Periodic Trajectories and Coincidence Points of Tuples of Set-Valued Maps

A fixed-point theorem is proved for a finite composition of set-valued Lipschitz maps such that the product of their Lipschitz constants is less than 1. The notion of a Lipschitz tuple of (finitely many) set-valued maps is introduced; it is proved that such a tuple has a periodic trajectory, which determines a fixed point of the given composition of set-valued Lipschitz maps. This result is applied to study the coincidence points of a pair of tuples (Lipschitz and covering).