Cyclical Mappings

In this chapter, we study cyclical mappings acting in a complete metric space. We study the asymptotic behavior of (inexact) iterates of cyclical mapping of contractive type. Using the Baire generic approach, it is shown that a typical cyclical mapping is of contractive type. More precisely, we consider a complete metric space of cyclical nonexpansive mappings acting on a union of two sets in a complete hyperbolic space. If these sets have a nonempty intersection, then using Baire category approach, we show that for most mappings, their iterates converge to a unique fixed point belonging to the intersection of the sets. If these two sets have an empty intersection, then using Baire category approach, we show for most mappings the distance between iterates converge to the distance between these two sets.