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A generalized metric space and related fixed point theorems
Fixed Point Theory and Applications volume 2015, Article number: 61 (2015)
Abstract
We introduce a new concept of generalized metric spaces for which we extend some wellknown fixed point results including Banach contraction principle, Ćirić’s fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and RodríguezLópez. This new concept of generalized metric spaces recover various topological spaces including standard metric spaces, bmetric spaces, dislocated metric spaces, and modular spaces.
Introduction
The concept of standard metric spaces is a fundamental tool in topology, functional analysis and nonlinear analysis. This structure has attracted a considerable attention from mathematicians because of the development of the fixed point theory in standard metric spaces.
In recent years, several generalizations of standard metric spaces have appeared. In 1993, Czerwik [1] introduced the concept of a bmetric space. Since then, several works have dealt with fixed point theory in such spaces; see [2–7] and references therein. In 2000, Hitzler and Seda [8] introduced the notion of dislocated metric spaces in which self distance of a point need not be equal to zero. Such spaces play a very important role in topology and logical programming. For fixed point theory in dislocated metric spaces, see [9–12] and references therein. The theory of modular spaces was initiated by Nakano [13] in connection with the theory of order spaces and was redefined and generalized by Musielak and Orlicz [14]. By defining a norm, particular Banach spaces of functions can be considered. Metric fixed theory for these Banach spaces of functions has been widely studied. Even though a metric is not defined, many problems in fixed point theory can be reformulated in modular spaces (see [15–20] and references therein).
In this work, we present a new generalization of metric spaces that recovers a large class of topological spaces including standard metric spaces, bmetric spaces, dislocated metric spaces, and modular spaces. In such spaces, we establish new versions of some known fixed point theorems in standard metric spaces including Banach contraction principle, Ćirić’s fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and RodíguezLópez.
A generalized metric space
Let X be a nonempty set and \(\mathcal{D}: X\times X\to[0,+\infty]\) be a given mapping. For every \(x\in X\), let us define the set
General definition
Definition 2.1
We say that \(\mathcal{D}\) is a generalized metric on X if it satisfies the following conditions:
 (\(\mathcal{D}_{1}\)):

for every \((x,y)\in X\times X\), we have
$$\mathcal{D}(x,y)=0\quad \Longrightarrow \quad x=y; $$  (\(\mathcal{D}_{2}\)):

for every \((x,y)\in X\times X\), we have
$$\mathcal{D}(x,y)=\mathcal{D}(y,x); $$  (\(\mathcal{D}_{3}\)):

there exists \(C>0\) such that
$$\mbox{if }(x,y)\in X\times X, \{x_{n}\}\in C(\mathcal{D},X,x), \mbox{then } \mathcal{D}(x,y)\leq C \limsup_{n\to\infty} \mathcal{D}(x_{n},y). $$
Remark 2.2
Obviously, if the set \(C(\mathcal{D},X,x)\) is empty for every \(x\in X\), then \((X,\mathcal{D})\) is a generalized metric space if and only if (\(\mathcal{D}_{1}\)) and (\(\mathcal{D}_{2}\)) are satisfied.
Topological concepts
Definition 2.3
Let \((X,\mathcal{D})\) be a generalized metric space. Let \(\{x_{n}\}\) be a sequence in X and \(x\in X\). We say that \(\{x_{n}\}\) \(\mathcal{D}\)converges to x if
Proposition 2.4
Let \((X,\mathcal{D})\) be a generalized metric space. Let \(\{x_{n}\}\) be a sequence in X and \((x,y)\in X\times X\). If \(\{x_{n}\}\) \(\mathcal{D}\)converges to x and \(\{x_{n}\}\) \(\mathcal{D}\)converges to y, then \(x=y\).
Proof
Using the property (\(\mathcal{D}_{3}\)), we have
which implies from the property (\(\mathcal{D}_{1}\)) that \(x=y\). □
Definition 2.5
Let \((X,\mathcal{D})\) be a generalized metric space. Let \(\{x_{n}\}\) be a sequence in X. We say that \(\{x_{n}\}\) is a \(\mathcal{D}\)Cauchy sequence if
Definition 2.6
Let \((X,\mathcal{D})\) be a generalized metric space. It is said to be \(\mathcal{D}\)complete if every Cauchy sequence in X is convergent to some element in X.
Examples
In this part of the paper, we will see that this new concept of generalized metric spaces recovers a large class of existing metrics in the literature.
Standard metric spaces
Recall that a standard metric on a nonempty set X is a mapping \(d: X\times X\to[0,+\infty)\) satisfying the following conditions:
 (d_{1}):

for every \((x,y)\in X\times X\), we have
$$d(x,y)=0 \quad \Longleftrightarrow\quad x=y; $$  (d_{2}):

for every \((x,y)\in X\times X\), we have
$$d(x,y)=d(y,x); $$  (d_{3}):

for every \((x,y,z)\in X\times X\times X\), we have
$$d(x,y)\leq d(x,z)+d(z,y). $$
bMetric spaces
In 1993, Czerwik [1] introduced the concept of bmetric spaces by relaxing the triangle inequality as follows.
Definition 2.7
Let X be a nonempty set and \(d: X\times X\to[0,+\infty)\) be a given mapping. We say that d is a bmetric on X if it satisfies the following conditions:
 (b_{1}):

for every \((x,y)\in X\times X\), we have
$$d(x,y)=0\quad \Longleftrightarrow\quad x=y; $$  (b_{2}):

for every \((x,y)\in X\times X\), we have
$$d(x,y)=d(y,x); $$  (b_{3}):

there exists \(s\geq1\) such that, for every \((x,y,z)\in X\times X\times X\), we have
$$d(x,y)\leq s\bigl[d(x,z)+d(z,y)\bigr]. $$
The concept of convergence in such spaces is similar to that of standard metric spaces.
Proposition 2.8
Any bmetric on X is a generalized metric on X.
Proof
Let d be a bmetric on X. We have just to proof that d satisfies the property (\(\mathcal {D}_{3}\)). Let \(x\in X\) and \(\{x_{n}\}\in C(d,X,x)\). For every \(y\in X\), by the property (b_{3}), we have
for every natural number n. Thus we have
The property (\(\mathcal{D}_{3}\)) is then satisfied with \(C=s\). □
HitzlerSeda metric spaces
Hitzler and Seda [8] introduced the notion of dislocated metric spaces as follows.
Definition 2.9
Let X be a nonempty set and \(d: X\times X\to[0,+\infty)\) be a given mapping. We say that d is a dislocated metric on X if it satisfies the following conditions:
 (HS_{1}):

for every \((x,y)\in X\times X\), we have
$$d(x,y)=0\quad \Longrightarrow\quad x=y; $$  (HS_{2}):

for every \((x,y)\in X\times X\), we have
$$d(x,y)=d(y,x); $$  (HS_{3}):

for every \((x,y,z)\in X\times X\times X\), we have
$$d(x,y)\leq d(x,z)+d(z,y). $$
The motivation of defining this new notion is to get better results in logic programming semantics.
The concept of convergence in such spaces is similar to that of standard metric spaces.
The following result can easily be established, so we omit its proof.
Proposition 2.10
Any dislocated metric on X is a generalized metric on X.
Modular spaces with the Fatou property
Let us recall briefly some basic concepts of modular spaces. For more details of modular spaces, the reader is advised to consult [19], and the references therein.
Definition 2.11
Let X be a linear space over \(\mathbb{R}\). A functional \(\rho: X\to [0,+\infty]\) is said to be modular if the following conditions hold:
 (\(\rho_{1}\)):

for every \(x\in X\), we have
$$\rho(x)=0\quad \Longleftrightarrow \quad x=0; $$  (\(\rho_{2}\)):

for every \(x\in X\), we have
$$\rho(x)=\rho(x); $$  (\(\rho_{3}\)):

for every \((x,y)\in X\times X\), we have
$$\rho(\alpha x+ \beta y)\leq\rho(x)+\rho(y), $$whenever \(\alpha,\beta\geq0\) and \(\alpha+\beta=1\).
Definition 2.12
If ρ is a modular on X, then the set
is called a modular space.
The concept of convergence in such spaces is defined as follows.
Definition 2.13
Let \((X,\rho)\) be a modular space.

(i)
A sequence \(\{x_{n}\}_{n\in\mathbb{N}}\subset X_{\rho}\) is said to be ρconvergent to \(x\in X_{\rho}\) if \(\lim_{n\to\infty}\rho(x_{n}x)=0\).

(ii)
A sequence \(\{x_{n}\}_{n\in\mathbb{N}}\subset X_{\rho}\) is said to be ρCauchy if \(\lim_{n,m\to\infty}\rho (x_{n}x_{n+m})=0\).

(iii)
\(X_{\rho}\) is said to be ρcomplete if any ρCauchy sequence is ρconvergent.
Definition 2.14
The modular ρ has the Fatou property if, for every \(y\in X_{\rho}\), we have
whenever \(\{x_{n}\}_{n\in\mathbb{N}}\subset X_{\rho}\) is ρconvergent to \(x\in X_{\rho}\).
Let \((X,\rho)\) be a modular space. Define the mapping \(D_{\rho}: X_{\rho}\times X_{\rho}\to[0,+\infty]\) by
for every \((x,y)\in X\times X\).
We have the following result.
Proposition 2.15
If ρ has the Fatou property, then \(D_{\rho}\) is a generalized metric on \(X_{\rho}\).
Proof
We have just to proof that \(D_{\rho}\) satisfies the property (\(\mathcal {D}_{3}\)). Let \(x\in X_{\rho}\) and \(\{x_{n}\}\in C(D_{\rho},X_{\rho},x)\), which means that
Using the Fatou property, for all \(y\in X_{\rho}\), we have
which yields
Then (\(\mathcal{D}_{3}\)) is satisfied with \(C=1\) and \(D_{\rho}\) is a generalized metric on \(X_{\rho}\). □
The following result is immediate.
Proposition 2.16
Let ρ be a modular on X having the Fatou property. Then

(i)
\(\{x_{n}\}\subset X_{\rho}\) is ρconvergent to \(x\in X_{\rho}\) if and only if \(\{x_{n}\}\) is \(D_{\rho}\)convergent to x;

(ii)
\(\{x_{n}\}\subset X_{\rho}\) is ρCauchy if and only if \(\{x_{n}\}\) is \(D_{\rho}\)Cauchy;

(iii)
\((X_{\rho},\rho)\) is ρcomplete if and only if \((X_{\rho},D_{\rho})\) is \(D_{\rho}\)complete.
The Banach contraction principle in a generalized metric space
In this section, we present an extension of the Banach contraction principle to the setting of generalized metric spaces introduced previously.
Let \((X,\mathcal{D})\) be a generalized metric space and \(f: X\to X\) be a mapping.
Definition 3.1
Let \(k\in(0,1)\). We say that f is a kcontraction if
First, we have the following observation.
Proposition 3.2
Suppose that f is a kcontraction for some \(k\in(0,1)\). Then any fixed point \(\omega\in X\) of f satisfies
Proof
Let \(\omega\in X\) be a fixed point of f such that \(\mathcal{D}(\omega,\omega)<\infty\). Since f is a kcontraction, we have
which implies that \(\mathcal{D}(\omega,\omega)=0\) since \(k\in(0,1)\) and \(\mathcal{D}(\omega,\omega)<\infty\). □
For every \(x\in X\), let
We have the following extension of the Banach contraction principle.
Theorem 3.3
Suppose that the following conditions hold:

(i)
\((X,\mathcal{D})\) is complete;

(ii)
f is a kcontraction for some \(k\in(0,1)\);

(iii)
there exists \(x_{0}\in X\) such that \(\delta(\mathcal {D},f,x_{0})<\infty\).
Then \(\{f^{n}(x_{0})\}\) converges to \(\omega\in X\), a fixed point of f. Moreover, if \(\omega'\in X\) is another fixed point of f such that \(\mathcal{D}(\omega,\omega')<\infty\), then \(\omega=\omega'\).
Proof
Let \(n\in\mathbb{N}\) (\(n\geq1\)). Since f is a kcontraction, for all \(i,j\in\mathbb{N}\), we have
which implies that
Then, for every \(n\in\mathbb{N}\), we have
Using the above inequality, for every \(n,m\in\mathbb{N}\), we have
Since \(\delta(\mathcal{D},f,x_{0})<\infty\) and \(k\in(0,1)\), we obtain
which implies that \(\{f^{n}(x_{0})\}\) is a \(\mathcal{D}\)Cauchy sequence.
Since \((X,\mathcal{D})\) is \(\mathcal{D}\)complete, there exists some \(\omega\in X\) such that \(\{f^{n}(x_{0})\}\) is \(\mathcal{D}\)convergent to ω.
On the other hand, since f is a kcontraction, for all \(n\in\mathbb {N}\), we have
Letting \(n\to\infty\) in the above inequality, we get
Then \(\{f^{n}(x_{0})\}\) is \(\mathcal{D}\)convergent to \(f(\omega)\). By the uniqueness of the limit (see Proposition 2.4), we get \(\omega =f(\omega)\), that is, ω is a fixed point of f.
Now, suppose that \(\omega'\in X\) is a fixed point of f such that \(\mathcal{D}(\omega,\omega')<\infty\). Since f is a kcontraction, we have
which implies by the property (\(\mathcal{D}_{1}\)) that \(\omega=\omega'\). □
Observe that we can replace condition (iii) in Theorem 3.3 by
 (iii)′:

there exists \(x_{0}\in X\) such that \(\sup\{\mathcal{D}(x_{0},f^{r}(x_{0})): r\in\mathbb{N}\}<\infty\).
So condition (iii)′ implies condition (iii).
The following result (see Kirk and Shahzad [6]) is an immediate consequence of Proposition 2.8 and Theorem 3.3.
Corollary 3.4
Let \((X,d)\) be a complete bmetric space and \(f: X\to X\) be a mapping. Suppose that for some \(k\in(0,1)\), we have
If there exists \(x_{0}\in X\) such that
then the sequence \(\{f^{n}(x_{0})\}\) converges to a fixed point of f. Moreover, f has one and only one fixed point.
Note that in [1], there is a better result than this given by Corollary 3.4.
The next result is an immediate consequence of Proposition 2.10 and Theorem 3.3.
Corollary 3.5
Let \((X,d)\) be a complete dislocated metric space and \(f: X\to X\) be a mapping. Suppose that for some \(k\in(0,1)\), we have
If there exists \(x_{0}\in X\) such that
then the sequence \(\{f^{n}(x_{0})\}\) converges to a fixed point of f. Moreover, f has one and only one fixed point.
The following result is an immediate consequence of Proposition 2.15, Proposition 2.16, and Theorem 3.3.
Corollary 3.6
Let \((X_{\rho},\rho)\) be a complete modular space and \(f: X\to X\) be a mapping. Suppose that for some \(k\in(0,1)\), we have
Suppose also that ρ satisfies the Fatou property. If there exists \(x_{0}\in X_{\rho}\) such that
then the sequence \(\{f^{n}(x_{0})\}\) ρconverges to some \(\omega\in X_{\rho}\), a fixed point of f. Moreover, if \(\omega'\in X_{\rho}\) is another fixed point of f such that \(\rho(\omega\omega')<\infty\), then \(\omega=\omega'\).
Observe that in Corollary 3.6, no \(\Delta_{2}\)condition is supposed.
Ćirić’s quasicontraction in a generalized metric space
In this section, we extend Ćirić’s fixed point theorem for quasicontraction type mappings [21] in the setting of generalized metric spaces.
Let \((X,\mathcal{D})\) be a generalized metric space and \(f: X\to X\) be a mapping.
Definition 4.1
Let \(k\in(0,1)\). We say that f is a kquasicontraction if
for every \((x,y)\in X\times X\).
Proposition 4.2
Suppose that f is a kquasicontraction for some \(k\in(0,1)\). Then any fixed point \(\omega\in X\) of f satisfies
Proof
Let \(\omega\in X\) be a fixed point of f such that \(\mathcal{D}(\omega,\omega)<\infty\). Since f is a kquasicontraction, we have
Since \(k\in(0,1)\), we get \(\mathcal{D}(\omega,\omega)=0\). □
We have the following result.
Theorem 4.3
Suppose that the following conditions hold:

(i)
\((X,\mathcal{D})\) is complete;

(ii)
f is a kquasicontraction for some \(k\in(0,1)\);

(iii)
there exists \(x_{0}\in X\) such that \(\delta(\mathcal {D},f,x_{0})<\infty\).
Then \(\{f^{n}(x_{0})\}\) converges to some \(\omega\in X\). If \(\mathcal {D}(x_{0},f(\omega))<\infty\) and \(\mathcal{D}(\omega,f(\omega))<\infty\), then ω is a fixed point of f. Moreover, if \(\omega'\in X\) is another fixed point of f such that \(\mathcal{D}(\omega,\omega')<\infty\) and \(\mathcal{D}(\omega',\omega')<\infty\), then \(\omega=\omega'\).
Proof
Let \(n\in\mathbb{N}\) (\(n\geq1\)). Since f is a kquasicontraction, for all \(i,j\in\mathbb{N}\), we have
which implies that
Hence, for any \(n\geq1\), we have
Using the above inequality, for every \(n,m\in\mathbb{N}\), we have
Since \(\delta(\mathcal{D},f,x_{0})<\infty\) and \(k\in(0,1)\), we obtain
which implies that \(\{f^{n}(x_{0})\}\) is a \(\mathcal{D}\)Cauchy sequence.
Since \((X,\mathcal{D})\) is \(\mathcal{D}\)complete, there exists some \(\omega\in X\) such that \(\{f^{n}(x_{0})\}\) is \(\mathcal{D}\)convergent to ω.
Now, we suppose that \(\mathcal{D}(x_{0},f(\omega))<\infty\). Using the inequality
for every \(n,m\in\mathbb{N}\), by the property (\(\mathcal{D}_{3}\)), there exists some constant \(C>0\) such that
for every \(n\in\mathbb{N}\).
On the other hand, we have
Again, using the above inequality, we have
Continuing this process, by induction we get
for every \(n\geq1\). Therefore, we have
since \(\mathcal{D}(x_{0},f(\omega))<\infty\) and \(\delta(\mathcal {D},f,x_{0})<\infty\). Using the property (\(\mathcal{D}_{3}\)), we get
which implies that \(\mathcal{D}(f(\omega),\omega)=0\) since \(\mathcal{D}(\omega,f(\omega))<\infty\) and \(k\in(0,1)\). Then ω is a fixed point of f. By Proposition 4.2, we have \(\mathcal{D}(\omega,\omega)=0\).
Finally, suppose that \(\omega'\in X\) is another fixed point of f such that \(\mathcal{D}(\omega,\omega')<\infty\) and \(\mathcal{D}(\omega ',\omega')<\infty\). By Proposition 4.2, we have \(\mathcal{D}(\omega ',\omega')=0\). Since f is a kquasicontraction, we get
which implies that \(\omega=\omega'\). □
As in the previous section, from Theorem 4.3, we can obtain fixed point results for Ćirić’s quasicontraction type mappings in various spaces including standard metric spaces, bmetric spaces, dislocated metric spaces, and modular spaces.
Banach contraction principle in a generalized metric space with a partial order
In this section, we extend the Banach contraction principle to the class of generalized metric spaces with a partial order.
Let \((X,\mathcal{D})\) be a generalized metric space and \(f: X\to X\) be a mapping. Let ⪯ be a partial order on X. We denote by \(E_{\preceq}\) the subset of \(X\times X\) defined by
Now, let us introduce some concepts.
Definition 5.1
We say that f is weak continuous if the following condition holds: if \(\{x_{n}\}\subset X\) is \(\mathcal{D}\)convergent to \(x\in X\), then there exists a subsequence \(\{x_{n_{q}}\}\) of \(\{x_{n}\}\) such that \(\{ f(x_{n_{q}})\}\) is \(\mathcal{D}\)convergent to \(f(x)\) (as \(q\to\infty\)).
Definition 5.2
We say that f is ⪯monotone if the following condition holds:
Definition 5.3
We say that the pair \((X,\preceq)\) is \(\mathcal{D}\)regular if the following condition holds: for every sequence \(\{x_{n}\}\subset X\) satisfying \((x_{n},x_{n+1})\in E_{\preceq}\), for every n large enough, if \(\{x_{n}\}\) is \(\mathcal{D}\)convergent to \(x\in X\), then there exists a subsequence \(\{x_{n_{q}}\}\) of \(\{x_{n}\}\) such that \((x_{n_{q}},x)\in E_{\preceq}\), for every q large enough.
Definition 5.4
We say that f is a weak kcontraction for some \(k\in(0,1)\) if the following condition holds:
Our first result holds under the weak continuity assumption.
Theorem 5.5
Suppose that the following conditions hold:

(i)
\((X,\mathcal{D})\) is complete;

(ii)
f is weak continuous;

(iii)
f is a weak kcontraction for some \(k\in(0,1)\);

(iv)
there exists \(x_{0}\in X\) such that \(\delta(\mathcal {D},f,x_{0})<\infty\) and \((x_{0},f(x_{0}))\in E_{\preceq}\);

(v)
f is ⪯monotone.
Then \(\{f^{n}(x_{0})\}\) converges to some \(\omega\in X\) such that ω is a fixed point of f. Moreover, if \(\mathcal{D}(\omega,\omega)<\infty \), then \(\mathcal{D}(\omega,\omega)=0\).
Proof
Since f is ⪯monotone and \((x_{0},f(x_{0}))\in E_{\preceq}\), then
Since ⪯ is a partial order (so it is transitive), then
Let \(n\in\mathbb{N}\) (\(n\geq1\)). Since f is a weak kcontraction and \(\mathcal{D}\) is symmetric, for all \(i,j\in\mathbb{N}\), we have
which implies that
Then, for every \(n\in\mathbb{N}\), we have
Using the above inequality, for every \(n,m\in\mathbb{N}\), we have
Since \(\delta(\mathcal{D},f,x_{0})<\infty\) and \(k\in(0,1)\), we obtain
which implies that \(\{f^{n}(x_{0})\}\) is a \(\mathcal{D}\)Cauchy sequence.
Since \((X,\mathcal{D})\) is \(\mathcal{D}\)complete, there exists some \(\omega\in X\) such that \(\{f^{n}(x_{0})\}\) is \(\mathcal{D}\)convergent to ω. Since f is weak continuous, there exists a subsequence \(\{ f^{n_{q}}(x_{0})\}\) of \(\{f^{n}(x_{0})\}\) such that \(\{f^{n_{q}+1}(x_{0})\}\) is \(\mathcal{D}\)convergent to \(f(\omega)\) (as \(q\to\infty\)). By the uniqueness of the limit, we get \(\omega=f(\omega)\), that is, ω is a fixed point of f.
Suppose now that \(\mathcal{D}(\omega,\omega)<\infty\), since \((\omega ,\omega)\in E_{\preceq}\), we have
which implies that \(\mathcal{D}(\omega,\omega)=0\) (since \(k\in(0,1)\)). □
Remark 5.6
Theorem 5.5 is an extension of Ran and Reurings fixed point result [22] established in the setting of metric spaces under the continuity of the mapping f.
Now, we replace the weak continuity assumption by the \(\mathcal {D}\)regularity of the pair \((X,\preceq)\). We have the following result.
Theorem 5.7
Suppose that the following conditions hold:

(i)
\((X,\mathcal{D})\) is complete;

(ii)
\((X,\preceq)\) is \(\mathcal{D}\)regular;

(iii)
f is a weak kcontraction for some \(k\in(0,1)\);

(iv)
there exists \(x_{0}\in X\) such that \(\delta(\mathcal {D},f,x_{0})<\infty\) and \((x_{0},f(x_{0}))\in E_{\preceq}\);

(v)
f is ⪯monotone.
Then \(\{f^{n}(x_{0})\}\) converges to some \(\omega\in X\) such that ω is a fixed point of f. Moreover, if \(\mathcal{D}(\omega,\omega)<\infty \), then \(\mathcal{D}(\omega,\omega)=0\).
Proof
Following the proof of the previous theorem, we know that \(\{f^{n}(x_{0})\}\) is \(\mathcal{D}\)convergent to some \(\omega\in X\) and
Since \((X,\preceq)\) is \(\mathcal{D}\)regular, there exists a subsequence \(\{f^{n_{q}}(x_{0})\}\) of \(\{f^{n}(x_{0})\}\) such that \((f^{n_{q}}(x_{0}),\omega)\in E_{\preceq}\), for every q large enough. On the other hand, f is a weak kcontraction, so we have
for every q large enough. Letting \(q\to\infty\) in the above inequality, we get
which implies that \(\{f^{n_{q}+1}(x_{0})\}\) is \(\mathcal{D}\)convergent to \(f(\omega)\). By uniqueness of the limit, we get \(f(\omega)=\omega\).
Similar to the proof in the previous theorem, we have \(\mathcal {D}(\omega,\omega)=0\). □
Remark 5.8
Theorem 5.7 is an extension of Nieto and RodíguezLópez fixed point result ([23], Theorem 4) established in the setting of metric spaces.
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This project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number (12MAT 289502).
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Jleli, M., Samet, B. A generalized metric space and related fixed point theorems. Fixed Point Theory Appl 2015, 61 (2015) doi:10.1186/s1366301503127
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MSC
 54H25
 47H10
Keywords
 generalized metric
 bmetric
 dislocated metric
 modular space
 fixed point
 partial order