Synchronous and asynchronous cyclic contractions on metric spaces

TL;DR

This paper introduces $r$-cyclic operators in metric spaces, analyzing their fixed point convergence under generalized contractions, with potential applications across multiple research fields.

Contribution

It presents the concept of $r$-cyclic operators and studies their fixed point properties under generalized contraction conditions, expanding cyclic operator theory.

Findings

Convergence of Picard iteration for $r$-cyclic operators established.

Generalized contraction conditions ensure fixed point existence.

Potential applications in diverse research areas identified.

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 different types of generalized contraction conditions. The obtained results may have interesting practical applications in various research areas.