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 Open Access
Common fixed point results in function weighted metric spaces
 Obaid Alqahtani^{1},
 Erdal Karapınar^{2}Email authorView ORCID ID profile and
 Priya Shahi^{3}
https://doi.org/10.1186/s1366001921236
© The Author(s) 2019
 Received: 15 April 2019
 Accepted: 31 May 2019
 Published: 7 June 2019
Abstract
In this paper, we consider a common fixed point result in the context of a very recently defined abstract space: “function weighted metric space”. We present also some examples to illustrate the validity of the given results.
Keywords
 Function weighted metric space
 Topological properties
 Banach contraction principle
MSC
 54E50
 54A20
 47H10
1 Introduction
One of the very natural trends of mathematical research is to refine the framework of the known theorems and results. For instance, Banach observed the first metric fixed point results in the setting of complete normed spaces. Immediate extension of this theorem was given by Caccioppoli who observed the characterization of Banach fixed point theorem in the context of complete metric spaces. After then, for the various abstract spaces, several analogs of the Banach contraction principle have been reported. Among them we can underline some of the interesting abstract structures such as modular metric space, symmetric space, semimetric space, quasimetric space, partial metric space, bmetric space, dislocated (metriclike) space, fuzzy metric space, probabilistic metric space, 2metric space, δmetric space, Gmetric space, Smetric space, function weighted metric space, and so on.
In this paper, we shall restrict ourselves to the recently introduced generalization of a metric space, namely, function weighted metric space [1]. Our aim is to obtain a fixed point result for two mappings. More precisely, we shall consider coincidence points and common fixed points for certain operators in the setting of the function weighted metric space.
The letter \(\mathfrak{F}\) denotes the set of all functions that are nondecreasing (in symbols, (\(\Delta _{1}\))) and logarithmiclike (in symbols, (\(\Delta _{2}\))).
By using the auxiliary functions of \(\mathfrak{F}\), Jleli–Samet [1] introduced a new metric space, more precisely, a function weighted metric space. Indeed, in this new metric space definition, Jleli–Samet [1] proposed a new condition instead of triangle inequality by using a function from the set \(\mathfrak{F}\). Henceforth, we presume that X is a nonempty set and avoid to repeat this in all statements. For the sake of the selfcontained text, we put the definition here:
Definition 1.1
 (\(\Delta _{1}\)):

(Selfdistance axiom) \(\delta (x,y)=0\Longleftrightarrow x=y\), for \(x,y \in X\);
 (\(\Delta _{2}\)):

(Symmetry axiom) For all \(x,y\in X\), we have \(\delta (x,y)=\delta (y,x)\);
 (\(\Delta _{3}\)):

(Generalized function fweighted triangle inequality axiom) For any pair \((x,y)\in X\times X\) and for any \(N\in \mathbb{N}\) with \(N\geq 2\), we havefor every \((u_{i})_{i=1}^{N}\subset X\) with \((u_{1},u_{N})=(x,y)\).$$ \delta (x,y)>0\quad \implies\quad f\bigl(\delta (x,y)\bigr)\leq f \Biggl(\sum _{i=1}^{N1} \delta (u_{i},u_{i+1}) \Biggr)+\mathcal{C}, $$
Throughout the text, we prefer to use the name “function weighted metric space” instead of “\(\mathcal{F}\)metric space”.
The main goal of this paper is to obtain some common fixed point result in the context of function weighted metric spaces.
2 Main results
In this section, we establish a common fixed point theorem in the setting of function weighted metric spaces.
Theorem 2.1
Proof
Example 2.1
Next, we present the notion of generalized θ–ψ contractive pair of mappings in the setting of function weighted metric spaces as follows:
Definition 2.1
Theorem 2.2
 (i)
T is θadmissible with respect to g;
 (ii)
There exists \(x_{0} \in X\) such that \(\theta (gx_{0}, Tx_{0}) \geq 1\);
 (iii)
If \(\{gx_{n}\}\) is a sequence in X such that \(\theta (gx_{n}, gx_{n+1}) \geq 1\) for all n and \(gx_{n} \rightarrow gz \in g(X)\) as \(n \rightarrow \infty \), then there exists a subsequence \(\{gx_{n(k)} \}\) of \(\{gx_{n}\}\) such that \(\theta (gx_{n(k)}, gz) \geq 1\) for all k.
Proof
We present the following example in support of our theorem:
Example 2.2
Theorem 2.3
If we assume that T or g is continuous, in addition to the axioms of in Theorem 2.2, then T and g possess a common point.
If we take \(\theta (gx, gy)=1\) in Theorem 2.2, then we find the following:
Theorem 2.4
If we take \(g(x)=x\) for all \(x \in X\) in Theorem 2.4, then we derive the following result.
Theorem 2.5
Declarations
Acknowledgements
The authors thanks anonymous referees for their remarkable comments, suggestion, and ideas that help to improve this paper. The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this group No. RG1440025.
Funding
Not applicable.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Jleli, M., Samet, B.: On a new generalization of metric spaces. J. Fixed Point Theory Appl. 20, 128 (2018). https://doi.org/10.1007/s1178401806066 MathSciNetView ArticleMATHGoogle Scholar