Is there a transformation $\mathcal{T}$ of maps $\mathbb{R}_{{\geq}0}^{n} \rightarrow \mathbb{R}_{{\geq}0}^{n}$ with the following property?

If a map $F : \mathbb{R}_{{\geq}0}^{n} \rightarrow \mathbb{R}_{{\geq}0}^{n}$ is smooth and order-reversing with respect to the product order and possesses a unique fixed point $\omega$, then

- The point $\omega$ is the unique fixed point of ${\mathcal{T}}\!F$; and
- For some (known) point $a \in \mathbb{R}_{{\geq}0}^{n}$, the sequence of iterates of ${\mathcal{T}}\!F$ starting at $a$ converges to the (unknown) point $\omega$.

For more on this question, please refer to the 3-page PDF document at this address: http://math.gillesgnacadja.info/files/FixedPointAlgo_OPEN.html. I would have liked to post everything here but I could not find a way to save and preview the question before posting it. The content of the document is as follows.

- The Question
- Why this Question?
- The Trivial Case $n = 1$
- Where Does This Question Come From?
- Satisfying the Hypotheses of the Question
- An Analogous Question