Convergence of Cauchy Sequences and Unique fixed points in MR-Metric Spaces

This paper investigates various properties of self-maps on metric spaces, particularly in the context of M R − m e t r i c spaces. We provide a range of examples to demonstrate the behavior of maps within such spaces, with a focus on those defined by specific criteria involving the metric M and a parameter R . The findings show that under certain conditions, the iterated distances of these self-maps converge, leading to the identification of unique fixed points. Additionally, we prove that sequences of iterated maps converge and form M − C a u c h y sequences, given the imposed conditions. The paper concludes with a theorem establishing the existence of a unique common fixed point for two partially commuting maps under appropriate assumptions. This research enhances the understanding of fixed point theory in metric spaces and offers potential applications in diverse areas of mathematics and computational science.