 Research
 Open Access
A generalized metric space and related fixed point theorems
 Mohamed Jleli^{1} and
 Bessem Samet^{1}Email author
https://doi.org/10.1186/s1366301503127
© Jleli and Samet; licensee Springer. 2015
 Received: 9 December 2014
 Accepted: 20 April 2015
 Published: 29 April 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.
Keywords
 generalized metric
 bmetric
 dislocated metric
 modular space
 fixed point
 partial order
MSC
 54H25
 47H10
1 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.
2 A generalized metric space
2.1 General definition
Definition 2.1
 (\(\mathcal{D}_{1}\)):

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.
2.2 Topological concepts
Definition 2.3
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
Definition 2.5
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.
2.3 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.
2.3.1 Standard metric spaces
 (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$$d(x,y)\leq d(x,z)+d(z,y). $$
2.3.2 bMetric spaces
In 1993, Czerwik [1] introduced the concept of bmetric spaces by relaxing the triangle inequality as follows.
Definition 2.7
 (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
2.3.3 HitzlerSeda metric spaces
Hitzler and Seda [8] introduced the notion of dislocated metric spaces as follows.
Definition 2.9
 (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.
2.3.4 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
 (\(\rho_{1}\)):

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 havewhenever \(\alpha,\beta\geq0\) and \(\alpha+\beta=1\).$$\rho(\alpha x+ \beta y)\leq\rho(x)+\rho(y), $$
Definition 2.12
The concept of convergence in such spaces is defined as follows.
Definition 2.13
 (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
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
The following result is immediate.
Proposition 2.16
 (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.
3 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
First, we have the following observation.
Proposition 3.2
Proof
We have the following extension of the Banach contraction principle.
Theorem 3.3
 (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 \(\delta(\mathcal {D},f,x_{0})<\infty\).
Proof
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.
 (iii)′:

there exists \(x_{0}\in X\) such that \(\sup\{\mathcal{D}(x_{0},f^{r}(x_{0})): r\in\mathbb{N}\}<\infty\).
The following result (see Kirk and Shahzad [6]) is an immediate consequence of Proposition 2.8 and Theorem 3.3.
Corollary 3.4
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
The following result is an immediate consequence of Proposition 2.15, Proposition 2.16, and Theorem 3.3.
Corollary 3.6
Observe that in Corollary 3.6, no \(\Delta_{2}\)condition is supposed.
4 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
Proposition 4.2
Proof
We have the following result.
Theorem 4.3
 (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 \(\delta(\mathcal {D},f,x_{0})<\infty\).
Proof
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.
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.
5 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.
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
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
Our first result holds under the weak continuity assumption.
Theorem 5.5
 (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.
Proof
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.
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
 (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.
Proof
Similar to the proof in the previous theorem, we have \(\mathcal {D}(\omega,\omega)=0\). □
Declarations
Acknowledgements
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).
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
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