**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.



Uniqueness of fixed points for rational transformations
I deleted the other answer in favor of a more elementary approach outlined below. Some details are missing, but they can be filled in easily.


$\newcommand{\R}{\mathbb{R}} \newcommand{\nlsum}{\sum\nolimits}$


We'll show that under fairly mild assumptions, your map has a unique fixed point towards which it converges.


We start by introducing the hyperbolic metric
\begin{equation\*}
 \delta : \R\_{++}^n \times \R\_{++}^n \to \R\_+\quad (x,y) \mapsto \|\log x - \log y\|\_{\infty},
\end{equation\*}
where $\log x$ denotes the vector $(\log x\_1,\ldots,\log x\_n)$.


A few key lemmas about this metric are listed below.


Symmetry under inversion.



> 
> **(Lemma)** Let $x, y > 0$, then $\delta(x^{-1},y^{-1}) = \delta(x,y)$, (the inverses are taken elementwise).
> 
> 
> 


Scale invariance.



> 
> **(Lemma)** Let $x, y \in \R\_{++}^n$, and $\alpha > 0$ be a scalar. Then, $\delta(\alpha x, \alpha y) = \delta(x,y)$.
> 
> 
> 


Negative curvature properties.



> 
> **(Lemma)** Let $x, y, w \in \R\_{++}^n$. Then,
>  \begin{equation\*}
>  \delta\left(\nlsum\_i w\_i x\_i, \nlsum\_i w\_iy\_i \right) \le \max\_{1 \le i \le n} \delta(x\_i,y\_i).
> \end{equation\*}
> 
> 
> 


Contraction.



> 
> **(Lemma)** Let $x, y > 0$ and $z \ge 0$. Then,
>  \begin{equation\*}
>  \delta(z+x,z+y) \le \gamma(x,y,z)\delta(x,y),
> \end{equation\*}
>  where $\gamma(x,y,z) \le 1$ (with strict inequality if $z > 0$).
> 
> 
> 


Now let $T : \R\_{++}^{n} \to \R\_{++}^{n}$ be the map of interest, so that in particular for a fixed $n \times n$ coefficient matrix $[a\_{ij}]$, we have
\begin{equation\*}
 Tx : x \mapsto \left(\frac{1}{1+a\_{i1}x\_{1}+a\_{i2}x\_{2}+\cdots + a\_{in}x\_n}\right)\_{i=1}^n.
\end{equation\*}



> 
> **(Theorem)** The map $T$ is a strict contraction in the metric $\delta$.
> 
> 
> 


*Proof:* The lemmas above are used in the various steps below.


\begin{eqnarray}
 \delta(Tu,Tv) &=& \delta((Tu)^{-1},(Tv)^{-1})\\\\
 &=& \left\|\log \left(1+\nlsum\_j a\_{ij}u\_j\right)-\log\left(1+\nlsum\_ja\_{ij}v\_{j}\right)\right\|\_{\infty}\\\\
 &\le& \max\_{1\le i \le n}\gamma\_i\left|\log (\nlsum\_j a\_{ij}u\_j)-\log(\nlsum\_j a\_{ij}v\_{j})\right|\\\\
 &\le& \max\_{1 \le i \le n}\gamma\_i\max\_{1\le j\le n}|\log u\_j-\log v\_j|\\\\
 &=& \max\_{1\le i \le n}\gamma\_i \delta(u,v)\\\\
 &\le& \gamma\delta(u,v),
\end{eqnarray}


where $\gamma = \max\_{1\le i \le n}\gamma\_i < 1$. Since our domain is compact, this suffices to ensure a unique fixed point via the Banach fixed-point theorem.


