Some New Results on Equivalent Cauchy Sequences and Their Applications to Meir-Keeler Contraction in Partial Rectangular Metric Space

Sidite Duraj,Eriola Sila,Elida Hoxha

doi:10.13189/ms.2021.090606

Abstract

The study of fixed points in the metric spaces plays a crucial role in the development of Functional Analysis. It is evolved by generalizing the metric space or improving the contractive conditions. Recently, the partial rectangular metric space and its topology have been the center of study for many researchers. They have defined open and closed balls the equivalent Cauchy sequences and Cauchy sequences, convergent sequences which are used as tools in many achieved results. In this paper, two facts for equivalent Cauchy sequences in a partial rectangular metric space are provided by using an ultra - altering distance function. Furthermore, some results of Cauchy sequences in a partial rectangular metric space are highlighted. There is proved that under some conditions the equivalent Cauchy sequences are Cauchy sequences in a partial rectangular metric space. Some fixed point results have been taken as applications of our new conditions of Cauchy sequences and equivalent Cauchy sequences in a partial rectangular metric space <img src=image/13424354_01.gif> for orbitally continuous functions <img src=image/13424354_02.gif>. To illustrate the obtained results some examples are given.

Highlights

IntroductionThe theory of the fixed point depends on either generalizing the metric type space or the contractive type mapping [1]
The theory of the fixed point depends on either generalizing the metric type space or the contractive type mapping [1].Firstly, the generalization of a metric space is based on reducing or modifying the metric axioms, for example, quasimetrics, b-metrics, partial metrics, rectangular metrics, etc., [2], [3], [4], [5].Secondly, many researchers have generalized Banach’s contraction principle in metric spaces and so in the generalized metric spaces [6], [7], [8].In [9] Matthew introduced a new abstract space called partial metric space
Two facts for equivalent Cauchy sequences in a partial rectangular metric space are provided by using an ultra - altering distance function

Summary

Introduction

The theory of the fixed point depends on either generalizing the metric type space or the contractive type mapping [1]. The partial metric space is a generalization of the usual metric spaces in which the distance of a point from itself may not be zero In these spaces, related to the theory of fixed point many authors have given their contribution such as Dosenovicand Radenovic [10], Radenovic [11], Kirk and Shahzad [1]. Another researcher Branciari [12] introduced a generalized metric, which is called a rectangular metric by replacing triangular inequality with similar one which involves four or more points instead of three points.

Preliminaries

Main Results

Fixed point theorem in a partial rectangular metric space

Conclusions