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Edit 1/25/23: convergence of iterates in a neighborhood of the fixed point

In the most recent version of the resulting paper, we show in Theorem 3.3 that iterates of this mapping converge in some (quantifiable) neighborhood of the fixed point. Global convergence is still neither known nor disproved.

This fixed-point formulation came from an optimization problem, to which we can show existence of a unique solution. Lemma 3.4 here provides the explicit mapping, and Corollary 3.2 shows that $\phi$ does indeed have a unique fixed point.

However, we have still not shown that iterating $\phi$ converges to this fixed point. An explicit formula for the fixed point of $\phi$ in terms of $A$ would be ideal, but may not be possible. There is a little discussion in the 'future work' section of some partial progress on fixed-points of $\phi$ that might be interesting. my co-authors and I would definitely still be interested in any fruitful directions.

This fixed-point formulation came from an optimization problem, to which we can show existence of a unique solution. Lemma 3.4 here provides the explicit mapping, and Corollary 3.2 shows that $\phi$ does indeed have a unique fixed point.

However, we have still not shown that iterating $\phi$ converges to this fixed point. An explicit formula for the fixed point of $\phi$ in terms of $A$ would be ideal, but may not be possible. There is a little discussion in the 'future work' section of some partial progress on fixed-points of $\phi$ that might be interesting. my co-authors and I would definitely still be interested in any fruitful directions.

Edit: partial answer

This fixed-point formulation came from an optimization problem, to which we can show existence of a unique solution. Lemma 3.4 here provides the explicit mapping, and Corollary 3.2 shows that $\phi$ does indeed have a unique fixed point.

However, we have still not shown that iterating $\phi$ converges to this fixed point. An explicit formula for the fixed point of $\phi$ in terms of $A$ would be ideal, but may not be possible. There is a little discussion in the 'future work' section of some partial progress on fixed-points of $\phi$ that might be interesting. my co-authors and I would definitely still be interested in any fruitful directions.