Bounded fixed point sets and Krasnoselskii iterates of Thompson metric nonexpansive maps

We consider maps defined on the interior of a normal, closed cone in a real Banach space that are nonexpansive with respect to Thompson’s metric. With mild compactness assumptions, we prove that the Krasnoselskii iterates of such maps converge to a fixed point when one exists. For maps that are also order-preserving, we give simple necessary and sufficient conditions in terms of upper and lower Collatz-Wielandt numbers for the fixed point set to be nonempty and bounded in Thompson’s metric. When the map is also real analytic, these conditions are both necessary and sufficient for the map to have a unique fixed point and for all iterates of the map to converge to the fixed point. We demonstrate how these results apply to certain nonlinear matrix equations on the cone of positive definite Hermitian matrices.