Abstract
We discuss the introduced concept of G-metric 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 quasi-metric 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 G-metric spaces, as generalization of a metric space . Subsequently, many fixed point results on such spaces appeared (see, for example, [3–7]).
Here, we present the necessary definitions and results in G-metric 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 is a function satisfying the following conditions:
-
(1)
if and only if ;
-
(2)
for all with ;
-
(3)
for all with ;
-
(4)
(symmetry in all three variables);
-
(5)
for all .
Then G is called a G-metric on X and is called a G-metric space.
Definition 1.2 A G-metric space is said to be symmetric if for all .
Definition 1.3 Let be a G-metric space. We say that is
-
(1)
a G-Cauchy sequence if, for any , there is (the set of all positive integers) such that for all , ;
-
(2)
a G-convergent sequence to if, for any , there is such that for all , .
A G-metric space is said to be complete if every G-Cauchy sequence in X is G-convergent in X.
Proposition 1.1 Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x;
-
(2)
as ;
-
(3)
as ;
-
(4)
as .
Proposition 1.2 Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy;
-
(2)
as .
An interesting observation is that any G-metric space induces a metric on X given by
Moreover, is G-complete if and only if is complete.
It was observed that in the symmetric case ( is symmetric), many fixed point theorems on G-metric spaces are particular cases of existing fixed point theorems in metric spaces. In this paper, we discuss the non-symmetric case. We will show that such spaces have a quasi-metric type structure and then many results on such spaces can be derived from fixed point theorems on quasi-metric spaces.
2 Basic definitions and results
As we mentioned earlier, G-metric spaces have a quasi-metric type structure. Indeed, we have the following result.
Theorem 2.1 Let be a G-metric space. The function defined by satisfies the following properties:
-
(1)
if and only if ;
-
(2)
for any points .
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 be a given function which satisfies
-
(1)
if and only if ;
-
(2)
for any points .
Then d is called a quasi-metric and the pair is called a quasi-metric space.
Note that any metric space is a quasi-metric space, but the converse is not true in general.
Now, we define convergence and completeness on quasi-metric spaces.
Definition 2.2 Let be a quasi-metric space, be a sequence in X, and . The sequence converges to x if and only if
Definition 2.3 Let be a quasi-metric space and be a sequence in X. We say that is left-Cauchy if and only if for every there exists a positive integer such that for all .
Definition 2.4 Let be a quasi-metric space and be a sequence in X. We say that is right-Cauchy if and only if for every there exists a positive integer such that for all .
Definition 2.5 Let be a quasi-metric space and be a sequence in X. We say that is Cauchy if and only if for every there exists a positive integer such that for all .
Obviously, a sequence in a quasi-metric space is Cauchy if and only if it is left-Cauchy and right-Cauchy.
Definition 2.6 Let be a quasi-metric space. We say that
-
(1)
is left-complete if and only if each left-Cauchy sequence in X is convergent;
-
(2)
is right-complete if and only if each right-Cauchy sequence in X is convergent;
-
(3)
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 be a G-metric space. Let be the function defined by . Then
-
(1)
is a quasi-metric space;
-
(2)
is G-convergent to if and only if is convergent to x in ;
-
(3)
is G-Cauchy if and only if is Cauchy in ;
-
(4)
is G-complete if and only if is complete.
Every quasi-metric induces a metric, that is, if is a quasi-metric space, then the function defined by
is a metric on X.
The following result is an immediate consequence of the above definitions and results.
Theorem 2.3 Let be a G-metric space. Let be the function defined by . Then
-
(1)
is a metric space;
-
(2)
is G-convergent to if and only if is convergent to x in ;
-
(3)
is G-Cauchy if and only if is Cauchy in ;
-
(4)
is G-complete if and only if is complete.
3 Discussion on fixed point results on G-metric spaces
3.1 From metric to G-metric: the linear case
In this section, we show that in the case of linear contractive conditions, the existing fixed point results on G-metric 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 be a complete G-metric space, and let be a mapping that satisfies the following condition: for all ,
where a, b, c, d are positive constants such that . 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 in (3.1), we have
for all . Also, from (3.1), we have
for all . Define the metric space by
It follows from (3.2) and (3.3) that
Now, T satisfies Ćirić’s contractive condition [9] in the complete metric space (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 quasi-metric to G-metric: 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 G-metric spaces from fixed point results on quasi-metric spaces. As a model example, we consider a weakly contractive condition. At first, we need the following fixed point theorem on quasi-metric spaces.
Theorem 3.2 Let be a complete quasi-metric space and be a mapping satisfying
for all , where is continuous with . Then T has a unique fixed point.
Proof Let be any point in X and be the sequence defined by for all . From (3.4), we have
This implies that is a decreasing sequence of positive numbers. Then there exists such that as . Letting in (3.5), we get that , that is, . Thus, we have
Using the same technique, we also have
Now, we shall prove that is a Cauchy sequence in the quasi-metric space , that is, is left-Cauchy and right-Cauchy. Suppose that is not a left-Cauchy sequence. Then there exists for which we can find subsequences and of with such that
for all k. Further, corresponding to , we can choose such that it is the smallest integer with satisfying the above inequality. Then
for all k. On the other hand, we have
Letting and using (3.7), we get
We have
and
Letting in the above inequalities, using (3.6), (3.7), and (3.8), we get
Now, from (3.4), for all k, we have
Letting in the above inequality, using (3.8) and (3.9), we obtain
which implies that : a contradiction with . Then we proved that is a left-Cauchy sequence. Similarly, we can show that is a right-Cauchy sequence. Then is a Cauchy sequence in the complete quasi-metric space . This implies that there exists such that
Now, we have
Letting and using (3.10), we have
Similarly, we have
Letting and using (3.10), we have
Then, we have
It follows from (3.10) and (3.11) that , 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 , that is, . Then a is the unique fixed point of T. □
Now, from Theorem 3.2, we deduce immediately the following fixed point theorem on G-metric spaces.
Theorem 3.3 Let be a G-complete metric space and be a mapping satisfying
for all , where is continuous with . Then T has a unique fixed point.
Proof Consider the quasi-metric for all . From (3.12), we have
for all . Then the result follows from Theorem 3.2. □
References
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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 G-metric spaces and fixed point theorems. Fixed Point Theory Appl 2012, 210 (2012). https://doi.org/10.1186/1687-1812-2012-210
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DOI: https://doi.org/10.1186/1687-1812-2012-210