Common Fixed Point Theorems for Two Mappings in Complete b-Metric Spaces

Lili Chen,Yanfeng Zhao,Xin Liu,Xin Xia

doi:10.3390/fractalfract6020103

Lili Chen, Yanfeng Zhao + Show 2 more

Open Access

https://doi.org/10.3390/fractalfract6020103

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Abstract

Our paper is devoted to the issue of the existence and uniqueness of common fixed points for two mappings in complete b-metric spaces by virtue of the new functions F and θ, respectively. Moreover, two specific examples to indicate the validity of our results are also given. Eventually, the generalized forms of Jungck fixed point theorem in the above spaces is investigated. Different from related literature, the conditions that the function F needs to satisfy are weakened, and F only needs to be non-decreasing in this paper. To some extent, our conclusions and methods improve the results of previous literature.

Highlights

The conceptual framework of b-metric spaces, as a meaningful generalization of metric spaces, was first formally proposed by Czerwik [1] who discussed the convergence of measurable functions and established the Banach contraction principle in b-metric spaces
Samet [2] fully certified that the class of (α, ψ)-type contractions contains a good deal of contraction-type operators, and the fixed points of the operators can be obtained in virtue of the Picard iteration
In [5], an interesting generalization of the Banach contraction principle was shown by introducing the notion of F-contractions, which as a new type of contraction, have been applied to obtain fixed point results for single-valued mappings and multi-valued mappings in b-metric spaces

Summary

Introduction

The conceptual framework of b-metric spaces, as a meaningful generalization of metric spaces, was first formally proposed by Czerwik [1] who discussed the convergence of measurable functions and established the Banach contraction principle in b-metric spaces. The Banach contraction principle plays an important role in b-metric spaces, and it is one of the most valid tools in the research fields of nonlinear analysis and its applications It is extensively regarded as the beginnings of metric fixed point theory. In [5], an interesting generalization of the Banach contraction principle was shown by introducing the notion of F-contractions, which as a new type of contraction, have been applied to obtain fixed point results for single-valued mappings and multi-valued mappings in b-metric spaces. Motivated by the above-mentioned discussions, we mainly study the existence and uniqueness of common fixed points for two mappings in complete b-metric spaces by virtue of the new functions F and θ, respectively. To some extent, our conclusions and methods improve the results of previous literature

Preliminaries

Existence and Uniqueness of Common Fixed Points for Two Mappings

A Generalized Form of Jungck Fixed Point Theorem

Conclusions