Synchronous and asynchronous cyclic contractions on metric spaces

Motivated by the existence of cyclic phenomena in which some characteristics are mapped into corresponding ones over more than one phase, we introduce the $r$-cyclic operators with respect to a covering of a metric space and investigate their behavior. We study the convergence of the Picard iteration to a fixed point of such an operator under different types of generalized contraction conditions. The obtained results may have interesting practical applications in various research areas.