Let $A$ be a real, invertible $n\times n$ matrix. I am interested in finding the vectors $\mathbf{x}\in\mathbb R^n$ that solve the following equation:
$$\mathbf x = A \tanh(\mathbf x)$$
where the $\tanh$ is applied element-wise. More generally, we can consider other kinds of non-linearities instead of the $\tanh$ (but always applied element-wise).
Is there a generic approach to studying the solutions of these type of equations? Probably exploiting the eigen decomposition of $A$?
I added the tag "reference-request" in case someone can suggest relevant references to the literature.
In the 2D case, the equation takes the form $$\begin{cases}x=a f(x)+bf(y),\\y=cf(x)+df(y)\end{cases}$$
and after elimintation of $y$, we get a univariate nonlinear equation $$\frac{x-af(x)}b=f\left(cf(x)+\frac db(x-af(x)\right).$$ We don't see any particular simplification nor connection with the Eigenvalues.
I have seen numerical cases with four distinct positive solutions.