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Common fixed point results in function weighted metric spaces
Journal of Inequalities and Applicationsvolume 2019, Article number: 164 (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.
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.
For the sake of the selfcontained text, we recall the definition of the newly introduced metric space. For this purpose, we first recall two basic notions for functions that we need: A function \(f:(0,+\infty ) \to \mathbb{R}\) is called logarithmiclike if each sequence \(\{t_{n}\}\subset (0,+\infty )\) satisfies
A function \(f:(0,+\infty )\to \mathbb{R}\) is called nondecreasing if
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
Let \(\delta : X\times X\to [0,+\infty )\) be a given mapping. Suppose that there exist an \(f \in \mathfrak{F}\) and a constant \(\mathcal{C} \in [0,+\infty )\) such that
 (\(\Delta _{1}\)):

(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 have
$$ \delta (x,y)>0\quad \implies\quad f\bigl(\delta (x,y)\bigr)\leq f \Biggl(\sum _{i=1}^{N1} \delta (u_{i},u_{i+1}) \Biggr)+\mathcal{C}, $$for every \((u_{i})_{i=1}^{N}\subset X\) with \((u_{1},u_{N})=(x,y)\).
Throughout the text, we prefer to use the name “function weighted metric space” instead of “\(\mathcal{F}\)metric space”.
As it seen clearly, the only difference between a “standard metric space” and a “function weighted metric space” is the last axiom: In a “function weighted metric space” instead of “the triangle inequality”, another axiom has been used, namely “generalized fweighted triangle inequality axiom.” Based on this observation, we also easily conclude that any metric on X is an \(\mathcal{F}\)metric on X by letting \(f(t)= \ln t\) for the axiom (\(\Delta _{3}\)). Indeed, on account of the triangle inequality, for all distinct \(x,y \in X\) and for each \(N\in \mathbb{N}\) with \(N\geq 2\), and for any \((u_{i})_{i=1}^{N}\subset X\) with \((u_{1},u_{N})=(x,y)\), we find
since \(d(x,y)\leq \sum_{i=1}^{N1} d(u_{i},u_{i+1})\), and \(f(t)= \ln t\) is nondecreasing. Here, we take \(\mathcal{C}=0\).
The main goal of this paper is to obtain some common fixed point result in the context of function weighted metric spaces.
Main results
In this section, we establish a common fixed point theorem in the setting of function weighted metric spaces.
Theorem 2.1
Let \(T, g: X\to X\) be selfmappings on a function weighted complete metric space \((X,\delta )\) such that \(T(X) \subseteq g(X)\). Suppose that there exists \(k\in (0,1)\) such that
Also suppose \(g(X)\) is closed. Then, T and g have a unique coincidence point.
Proof
Since \(T(X) \subseteq g(X)\), we can choose a point \(x_{1} \in X\) such that \(Tx_{0} = gx_{1}\). We shall construct a sequence \(x_{n}\) in X such that
First, observe that T and g possess a unique coincidence point. Indeed, suppose on the contrary that \((u, v) \in X \times X\) are two distinct coincidence points of T and g. Thus, \(\delta (u, v) > 0\), \(g(u) = T(u)\) and \(g(v) = T(v)\). Then from (ii), we have
a contradiction.
Suppose \((f, \mathcal{C}) \in \mathfrak{F} \times [0, +\infty )\) so that (\(\Delta _{3}\)) is fulfilled. For a given \(\epsilon > 0\) and on account of \((\Delta _{2})\), there exists \(\gamma > 0\) such that
Consider the sequence \(\{y_{n}\} \subset X\) defined in (2.1). Now, without loss of generality, we assume that \(\delta (Tx_{0}, Tx_{1}) > 0\). Otherwise, \(x_{1}\) will be a coincidence point of T and g. By the contraction condition,
thereby implying that
So we have
Since
there exists some \(N \in N\) such that
Hence, by (2.2) and \((\Delta _{1})\), we have
Employing (\(\Delta _{3}\)) together with (2.4), we find
which implies by \((\Delta _{1})\) that
This proves that \(\{Tx_{n}\}\) is Cauchy. Since \(\{Tx_{n}\} = \{gx_{n+1} \} \subseteq g(X)\) and \(g(X)\) is closed, there exists \(z \in X\) such that
As a next step, we shall indicate that z is a coincidence point of T and g. On the contrary, assume that \(\delta (Tz, gz) > 0\). We have
As \(n \rightarrow \infty \) in the inequality above, and due to (2.5), we get
which is a contradiction. Therefore, we conclude that \(\delta (Tz, gz) = 0\), and hence z is a unique coincidence point of T and g. □
Example 2.1
Consider \(X = \mathbb{R}_{0}^{+}\). Define \(\delta : X \times X \rightarrow [0, \infty )\) as
So δ is a function weighted metric with \(f(t) = \ln (t)\) and \(a = 0\). Consider,
and
Clearly, \(x = 4\) is a coincidence point of T and g. We shall show that there exists a \(k \in (0, 1)\) such that
Therefore, all the hypothesis of Theorem 2.1 are satisfied. Furthermore, \(x = 4\) is a unique coincidence point of T and g.
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
Let T, g be selfmappings on a function weighted metric space \((X, \delta )\). We say that the pair \((T, g)\) is a generalized θ–ψ contractive pair of mappings if there exist \(\theta : X \times X \rightarrow [0, +\infty )\) and \(\psi \in \varPsi \) such that
for all \(x, y \in X\), where
Theorem 2.2
Let \((X, \delta )\) be a complete function weighted metric space and \(T, g : X \rightarrow X\) be such that \(T(X) \subseteq g(X)\) where \(g(X)\) is closed. Assume that the pair \((T, g)\) is a generalized θ–ψ contractive pair of mappings and the following conditions hold:

(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.
Then, T and g have a coincidence point.
Proof
In view of condition (ii), let \(x_{0} \in X\) be such that \(\theta (gx _{0}, Tx_{0}) \geq 1\). On account of \(T(X) \subseteq g(X)\), we choose a point \(x_{1} \in X\) in a way that \(Tx_{0} = gx_{1}\). Iteratively, we build a sequence \(\{x_{n}\}\) in X so that
Since T is θadmissible with respect to g, we have
Using mathematical induction, we get
If \(Tx_{n+1} = Tx_{n}\) for some n, then by (2.7),
and then \(x = x_{n+1}\) forms a coincidence point of T and g, which completes the proof. Accordingly, we suppose that \(\delta (Tx_{n}, Tx _{n+1}) > 0\) for all n. Applying inequality (2.6) and using (2.8), we obtain
On the other hand, we have
Due to the monotonicity of function ψ and using inequalities (2.9), we have for all \(n \geq 1\)
If, the inequality \(\delta (Tx_{n1}, Tx_{n}) \leq \delta (Tx_{n}, Tx _{n+1})\) is satisfied for some \(n \in \mathbb{N}\), from (2.10), we obtain that
a contradiction. Thus, for all \(n \geq 1\), we have
Notice that in view of (2.10) and (2.11), we get for all \(n \geq 1\) that
Continuing this process inductively, we obtain
So,
Since
there exists some \(N \in N\) such that
Next, let \((f, \mathcal{C}) \in \mathfrak{F} \times [0, +\infty )\) be such that (\(\Delta _{3}\)) is satisfied. Let \(\epsilon > 0\) be fixed. By \((\Delta _{2})\), \(\exists \gamma > 0\) such that
Hence, by (2.15) and \((\Delta _{1})\), we have
Using (\(\Delta _{3}\)) and (2.16), we have
which implies by \((\Delta _{1})\) that
It proves that \(\{y_{n}\}=\{Tx_{n}\}\) is Cauchy sequence. Due to the completeness of the considered space, there is \(y \in X\) such that
Since \(\{Tx_{n}\} = \{gx_{n+1}\} \subseteq g(X)\) and \(g(X)\) is closed, there exists \(z \in X\) such that \(gz=y\) and
Next, we show that z is a coincidence point of T and g. On the contrary, assume that \(\delta (Tz, gz) > 0\). We have from (iii) that
where
Keeping (2.17) in mind and letting \(n \rightarrow \infty \) in (2.18), we conclude that the righthand side tends to ∞. This is a contradiction, and hence z is coincidence point of point T and g. □
We present the following example in support of our theorem:
Example 2.2
Consider \(X = \mathbb{R}\). Define \(\delta : X \times X \rightarrow [0, \infty )\) as
So δ is a complete function weighted metric with \(f(t) =  \frac{1}{t}\) and \(a = 1\). Consider the following selfmappings:
and
Choose \(\psi (t) = \sqrt{t}\) for all \(t \geq 0\). So all the hypothesis of Theorem 2.2 are satisfied and \(x = \frac{3}{2}\) is a unique coincidence point of T and g.
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
Let \((X, \delta )\) be a complete function weighted metric space and \(T, g : X \rightarrow X\) be such that \(T(X) \subseteq g(X)\) where \(g(X)\) is closed. Assume that the pair \((T, g)\) satisfies
for all \(x,y \in X\) with \(\delta (Tx,Ty)>0\) where
Then, T and g possess a coincidence point.
If we take \(g(x)=x\) for all \(x \in X\) in Theorem 2.4, then we derive the following result.
Theorem 2.5
Let \((X, \delta )\) be a complete function weighted metric space and assume \(T: X \rightarrow X\) satisfies
for all \(x,y \in X\) with \(\delta (Tx,Ty)>0\) where
Then, T possesses a fixed point.
References
 1.
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
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.
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Correspondence to Erdal Karapınar.
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MSC
 54E50
 54A20
 47H10
Keywords
 Function weighted metric space
 Topological properties
 Banach contraction principle