \mu(x) = \{ 8, 20 \} $$
So much to have a framework for talking.
You probably want to have a method which calculates $(*)$ for a given family of function $y_x$. This seams hard because even the mapping $x \mapsto F_x$ is usually hard to determine for individual $x$ for general $y_x$. The resulting subject $\mu$ of argument maximization is thus hard too.
So if you can not exploit special properties of $y_x$ I seems like one ends up with some kind of search over $X$.