Common Fixed-Point Theorems in the Partial  b -Metric Spaces and an Application to the System of Boundary Value Problems

Muhammad Nazam,Zahida Hamid,Aftab Hussain,Hamed Al Sulami,Rich Avery

doi:10.1155/2021/7777754

Abstract

In this paper, we investigate the conditions for the existence of the common fixed points of generalized contractions in the partial b -metric spaces endowed with an arbitrary binary relation. We establish some unique common fixed-point theorems. The obtained results may generalize and improve earlier fixed-point results. We provide examples to illustrate our findings. As an application, we discuss the common solution to the system of boundary value problems.

Highlights

We investigate the conditions for the existence of the common fixed points of generalized contractions in the partial b-metric spaces endowed with an arbitrary binary relation
The b-metric space was introduced by Czerwik [1]. It is obtained by modifying the triangle property of the metric space
Shukla [13] introduced the concept of partial b-metric by modifying the triangle property of the partial metric and investigated fixed points of Banach contraction and Kannan contraction in the partial b-metric spaces

Summary

Basic Notions in the Partial b-Metric Spaces

Let X be a nonempty set, and the mapping P : X × X ⟶ ⟶1⁄20,∞Þ satisfies the following axioms:. According to Matthews [11], if the mapping P satisfies axioms (1-4), we say that it is a partial metric on the set X and ðX, PÞ is called partial metric space. According to Shukla [13], if P satisfies axioms (1, 2, 3, and 5), it is a partial b -metric on the set X and ðX, PbÞ is called partial b -metric space. A sequence fxngn∈N in the partial b-metric space ðX, Pb, sÞ is called a convergent sequence if there exists x ∈ X such that limPb ðxn, xÞ = Pbðx, xÞ: ð10Þ n⟶∞. A sequence fxngn∈N in a partial b -metric space ðX, Pb, sÞ is called the Cauchy sequence if limPbðxn,m,n⟶∞xmÞ = Pbðx, xÞ: ð11Þ. Partial b-metric space (1) Every Cauchy sequence in the b -metric space is Cauchy in the partial b-metric space and vice versa (2) The partial b-metric space is complete if and only if b -metric space (induced b-metric space) is complete (3)

Common Fixed-Point Theorems in the Partial b-Metric Spaces

Application to the System of Boundary Value Problems

Þ are defined by ð1