**Background**

Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by

$$ T(x,y) := \left( \frac{1}{1 + ax + by}, \frac{1}{1 + cx + dy} \right), \quad (x,y) \in [0,1] \times [0,1]. $$

I'm interested in this map (and other similar ones) in the context of monotone dynamical systems. They come up, for instance, as the *input to output characteristic* of controlled dynamical systems with outputs modeling simple biochemical networks.

**Question**

By Brower Fixed Point Theorem, $T^2:= T \circ T$ has a fixed point. Now **is this fixed point unique?** If it is not unique, is there a counterexample? Are there reasonable hypotheses on $a,b,c,d$ which would guarantee that to be true?

Of course I'm also interested in the general finite-dimensional case

$$ T(x_1, \ldots, x_n) := \left( \frac{1}{1 + a_{11}x_1 + \cdots + a_{1n}x_n}, \ldots, \frac{1}{1 + a_{n1}x_1 + \cdots + a_{nn}x_n} \right), $$ as well as replacing the $1$'s in the numerator and denominator by general constants. But I'm hoping to find something conceptual in the simplest case which could then be applied in the general scenario.

**Progress**

(1) Computer simulations with randomly generated coefficients $a,b,c,d$ seem to indicate that this is true in arbitrary finite dimensions, with any nonnegative $a,b,c,d$.

(2) If $a + c < 1$ and $b + d < 1$, then $T$ is a contraction (with respect to the *sum-norm* $|(x,y)| := |x| + |y|$. In particular, it has no period-2 points, and so $T^2$ has a unique equilibrium. But these hypotheses seem too restrictive. In fact, taking into consideration where $a,b,c,d$ come from, this is not always true.

(3) This is also true if $b = c$ and $a = d = 0$. This makes me wonder whether there could be some sort of diagonalisation argument. Though I have no idea what to do with the nonlinearity.

(4) I'm also attempting to approach this as a global optimization problem. More precisely, I'm looking at the map $$ (x,y) \longmapsto \|T^2(x,y) - (x,y)\|^2 $$ and tried to compute the Hessian using Maple but haven't gotten anywhere so far.