Some Globally Stable Fixed Points in b-Metric Spaces

Umar Batsari,Kanokwan Sitthithakerngkiet,Poom Kumam

doi:10.3390/sym10110555

Abstract

In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction mapping that generalizes that of Banach, Kanan, and Chatterjea. Using our new introduced contraction mapping, we establish some results on the existence and uniqueness of fixed points. In obtaining some of our results, we assume that the space is associated with a partial order, and the b-metric function has the regularity property. Our results improve, and generalize some current results in the literature.

Highlights

The research area of fixed point theory is playing an important role in finding solutions for some nonlinear equations
We investigated the global stability of the fixed points of an asymptotically contractive mapping
T : X → X in a complete b-metric space is guided upon the existence of some important two elements u, v ∈ X, satisfying the conditions provided in Theorem 1

Summary

Introduction

The research area of fixed point theory is playing an important role in finding solutions for some nonlinear equations (differential equations). The concept was improved by many authors [20], others generalized the concept [21,22] and established some fixed point existence results in b-metric spaces. In 2013, Kamihigashi and Stachurski proved some existence and uniqueness theorems of a fixed point in a complete metric space [8]. In this paper, motivated by Kamihigashi et al [8], Du et al [24], Zhou et al [17], and Yusuf et al [9], we establish the existence of fixed points in a complete b-metric space associated with a partial order. We investigated the global stability of the fixed points of an asymptotically contractive mapping

Preliminaries

Main Results

Conclusions