Contractive mappings and existence of cycle times for a monotone and homogeneous function

Given a monotone and homogeneous self-mapping ff of the nn-dimensional positive cone, a family of contractive mappings is used to define an equivalence relation in the index set, as well as a total order among the equivalence classes. Then, it is shown (i) that the cycle times are well-defined at each index belonging to the maximal and minimal classes, and (ii) that the cycle times of ff exist at every index whenever a weak convexity condition is satisfied.