Dynamics of Cyclic Contractions

TL;DR

This paper extends fixed point theorems for various cyclic contractions in b-metric spaces, analyzing their dynamics, fixed and periodic points, and convergence of iterations, broadening the understanding of these mappings.

Contribution

It introduces new fixed point results for cyclic contractions in b-metric spaces, including local versions and a wider range of ratio values, with detailed analysis of dynamics and convergence.

Findings

Expanded fixed and periodic point results for cyclic contractions.

Established convergence of Picard iterations to fixed points.

Analyzed dynamics of cyclic contractions in b-metric spaces.

Abstract

Cyclic contractions generalize the usual contractivities in metric spaces and $b$-MSs. In this paper, we enhance several fixed point theorems related to cyclic (i) Banach self-maps, (ii) Chatterjea contractivities, (iii) Kannan self-mappings, (iv) \'{C}iri\'c and Hardy-Rogers, and (v) Reich contractions including local versions in $b$-metric spaces while also delineating the associated dynamics. Especially noteworthy is the expansion of the results concerning both fixed and periodic points, which are substantiated across a wider spectrum of ratio values for the aforementioned cyclic contractions within this class of spaces. Additionally, the convergence of Picard iterations towards the fixed point is rigorously established.