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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.

  1. The Question
  2. Why this Question?
  3. The Trivial Case $n = 1$
  4. Where Does This Question Come From?
  5. Satisfying the Hypotheses of the Question
  6. An Analogous Question
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What do you mean by "order-reversing" if $n>1$? –  fedja Dec 13 '10 at 2:47
    
My guess is the product order, although it might be lexicographic order. –  Ricky Demer Dec 13 '10 at 3:25
    
Thanks to fedja for the question and to Ricky Demer for the answer. The order is indeed the product order. I added the precision in the question. –  Gilles Gnacadja Dec 13 '10 at 16:58
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