Upper and lower bounds for the iterates of order-preserving homogeneous maps on cones

We define upper bound and lower bounds for order-preserving homogeneous of degree one maps on a proper closed cone in Rn in terms of the cone spectral radius. We also define weak upper bounds for these maps. For a proper closed cone C⊂Rn, we prove that any order-preserving homogeneous of degree one map f:intC→intC has a lower bound. If C is polyhedral, we prove that the map f has a weak upper bound. We give examples of weak upper bounds for certain order-preserving homogeneous of degree one maps defined on the interior of .