Abstract
We discuss the introduced concept of Gmetric spaces and the fixed point existing results of contractive mappings defined on such spaces. In particular, we show that the most obtained fixed point theorems on such spaces can be deduced immediately from fixed point theorems on metric or quasimetric spaces.
MSC:47H10, 11J83.
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1 Introduction
In 2005, Mustafa and Sims introduced a new class of generalized metric spaces (see [1, 2]), which are called Gmetric spaces, as generalization of a metric space (X,d). Subsequently, many fixed point results on such spaces appeared (see, for example, [3–7]).
Here, we present the necessary definitions and results in Gmetric spaces, which will be useful for the rest of the paper. However, for more details, we refer to [1, 2].
Definition 1.1 Let X be a nonempty set. Suppose that G:X\times X\times X\to [0,+\mathrm{\infty}) is a function satisfying the following conditions:

(1)
G(x,y,z)=0 if and only if x=y=z;

(2)
0<G(x,x,y) for all x,y\in X with x\ne y;

(3)
G(x,x,y)\le G(x,y,z) for all x,y,z\in X with y\ne z;

(4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots (symmetry in all three variables);

(5)
G(x,y,z)\le G(x,a,a)+G(a,y,z) for all x,y,z,a\in X.
Then G is called a Gmetric on X and (X,G) is called a Gmetric space.
Definition 1.2 A Gmetric space (X,G) is said to be symmetric if G(x,y,y)=G(y,x,x) for all x,y\in X.
Definition 1.3 Let (X,G) be a Gmetric space. We say that \{{x}_{n}\} is

(1)
a GCauchy sequence if, for any \epsilon >0, there is N\in \mathbb{N} (the set of all positive integers) such that for all n,m,l\ge N, G({x}_{n},{x}_{m},{x}_{l})<\epsilon;

(2)
a Gconvergent sequence to x\in X if, for any \epsilon >0, there is N\in \mathbb{N} such that for all n,m\ge N, G(x,{x}_{n},{x}_{m})<\epsilon.
A Gmetric space (X,G) is said to be complete if every GCauchy sequence in X is Gconvergent in X.
Proposition 1.1 Let (X,G) be a Gmetric space. The following are equivalent:

(1)
\{{x}_{n}\} is Gconvergent to x;

(2)
G({x}_{n},{x}_{n},x)\to 0 as n\to +\mathrm{\infty};

(3)
G({x}_{n},x,x)\to 0 as n\to +\mathrm{\infty};

(4)
G({x}_{n},{x}_{m},x)\to 0 as n,m\to +\mathrm{\infty}.
Proposition 1.2 Let (X,G) be a Gmetric space. Then the following are equivalent:

(1)
the sequence \{{x}_{n}\} is GCauchy;

(2)
G({x}_{n},{x}_{m},{x}_{m})\to 0 as n,m\to +\mathrm{\infty}.
An interesting observation is that any Gmetric space (X,G) induces a metric {d}_{G} on X given by
Moreover, (X,G) is Gcomplete if and only if (X,{d}_{G}) is complete.
It was observed that in the symmetric case ((X,G) is symmetric), many fixed point theorems on Gmetric spaces are particular cases of existing fixed point theorems in metric spaces. In this paper, we discuss the nonsymmetric case. We will show that such spaces have a quasimetric type structure and then many results on such spaces can be derived from fixed point theorems on quasimetric spaces.
2 Basic definitions and results
As we mentioned earlier, Gmetric spaces have a quasimetric type structure. Indeed, we have the following result.
Theorem 2.1 Let (X,G) be a Gmetric space. The function d:X\times X\to [0,\mathrm{\infty}) defined by d(x,y)=G(x,y,y) satisfies the following properties:

(1)
d(x,y)=0 if and only if x=y;

(2)
d(x,y)\le d(x,z)+d(z,y) for any points x,y,z\in X.
Proof The proof of (1) follows immediately from the property (1) in Definition 1.1. Now, let x, y, z be any points in X. Using the property (5) in Definition 1.1, we have
Thus, (2) holds. □
The above result suggests the following definition.
Definition 2.1 Let X be a nonempty set and d:X\times X\to [0,\mathrm{\infty}) be a given function which satisfies

(1)
d(x,y)=0 if and only if x=y;

(2)
d(x,y)\le d(x,z)+d(z,y) for any points x,y,z\in X.
Then d is called a quasimetric and the pair (X,d) is called a quasimetric space.
Note that any metric space is a quasimetric space, but the converse is not true in general.
Now, we define convergence and completeness on quasimetric spaces.
Definition 2.2 Let (X,d) be a quasimetric space, \{{x}_{n}\} be a sequence in X, and x\in X. The sequence \{{x}_{n}\} converges to x if and only if
Definition 2.3 Let (X,d) be a quasimetric space and \{{x}_{n}\} be a sequence in X. We say that \{{x}_{n}\} is leftCauchy if and only if for every \epsilon >0 there exists a positive integer N=N(\epsilon ) such that d({x}_{n},{x}_{m})<\epsilon for all n\ge m>N.
Definition 2.4 Let (X,d) be a quasimetric space and \{{x}_{n}\} be a sequence in X. We say that \{{x}_{n}\} is rightCauchy if and only if for every \epsilon >0 there exists a positive integer N=N(\epsilon ) such that d({x}_{n},{x}_{m})<\epsilon for all m\ge n>N.
Definition 2.5 Let (X,d) be a quasimetric space and \{{x}_{n}\} be a sequence in X. We say that \{{x}_{n}\} is Cauchy if and only if for every \epsilon >0 there exists a positive integer N=N(\epsilon ) such that d({x}_{n},{x}_{m})<\epsilon for all m,n>N.
Obviously, a sequence \{{x}_{n}\} in a quasimetric space is Cauchy if and only if it is leftCauchy and rightCauchy.
Definition 2.6 Let (X,d) be a quasimetric space. We say that

(1)
(X,d) is leftcomplete if and only if each leftCauchy sequence in X is convergent;

(2)
(X,d) is rightcomplete if and only if each rightCauchy sequence in X is convergent;

(3)
(X,d) is complete if and only if each Cauchy sequence in X is convergent.
The following result is an immediate consequence of the above definitions and results.
Theorem 2.2 Let (X,G) be a Gmetric space. Let d:X\times X\to [0,\mathrm{\infty}) be the function defined by d(x,y)=G(x,y,y). Then

(1)
(X,d) is a quasimetric space;

(2)
\{{x}_{n}\}\subset X is Gconvergent to x\in X if and only if \{{x}_{n}\} is convergent to x in (X,d);

(3)
\{{x}_{n}\}\subset X is GCauchy if and only if \{{x}_{n}\} is Cauchy in (X,d);

(4)
(X,G) is Gcomplete if and only if (X,d) is complete.
Every quasimetric induces a metric, that is, if (X,d) is a quasimetric space, then the function \delta :X\times X\to [0,\mathrm{\infty}) defined by
is a metric on X.
The following result is an immediate consequence of the above definitions and results.
Theorem 2.3 Let (X,G) be a Gmetric space. Let \delta :X\times X\to [0,\mathrm{\infty}) be the function defined by \delta (x,y)=max\{G(x,y,y),G(y,x,x)\}. Then

(1)
(X,\delta ) is a metric space;

(2)
\{{x}_{n}\}\subset X is Gconvergent to x\in X if and only if \{{x}_{n}\} is convergent to x in (X,\delta );

(3)
\{{x}_{n}\}\subset X is GCauchy if and only if \{{x}_{n}\} is Cauchy in (X,\delta );

(4)
(X,G) is Gcomplete if and only if (X,\delta ) is complete.
3 Discussion on fixed point results on Gmetric spaces
3.1 From metric to Gmetric: the linear case
In this section, we show that in the case of linear contractive conditions, the existing fixed point results on Gmetric spaces are immediate consequences of existing fixed point theorems on metric spaces.
As a model example, we consider the following result of Mustafa et al. [8].
Theorem 3.1 (Mustafa et al. [8])
Let (X,G) be a complete Gmetric space, and let T:X\to X be a mapping that satisfies the following condition: for all x,y,z\in X,
where a, b, c, d are positive constants such that k=a+b+c+d<1. Then T has a unique fixed point.
Now, we will show that the above result is an immediate consequence of Ćirić’s fixed point theorem [9]. Indeed, taking z=y in (3.1), we have
for all x,y\in X. Also, from (3.1), we have
for all x,y\in X. Define the metric space \delta :X\times X\to [0,\mathrm{\infty}) by
It follows from (3.2) and (3.3) that
Now, T satisfies Ćirić’s contractive condition [9] in the complete metric space (X,\delta ) (see Theorem 2.3), then T has a unique fixed point.
For more details about the linear case, we refer the reader to [10].
3.2 From quasimetric to Gmetric: the nonlinear case
In some cases, when the contractive condition is of nonlinear type, the above strategy cannot be used. However, we will show that we can deduce fixed point results on Gmetric spaces from fixed point results on quasimetric spaces. As a model example, we consider a weakly contractive condition. At first, we need the following fixed point theorem on quasimetric spaces.
Theorem 3.2 Let (X,d) be a complete quasimetric space and T:X\to X be a mapping satisfying
for all x,y\in X, where \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is continuous with {\phi}^{1}(\{0\})=\{0\}. Then T has a unique fixed point.
Proof Let {x}_{0} be any point in X and \{{x}_{n}\} be the sequence defined by {x}_{n+1}=T{x}_{n} for all n\ge 0. From (3.4), we have
This implies that \{d({x}_{n},{x}_{n+1})\} is a decreasing sequence of positive numbers. Then there exists r\ge 0 such that d({x}_{n},{x}_{n+1})\to r as n\to \mathrm{\infty}. Letting n\to \mathrm{\infty} in (3.5), we get that \phi (r)=0, that is, r=0. Thus, we have
Using the same technique, we also have
Now, we shall prove that \{{x}_{n}\} is a Cauchy sequence in the quasimetric space (X,d), that is, \{{x}_{n}\} is leftCauchy and rightCauchy. Suppose that \{{x}_{n}\} is not a leftCauchy sequence. Then there exists \epsilon >0 for which we can find subsequences \{{x}_{n(k)}\} and \{{x}_{m(k)}\} of \{{x}_{n}\} with n(k)>m(k)>k such that
for all k. Further, corresponding to m(k), we can choose n(k) such that it is the smallest integer with n(k)>m(k) satisfying the above inequality. Then
for all k. On the other hand, we have
Letting k\to \mathrm{\infty} and using (3.7), we get
We have
and
Letting k\to \mathrm{\infty} in the above inequalities, using (3.6), (3.7), and (3.8), we get
Now, from (3.4), for all k, we have
Letting k\to \mathrm{\infty} in the above inequality, using (3.8) and (3.9), we obtain
which implies that \epsilon =0: a contradiction with \epsilon >0. Then we proved that \{{x}_{n}\} is a leftCauchy sequence. Similarly, we can show that \{{x}_{n}\} is a rightCauchy sequence. Then \{{x}_{n}\} is a Cauchy sequence in the complete quasimetric space (X,d). This implies that there exists a\in X such that
Now, we have
Letting n\to \mathrm{\infty} and using (3.10), we have
Similarly, we have
Letting n\to \mathrm{\infty} and using (3.10), we have
Then, we have
It follows from (3.10) and (3.11) that a=Ta, that is, a is a fixed point of T.
To show the uniqueness of the fixed point, suppose that b is also a fixed point of T. From (3.4), we have
which implies that d(a,b)=0, that is, a=b. Then a is the unique fixed point of T. □
Now, from Theorem 3.2, we deduce immediately the following fixed point theorem on Gmetric spaces.
Theorem 3.3 Let (X,G) be a Gcomplete metric space and T:X\to X be a mapping satisfying
for all x,y\in X, where \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is continuous with {\phi}^{1}(\{0\})=\{0\}. Then T has a unique fixed point.
Proof Consider the quasimetric \delta (x,y)=G(x,y,y) for all x,y\in X. From (3.12), we have
for all x,y\in X. Then the result follows from Theorem 3.2. □
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Acknowledgements
This work is supported by the Research Center, College of Science, King Saud University.
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Jleli, M., Samet, B. Remarks on Gmetric spaces and fixed point theorems. Fixed Point Theory Appl 2012, 210 (2012). https://doi.org/10.1186/168718122012210
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DOI: https://doi.org/10.1186/168718122012210