Synchronous r-cyclic contractions on metric spaces

Abstract

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 a Banach type generalized contraction condition. The obtained results may have interesting practical applications in various research areas.