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We consider maps defined on the interior of a normal, closed cone in a realBanach space that are nonexpansive with respect to Thompson's metric. With mildcompactness assumptions, we prove that the Krasnoselskii iterates of such mapsconverge to a fixed point when one exists. For maps that are alsoorder-preserving, we give simple necessary and sufficient conditions in termsof upper and lower Collatz-Wielandt numbers for the fixed point set to benonempty 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 uniquefixed point and for all iterates of the map to converge to the fixed point. Wedemonstrate how these results apply to certain nonlinear matrix equations onthe cone of positive definite Hermitian matrices.

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