Weak Partial b-Metric Spaces and Nadler’s Theorem

Tanzeela Kanwal,Poom Kumam,Ekrem Savas,Azhar Hussain

doi:10.3390/math7040332

Abstract

The purpose of this paper is to define the notions of weak partial b-metric spaces and weak partial Hausdorff b-metric spaces along with the topology of weak partial b-metric space. Moreover, we present a generalization of Nadler’s theorem by using weak partial Hausdorff b-metric spaces in the context of a weak partial b-metric space. We present a non-trivial example which show the validity of our result and an application to nonlinear Volterra integral inclusion for the applicability purpose.

Highlights

The famous Banach contraction principle has been generalized in many directions, whether by generalizing the contractive condition or by extending the domain of the function
Czerwik [2] introduced b-metric spaces generalizing the ordinary metric space and considering the problem of convergence of measurable functions with respect to measure; Czerwik [2] proved the variant of Banach contraction in b-metric spaces
Matthews [14] established the notion of a partial metric space and proved an analogue of Banach’s principle in such spaces

Summary

Introduction

The famous Banach contraction principle has been generalized in many directions, whether by generalizing the contractive condition or by extending the domain of the function. The concept of partial Hausdorff metric was given by Aydi et al [6] and they established a fixed point theorem for multivalued mappings in partial metric spaces. Shukla [23] introduced the concept of the partial b-metric and proved some fixed point results. Beg [7] presented the idea of the almost partial Hausdorff metric and extended Nadler’s theorem (Nadler [19]) to weak partial metric spaces. The aim of this paper is to introduce the notion of the weak partial b-metric space, the H + -type partial Hausdorff b-metric and prove Nadler’s theorem to weak partial b-metric spaces.

Preliminaries

Weak Partial b-Metric Space

Fixed Point Result

Application

Conclusions