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G-Metric spaces in any number of arguments and related fixed-point theorems
Fixed Point Theory and Applications volume 2014, Article number: 13 (2014)
Abstract
Inspired by the notion of Mustafa and Sims’ G-metric space and the attention that this kind of metric has received in recent times, we introduce the concept of a G-metric space in any number of variables, and we study some of the basic properties. Then we prove that the family of this kind of metric is closed under finite products. Finally, we show some fixed-point theorems that improve and extend some well-known results in this field.
MSC:46T99, 47H10, 47H09, 54H25.
1 Introduction
In the 1960s, Gähler [1, 2] tried to generalize the notion of metric and introduced the concept of 2-metric spaces inspired by the mapping that associated the area of a triangle to its three vertices. Later, Dhage [3] changed the axioms and presented the concept of a D-metric. Unfortunately, both kinds of metrics appear not to have as good properties as their authors announced (see [4–10]). To overcome these drawbacks, Mustafa and Sims [11] presented the notion of a G-metric space, which have received much attention since then. The literature on this topic, especially in related fixed point theory, has grown a lot in recent times (see, for instance, [12–26] and references therein).
The main aim of the present paper is to introduce the notion of a G-metric space in any number of variables. To do that, we have been inspired by the perimeter of a triangle, as well as Dhage, which in the multidimensional case can be seen as the sum of all distances between any pair of points. In this sense, the axioms we present and the properties we deduce are very natural. We also prove two relevant facts: the product of metrics of this kind is also a metric in this sense, and there is no a trivially way to reduce the number of variables (which, for instance, permits us to reduce a -metric to the Mustafa and Sims’ spaces). Later, we demonstrate some fixed-point theorems distinguishing between the axioms that the metric verifies (-metrics and -metrics). As a consequence, our main results are, obviously, valid in the context of G-metric spaces.
2 Preliminaries
Let n be a positive integer such that . Henceforth, X will denote a non-empty set and will denote the product space . Throughout this manuscript, m and k will denote non-negative integers. Unless otherwise stated, ‘for all m’ will mean ‘for all ’. Let and let .
Proposition 1 If are N sequences of non-negative real numbers and we define for all , then if, and only if, for all .
Definition 2 (Khan et al. [27])
An altering distance function is a continuous, non-decreasing mapping such that . Let Ψ denote the family of all altering distance functions.
Lemma 3 If and verifies , then .
Corollary 4 If and verifies for all m, then .
The following kind of mapping was introduced by Popescu in [28] and Moradi and Farajzadeh in [29].
Definition 5 We will denote by Φ the family of all mappings verifying:
Remark 6 Obviously, .
Definition 7 We will say that ≼ is a partial preorder on X (or is a preordered set or is a partially preordered space) if the following properties hold.
-
Reflexivity: for all .
-
Transitivity: If verify and , then .
Definition 8 (Mustafa and Sims [11])
A generalized metric (or a G-metric) on X is a mapping verifying, for all :
(G1) .
(G2) if .
(G3) if .
(G4) (symmetry in all three variables).
(G5) (rectangle inequality).
In [11], the authors proved that, in general, the product space of G-metric spaces is not a G-metric space (unless the factors are symmetric, that is, that they can be reduced to metric spaces). To overcome this drawback, Roldán and Karapınar introduced the concept of -metric spaces, in which the axiom (G3) is omitted.
Definition 9 (Roldán and Karapınar [20])
A -metric on X is a mapping verifying (G1), (G2), (G4) and (G5).
Using this class of spaces, these authors proved that the product of -metric spaces is also a -metric space, and they also showed some related fixed point results. This is the case of the generalized metrics that we are going to introduce in the following section.
3 Multidimensional -metric spaces
The following definition is a natural extension of the concept of a G-metric space. Note that, for convenience, we change the order in which we present the axioms with respect to the order Mustafa and Sims chose.
Definition 10 A -metric on X is a mapping verifying, for all :
(A1) .
(A2) If then .
(A3) If is a permutation, then (symmetry in all its variables).
(A4) (multidimensional inequality).
We will say that is a -metric (or a G-metric on n variables) if it also verifiesa:
(A5) If , then .
Remark 11 Following the previous definition, it is not difficult to prove that a -metric space is a classical metric space, a -metric space if a G-metric space in the sense of Mustafa and Sims [11], and a -metric space if a -metric space in the sense of Roldán and Karapınar [20].
However, a -metric does not generate a -metric in a trivial way since, if is a -metric on X and we define , for all , by
(where is fixed), then and are not -metrics ( does not have to verify (A3) and need not verify (A1)).
Example 12 Each metric space can be provided with a -metric defining by
Lemma 13 If is a -metric space and we define by
then is a metric on X. Furthermore, if d is a metric on X, then .
For brevity, when the last arguments are repeated, we will denote
Corollary 14 If is a -metric space and we define by
then is also a metric on X. Moreover, .
The metric generates a unique Hausdorff topology on each -metric space such that is a neighborhood system at each , where denotes the ball . This topology yields the following notions of convergence, Cauchy sequence, completeness, and continuity.
Definition 15 Let be a -metric space, let be a sequence and let be a point. We will say that:
-
-converges to x (we will denote this by ) if
that is, for all there is such that if , then ;
-
is a -Cauchy sequence if (that is, for all there is such that if , then );
-
a subset is -complete is every -Cauchy sequence in A is -convergent in A;
-
a mapping is -continuous if for all N sequences such that for all , we have .
Notice that, by the symmetry condition (A3), we could reduce the previous definitions to the case in which . When the -metric space is preordered, we can also consider the following class of spaces.
Definition 16 Let be a -metric space and let ≼ be a preorder on X. We will say that is regular-non-decreasing if it verifies the following property:
⊳ If is a ≼-non-decreasing sequence ( for all m) that -converges to , then for all m.
We will say that is regular-non-increasing when:
⊳ If is a ≼-non-increasing sequence ( for all m) that -converges to , then for all m.
The space is regular if it is both regular-non-decreasing and regular-non-increasing.
Some properties of a -metric space are listed in the following result.
Lemma 17 Let be a -metric space, let be a sequence and let .
-
(1)
If there exist such that , then .
-
(2)
If , then .
-
(3)
.
-
(4)
for all .
-
(5)
-converges to x if, and only if, , which is equivalent to -converges to x.
-
(6)
is -Cauchy if, and only if, it is -Cauchy.
Proof (1) Suppose that () and let be any permutation such that and . Then, by axiom (A2)
(2) The first item establishes that if two points are different, then the -metric is strictly positive. Then, if the -metric takes the value zero, then all points must be equal.
(3) Taking into account that is symmetric in all its variables, we can apply times the axiom (A4) using to deduce
(4) By (A4) and (A3),
(5) Suppose that -converges to x, and let be arbitrary. Using , by hypothesis, there exists such that if , then . In particular, if , then, by item (3),
Therefore, .
Conversely, suppose that and fix arbitrary. Using , let be such that if , then . Therefore, by item (4) with , if , we have
-
(6)
Suppose that is -Cauchy and let . There is such that if , then . Let . Hence and and we deduce that . Therefore, is -Cauchy.
Conversely, suppose that is -Cauchy and let . Given , there is such that if , then . Therefore, if , item (4) ensures us that
Hence, is -Cauchy. □
Corollary 18 A sequence on a -metric space is not -Cauchy if, and only if, there exist a positive and two subsequences and of such that ,
To prove our main results, the following refinement of Corollary 18 plays a key role.
Lemma 19 Suppose that a sequence in a -metric space is not -Cauchy and verifies . Then there exist and two subsequences and such that, for all ,
Furthermore, for all , we have
Moreover,
In particular,
Proof First part is Corollary 18. Now suppose that verifies
By item (4) of Lemma 17, we have for all m, so
Notice that, using (A4),
Using equation (3) and taking the limit as we deduce that
Next we show, by induction, that for all ,
If , the claim holds by equation (5). Suppose that equation (6) holds for some ; we will prove it for . On the one hand,
and, on the other hand,
Joining the two inequalities, for all ,
taking the limit as , and applying equations (3) and (4), we deduce that
that is, equation (6) holds for . This completes the induction and equation (6) is valid for any . Next we prove that, for all ,
Fix arbitrarily. If , equation (7) is true by equation (6). Suppose that equation (7) is true for some and we will prove it for . Indeed, on the one hand,
and, on the other hand,
Combining the two inequalities, for all ,
Taking the limit as and applying equations (3) and (4), we deduce that
that is, equation (7) also holds for . This completes the second induction. Repeating this reasoning in all arguments, we conclude that equation (2) holds. Exactly the same argument lets us prove that
□
4 Product of -metric spaces
In [11], the authors proved that, in general, the product space of G-metric spaces is not a G-metric space (unless the factors are symmetric, that is, that they can be reduced to metric spaces). Later, Roldán and Karapınar [20] introduced the concept of -metric spaces, in which the axiom (G3) is omitted. Then they succeeded in proving that the product of -metric spaces is also a -metric space. This is the case of -metric spaces.
Theorem 20 Let be a family of -metric spaces, consider the product space and define and on by
for all . Then the following statements hold.
-
(1)
and are -metrics on . Also they are equivalent since .
-
(2)
If for all m and , then -converges (respectively, -converges) to W if, and only if, each -converges to .
-
(3)
is -Cauchy if, and only if, each is -Cauchy for all .
-
(4)
(respectively, ) is complete if, and only if, every is complete.
Proof We only reason with since the other case is similar. Note that if , then for all and all .
-
(1)
We prove four axioms.
(A1) If , then .
(A2) Suppose that , that is, there is such that . Then and we deduce that
(A3) If is a permutation, then
(A4) Let . Then
(2) It follows from Proposition 1, item (5) of Lemma 17, and the fact that
(3) It is the same reasoning as taking into account that
(4) Assume that is complete for all and let be a -Cauchy sequence in . By item (3), each sequence is -Cauchy for all . Since is complete, there is such that , for all . If , item (2) guarantees that . Therefore, is complete. The converse is similar. □
5 Some fixed point results on -metric spaces
The following result is a natural extension of Theorem 26 in [20] and of Theorem 3.1 in [22]. We highlight two facts: on the one hand, if , the following result cannot be reduced to metric spaces since the role of x and y is not symmetric; on the other hand, it cannot be reduced to G-metric spaces (three arguments) because repeating some arguments does not yield G-metric spaces (see Remark 11 and also Remark 2 in [22]).
Theorem 21 Let be a preordered set endowed with a -metric and let be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is non-decreasing (w.r.t. ≼).
-
(c)
Either T is -continuous or is regular-non-decreasing.
-
(d)
There exists such that .
-
(e)
There exist two mappings such that, for all with ,
(8)
Then T has a fixed point. Furthermore, if for all fixed points of T there exists such that and , we obtain uniqueness of the fixed point.
Proof Define for all . Since T is ≼-non-decreasing, then for all . Then
Applying Lemma 4, . By item (4) of Lemma 17, , so
Let us show that is -Cauchy. Reasoning by contradiction, if is not -Cauchy, by Lemma 19, there exist and two subsequences and verifying ,
Therefore, by (8),
Consider the sequence of non-negative real numbers . If this sequence has a subsequence converging to zero, then we can take the limit in equation (10) using this subsequence and we would deduce , which is impossible. Then cannot have a subsequence converging to zero. This means that there exist and such that
Since φ is non-decreasing, . We also notice that, using (A4),
By equations (10), (11), and (12), and taking into account that ψ is non-decreasing, it follows that, for all ,
Using equation (9), the fact that ψ is continuous and taking the limit when in equation (13), we deduce that , which is impossible since and . This contradiction shows us that is a -Cauchy sequence. Since is complete, there exists such that .
Now suppose that T is -continuous. Then . By the unicity of the limit, , and is a fixed point of T.
On the other case, suppose that is regular. Since and is monotone non-decreasing (w.r.t. ≼), it follows that for all m. Hence
Since , then . Taking the limit when we deduce that
By Lemma 3, , so and we also conclude that is a fixed point of T.
To prove the uniqueness, let two fixed points of T. By hypothesis, there exists such that and . Let us show that . Indeed,
By Lemma 4, we deduce , that is, . The same reasoning proves that , so . □
If we particularize the previous result to the case in which , we obtain the following consequence.
Corollary 22 (Roldán and Karapınar [20], Theorem 26)
Let be a preordered set endowed with a -metric G and be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is non-decreasing (w.r.t. ≼).
-
(c)
Either T is G-continuous or is regular-non-decreasing.
-
(d)
There exists such that .
-
(e)
There exist two mappings such that, for all with ,
Then T has a fixed point. Furthermore, if for all fixed points of T there exists such that and , we obtain uniqueness of the fixed point.
Taking for all in the previous theorem, we deduce the following result.
Corollary 23 Let be a complete -metric space endowed with a preorder ≼ and let be a ≼-non-decreasing mapping such that there exists verifying, for all with ,
Assume that T is -continuous or is regular. Then T has a fixed point provided that there is such that .
In addition to this, taking for all in the previous corollary, we deduce the following result.
Corollary 24 Let be a complete -metric space endowed with a preorder ≼ and let be a ≼-non-decreasing mapping such that there exists verifying
Assume that T is -continuous or is regular. Then T has a fixed point provided that there is such that .
6 Some fixed point results on -metric spaces
The following result is a variation of Theorem 21 in the setting of -metric spaces.
Theorem 25 Let be a preordered set endowed with a -metric and let be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is non-decreasing (w.r.t. ≼).
-
(c)
Either T is -continuous.
-
(d)
There exists such that .
-
(e)
There exist two mappings and such that, for all with ,
(14)
Then T has a fixed point. Furthermore, if for all fixed points of T there exists such that and , we obtain uniqueness of the fixed point.
Proof Define for all . If there exists some such that , then is a fixed point of T. On the contrary, suppose that
Since T is ≼-non-decreasing and , then for all . Therefore, taking and in equation (14), we have
Applying Lemma 4, . Taking into account equation (15) and (A5),
Therefore,
Let us show that is -Cauchy. Reasoning by contradiction, if is not -Cauchy, by Lemma 19, there exist and two subsequences and verifying ,
In particular, taking appropriate values for , we have
From and by equation (14),
for all . Using that ψ is a continuous mapping, and equations (16) and (17), we deduce that the sequence
has a finite limit and, more precisely,
Hence
It follows from equation (1) that
which contradicts that . This contradiction shows us that is a -Cauchy sequence. Since is complete, there exists such that . Furthermore, since T is -continuous, then . By the unicity of the limit, , and is a fixed point of T.
To prove the uniqueness, let two fixed points of T. By hypothesis, there exists such that and . Let us show that . On the one hand, since T is non-decreasing, . On the other hand, for all ,
By Lemma 4, we deduce , that is, by item (5) of Lemma 17, . The same reasoning proves that , so . □
In the following result, we suppose that , and we analyze the case in which T is not necessarily continuous.
Theorem 26 Let be a preordered set endowed with a -metric and let be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is non-decreasing (w.r.t. ≼).
-
(c)
Either T is -continuous or is regular.
-
(d)
There exists such that .
-
(e)
There exist two mappings such that, for all with ,
Then T has a fixed point. Furthermore, if for all fixed points of T there exists such that and , we obtain uniqueness of the fixed point.
Proof By Remark 6, , so this result holds when T is continuous. Now suppose that is regular. Repeating the previous proof, we know that . Since is ≼-non-decreasing and is regular, it follows that for all m. Hence
As , we know that . Since ψ and φ are continuous, then , so . By Lemma 3,
and this means that . The unicity of the limit leads us to conclude that . □
Theorems 25 and 26 can be particularized to the case in which for all as follows.
Corollary 27 Let be a complete -metric space endowed with a preorder ≼ and let be a ≼-non-decreasing, -continuous mapping such that there exists verifying, for all with ,
Then T has a fixed point provided that there is such that .
Corollary 28 Let be a complete -metric space endowed with a preorder ≼ and let be a ≼-non-decreasing mapping such that there exists verifying, for all with ,
Assume that T is -continuous or is regular. Then T has a fixed point provided that there is such that .
The following result is a particularization to the case in which for all , where .
Corollary 29 Let be a complete -metric space endowed with a preorder ≼ and let be a ≼-non-decreasing mapping such that there exists verifying, for all with ,
Assume that T is -continuous or is regular. Then T has a fixed point provided that there is such that .
7 Consequences
Similar (but easier) techniques permit us to prove the following result.
Theorem 30 Let be a preordered set endowed with a -metric and let be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is non-decreasing (w.r.t. ≼).
-
(c)
Either T is -continuous or is regular-non-decreasing.
-
(d)
There exists such that .
-
(e)
There exist two mappings such that, for all with ,
Then T has a fixed point. Furthermore, if for all fixed points of T there exists such that and , we obtain uniqueness of the fixed point.
In particular, taking , we can conclude the following result.
Corollary 31 (Choudhury and Maity [22], Theorem 3.1)
Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists a such that for , the following holds:
for all and where either or .
If there exist such that and , then F has a coupled fixed point in X, that is, there exist such that and .
Endnote
a Notice that in this axiom, it is possible that or .
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Acknowledgements
The first author has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE. The third author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (Under NRU-CSEC Project No. NRU56000508).
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Roldán, A., Karapınar, E. & Kumam, P. G-Metric spaces in any number of arguments and related fixed-point theorems. Fixed Point Theory Appl 2014, 13 (2014). https://doi.org/10.1186/1687-1812-2014-13
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DOI: https://doi.org/10.1186/1687-1812-2014-13