Partial extended b-metric and some fixed point theorem

TL;DR

This paper introduces partial extended b-metric spaces (PEBMS), a new generalized framework unifying and extending existing metric space concepts, and proves fixed point theorems within this setting.

Contribution

The paper defines PEBMS, establishes fundamental properties, and extends fixed point theorems, broadening the scope of generalized metric space theories.

Findings

PEBMS unifies extended and partial b-metric spaces.

Fixed point theorems are proved for contractive mappings in PEBMS.

Application to stability analysis of discrete dynamical systems.

Abstract

In this paper, we introduce the concept of partial extended b-metric spaces (PEBMS) as a unification and generalization of extended b-metric spaces and partial b-metric spaces. This new structure incorporates a point-dependent control function together with the possibility of non-zero self-distance, providing a more flexible framework for the study of generalized metric spaces. We establish several fundamental properties of PEBMS, including convergence, Cauchy sequences, and 0-completeness. By introducing the notion of 0-Cauchy sequences, we extend various results from extended b-metric spaces to the PEBMS setting. In particular, we prove fixed point theorems for contractive mappings and show the existence and uniqueness of fixed points under suitable conditions. Furthermore, we demonstrate that every extended b-metric space can be viewed as a special case of a PEBMS. As an application, we study the stability of discrete dynamical systems within this framework. The results presented here generalize and enrich existing theories in metric-type spaces and open new directions for further research.