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Some multidimensional fixed point theorems on partially preordered -metric spaces under -contractivity conditions
Fixed Point Theory and Applications volume 2013, Article number: 158 (2013)
Abstract
In this paper we present some (unidimensional and) multidimensional fixed point results under -contractivity conditions in the framework of -metric spaces, which are spaces that result from G-metric spaces (in the sense of Mustafa and Sims) omitting one of their axioms. We prove that these spaces let us consider easily the product of -metrics. Our result clarifies and improves some recent results on this topic because, among other different reasons, we will not need a partial order on the underlying space. Furthermore, the way in which several contractivity conditions are proposed imply that our theorems cannot be reduced to metric spaces.
MSC: 46T99, 47H10, 47H09, 54H25.
1 Introduction
In the sixties, inspired by the mapping that associated the area of a triangle to its three vertices, Gähler [1, 2] introduced the concept of 2-metric spaces. Gähler believed that 2-metric spaces can be interpreted as a generalization of usual metric spaces. However, some authors demonstrated that there is no clear relationship between these notions. For instance, Ha et al. [3] showed that a 2-metric does not have to be a continuous function of its three variables. Later, inspired by the perimeter of a triangle rather than the area, Dhage [4] changed the axioms and presented the concept of D-metric. Different topological structures (see [5–7]) were considered in such spaces and, subsequently, several fixed point results were established. Unfortunately, most of their properties turned out to be false (see [8–10]). These considerations led to the concept of G-metric space introduced by Mustafa and Sims [11]. Since then, this theory has been expansively developed, paying a special attention to fixed point theorems (see, for instance, [12–28] and references therein).
The main aim of the present paper is to prove new unidimensional and multidimensional fixed point results in the framework of the G-metric spaces provided with a partial preorder (not necessarily a partial order). However, we need to overcome the well-known fact that the usual product of G-metrics is not necessarily a G-metric unless it comes from classical metrics (see [11], Section 4). Hence, we will omit one of the axioms that define a G-metric and we consider a new class of metrics, called -metrics. As a consequence, our main results are valid in the context of G-metric spaces.
2 Preliminaries
Let n be a positive integer. 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 and . Unless otherwise stated, ‘for all m’ will mean ‘for all ’ and ‘for all i’ will mean ‘for all ’. Let .
Definition 1 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 .
Henceforth, let be a partition of , that is, and such that A and B are non-empty sets. In the sequel, we will denote
From now on, let be an n-tuple of mappings from into itself verifying if and if .
If is a partially preordered space, and , we will use the following notation:
Consider on the product space the following partial preorder: for ,
Notice that ⊑ depends on A and B. We say that two points X and Y are ⊑-comparable if or .
Proposition 2 If and , then and are ⊑-comparable. In particular,
Proof Suppose that for all i. Hence for all i. Fix . If , then , so implies that , which means that . If , then , so implies that , which means that . In any case, if , then for all i. It follows that .
Now fix . If , then , so implies that , which means that . If , then , so implies that , which means that . □
Let be a mapping.
Definition 3 (Roldán et al. [20])
A point is called an ϒ-fixed point of the mapping F if
Definition 4 (Roldán et al. [20])
Let be a partially preordered space. We say that F has the mixed monotone property (w.r.t. ) if F is monotone non-decreasing in the arguments of A and monotone non-increasing in the arguments of B, i.e., for all and all i,
We will use the following results about real sequences in the proof of our main theorems.
Lemma 5 Let be n real lower bounded sequences such that . Then there exist and a subsequence such that .
Proof Let for all m. As is convergent, it is bounded. As for all m and i, then every is bounded. As is a real bounded sequence, it has a convergent subsequence . Consider the subsequences , that are real bounded sequences, and the sequence that also converges to δ. As is a real bounded sequence, it has a convergent subsequence . Then the sequences also are real bounded sequences and and . Repeating this process n times, we can find n subsequences (where ) such that for all i. And . But
so and there exists such that . Therefore, there exist and a subsequence such that . □
Lemma 6 Let be a sequence of non-negative real numbers which has not any subsequence converging to zero. Then, for all , there exist and such that for all .
Proof Suppose that the conclusion is not true. Then there exists such that, for all , there exists verifying . Let be such that . For all , take . Then there exists verifying . Taking limit when , we deduce that . Then has a subsequence converging to zero (maybe, reordering ), but this is a contradiction. □
Let
Lemma 7 If and verifies , then .
Proof If the conclusion does not hold, there exists such that, for all , there exists verifying . This means that has a partial subsequence such that . As ψ is non-decreasing, for all . Therefore, has a subsequence lower bounded by , but this is impossible since . □
Lemma 8 Let be 2n sequences of non-negative real numbers and suppose that there exist such that
Then for all i.
Proof Let for all m. Then, for all m,
Therefore, is a non-increasing, bounded below sequence. Then it is convergent. Let be such that and . Let us show that . Since
Lemma 5 guarantees that there exist and a partial subsequence such that . Moreover,
Consider the sequence . If this sequence has a partial subsequence converging to zero, then we can take limit in (3) when using that partial subsequence, and we deduce . On the contrary, if has not any partial subsequence converging to zero, Lemma 6 assures us that there exist and such that for all . Since φ is non-decreasing, . Then, by (3), for all ,
Taking limit as , we deduce , which is impossible. This proves that . Since , Lemma 7 implies that , which is equivalent to for all i. □
Corollary 9 If and verify and for all m, then .
Corollary 10 If and verifies for all m, then .
Definition 11 (Mustafa and Sims [11])
A generalized metric (or a G-metric) on X is a mapping verifying, for all :
-
() .
-
() if .
-
() if .
-
() (symmetry in all three variables).
-
() (rectangle inequality).
Let be a family of G-metric spaces, consider the product space and define and on by
for all .
A classical example of G-metric comes from a metric space , where measures the perimeter of a triangle. In this case, property has an obvious geometric interpretation: the length of an edge of a triangle is less than or equal to its semiperimeter, that is, . However, property implies that, in general, the major structures and are not necessarily G-metrics on . Only when each is symmetric (that is, for all x, y), the product is also a G-metric (see [11]). But in this case, symmetric G-metrics can be reduced to usual metrics, which limits the interest in this kind of spaces.
In order to prove our main results, that are also valid in G-metric spaces, we will not need property . Omitting this property, we consider a class of spaces for which and have the same initial metric structure. Then we present the following spaces.
3 -metric spaces
Definition 12 A -metric on X is a mapping verifying , , and .
The open ball of center and radius in a -metric space is
The following lemma is a characterization of the topology generated by a neighborhood system at each point.
Lemma 13 Let X be a set and, for all , let be a non-empty family of subsets of X verifying:
-
1.
for all .
-
2.
For all , there exists such that .
-
3.
For all , there exists such that for all , there exists verifying .
Then there exists a unique topology τ on X such that is a neighborhood system at x.
Let be a -metric space and consider the family . It is clear that (by , ) and . Next, let and let . We have to prove that there exists such that . Indeed, if , then we can take . On the contrary, if , then by . Let arbitrary and let (that is, ). Now we prove that . Let . Then, using and ,
Then and, as a consequence, . Lemma 13 guarantees that there exists a unique topology on X such that is a neighborhood system at each .
Next, let us show that is Hausdorff. Let be two points such that . By , . We claim that . We reason by contradiction. Let , that is, and . Using and twice
which is impossible. Then and is Hausdorff.
A subset is G-open if for all there exists such that . Following classic techniques, it is possible to prove that there exists a unique topology on X such that is a neighborhood system at each . Furthermore, is a Hausdorff topology. In this topology, we characterize the notions of convergent sequence and Cauchy sequence in the following way. Let be a -metric space, let be a sequence and let .
-
G-converges to x, and we will write if , that is, for all , there exists verifying that for all such that .
-
is G-Cauchy if , that is, for all , there exists verifying that for all such that .
Lemma 14 Let be a -metric space, let be a sequence and let . Then the following conditions are equivalent.
-
(a)
G-converges to x.
-
(b)
.
-
(c)
.
-
(d)
and .
-
(e)
and .
Notice that the condition is not strong enough to prove that G-converges to x.
Proposition 15 The limit of a G-convergent sequence in a -metric space is unique.
Lemma 16 If is a -metric space and is a sequence, then the following conditions are equivalent.
-
(a)
is G-Cauchy.
-
(b)
.
-
(c)
.
Remark 17 As a consequence, a sequence is not G-Cauchy if and only if there exist and two partial subsequences and such that , and for all k.
Definition 18 Let be a -metric space and let ≼ be a preorder on X. We will say that is regular non-decreasing (respectively, regular non-increasing) if for all ≼-monotone non-decreasing (respectively, non-increasing) sequence such that , we have that (respectively, ) for all m. We will say that is regular if it is both regular non-decreasing and regular non-increasing.
Some authors said that verifies the sequential monotone property if is regular (see [20]). The notion of G-continuous mapping follows considering on X the topology and in the product topology.
Definition 19 If is a -metric space, we will say that a mapping is G-continuous if for all n sequences such that for all i, we have that .
In this topology, the notion of convergence is the following.
This property can be characterized as follows.
Lemma 20 Let be a -metric space, let be a sequence and let . Then the following conditions are equivalent.
-
(a)
G-converges to x (that is, , which means that for all , there exists such that for all ).
-
(b)
.
-
(c)
.
-
(d)
and .
-
(e)
and .
Proof [(a) ⇒ (c)] It is apparent using .
-
[(c) ⇒ (b)] Using , .
-
[(b) ⇒ (a)] Using and ,
-
[(a) ⇒ (d),(e)] It is apparent using and .
-
[(d) ⇒ (c)] It is evident.
-
[(e) ⇒ (b)] It is evident. □
Corollary 21 If is a G-metric space, then if and only if .
Proof We only need to prove that the condition is sufficient. Suppose that . In a G-metric space, the following property holds (see [11]):
Then, using ,
This proves (b) in the previous lemma. □
Proposition 22 The limit of a G-convergent sequence in a -metric space is unique.
Proof Suppose that and . Then
By items (a) and (c) of Lemma 20, we deduce that , which means that by . □
In the topology , the notion of Cauchy sequence is the following.
This definition can be characterized as follows.
Lemma 23 If is a -metric space and is a sequence, then the following conditions are equivalent.
-
(a)
is G-Cauchy.
-
(b)
.
-
(c)
.
Proof [(b) ⇒ (a)] Using , .
-
[(a) ⇒ (c)] It is apparent using .
-
[(c) ⇒ (b)] Let and let be such that for all . Then
Therefore, using and ,
Therefore, . □
4 Product of -metric spaces
Lemma 24 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 .
-
2.
If for all m and , then -converges (respectively, -converges) to A if and only if each -converges to .
-
3.
is -Cauchy if and only if each is -Cauchy.
-
4.
(respectively, ) is complete if and only if every is complete.
-
5.
For all i, let be a preorder on and define if and only if for all i. Then is regular (respectively, regular non-decreasing, regular non-increasing) if and only if each factor is also regular (respectively, regular non-decreasing, regular non-increasing).
Proof Let us denote . Taking into account that , we will only develop the proof using G.
(1) It is a straightforward exercise to prove the following statements.
-
.
-
If , there exists such that . Then .
-
Symmetry in all three variables of G follows from symmetry in all three variables of each .
-
We have that
(2) We use Lemma 20. Suppose that G-converges to A and let . Then, for all and all m,
Therefore, -converges to . Conversely, assume that each -converges to . Let and let be such that if , then . If and , then , so G-converges to A.
(3) We use Lemma 23. Suppose that is G-Cauchy and let . Then, for all and all m, ,
Therefore, is -Cauchy. Conversely, assume that each is -Cauchy. Let and let be such that if , then . If and , then , so is G-Cauchy.
(4) It is an easy consequence of items 2 and 3 since
(5) A sequence on is ⪯-monotone non-decreasing if and only if each sequence is ⪯-monotone non-decreasing. Moreover, G-converges to if and only if each -converges to . Finally, if and only if for all i. Therefore, is regular non-decreasing if and only if each factor is also regular non-decreasing. Other statements may be proved similarly. □
Taking for all i, we derive the following result.
Corollary 25 Let be a -metric space and consider on the product space the mappings and defined by
for all .
-
1.
and are -metrics on .
-
2.
If for all m and , then -converges (respectively, -converges) to A if and only if each G-converges to .
-
3.
is -Cauchy (respectively, -Cauchy) if and only if each is G-Cauchy.
-
4.
(respectively, ) is complete if and only if is complete.
-
5.
If is ≼-regular, then is ⊑-regular.
5 Unidimensional fixed point result in partially preordered -metric spaces
Theorem 26 Let be a preordered set endowed with a -metric G 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 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.
Proof Define for all . Since T is non-decreasing (w.r.t. ≼), then for all . Then
Applying Lemma 10, . Let us show that is G-Cauchy. Reasoning by contradiction, if is not G-Cauchy, by Remark 17, there exist and two partial subsequences and verifying ,
Therefore
Consider the sequence of non-negative real numbers . If this sequence has a partial subsequence converging to zero, then we can take the limit in (5) using this partial subsequence and we would deduce , which is impossible. Then cannot have a partial subsequence converging to zero. This means that there exist and such that
Since φ is non-decreasing, . By (G5) and (4),
Since ψ is non-decreasing, it follows from (5) that
Taking limit when , we deduce that , which is impossible. This contradiction finally proves that is G-Cauchy. Since is complete, there exists such that .
Now suppose that T is G-continuous. Then . By the unicity of the limit, and is a fixed point of T.
On the contrary, suppose that is regular non-decreasing. Since and is monotone non-decreasing (w.r.t. ≼), it follows that for all m. Hence
Since , then . Taking limit when , we deduce that . By Lemma 7, , so and we also conclude that is a fixed point of T.
To prove the uniqueness, let be two fixed points of T. By hypothesis, there exists such that and . Let us show that . Indeed,
By Lemma 10, we deduce , that is, . The same reasoning proves that , so . □
We particularize the previous theorem in two cases. If take in Theorem 26, then we get the following results.
Corollary 27 Let be a preordered set endowed with a -metric G 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 G-continuous or is regular non-decreasing.
-
(d)
There exists such that .
-
(e)
There exists a mapping 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.
If take with in Corollary 27, then we derive the following result.
Corollary 28 Let be a preordered set endowed with a -metric G 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 G-continuous or is regular non-decreasing.
-
(d)
There exists such that .
-
(e)
There exists a constant 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.
6 Multidimensional ϒ-fixed point results in partially preordered -metric spaces
In this section we extend Theorem 26 to an arbitrary number of variables. To do that, it is necessary to introduce the following notation. Given a mapping , we define by
and will be
for all .
Lemma 29
-
1.
is a ϒ-fixed point of F if and only if Z is a fixed point of (that is, ).
-
2.
If F has the mixed monotone property, then is ⊑-monotone non-decreasing on .
-
3.
If is a -metric space and F is G-continuous, then is -continuous and is G-continuous.
6.1 A first multidimensional contractivity result
In this subsection we apply Theorem 26 considering defined on . In order to do that, we notice that joining some of the previous results, we obtain the following consequences.
-
If is complete, it follows from Corollary 25 that is also complete.
-
By item 2 of Lemma 29, if F has the mixed monotone property, then is ⊑-monotone non-decreasing on .
-
By item 3 of Lemma 29, if F is G-continuous, then is -continuous and is G-continuous.
-
If is regular, it follows from Corollary 25 that is also regular.
-
If are such that for all i, then verifies .
We study how the contractivity condition
may be equivalently established. Let and let for all i. Then
It follows that
Therefore, a possible version of Theorem 26 applied to taking is the following.
Theorem 30 Let be a complete -metric space and let ≼ be a partial preorder on X. Let be an n-tuple of mappings from into itself verifying if and if . Let be a mapping verifying the mixed monotone property on X. Assume that there exist such that
for which for all i. Suppose either F is continuous or is regular. If there exist verifying for all i, then F has, at least, one ϒ-fixed point.
6.2 A second multidimensional contractivity result
In this section we introduce a slightly different contractivity condition that cannot be directly deduced applying Theorem 26 to taking , because the contractivity condition is weaker. Then we need to show a classical proof.
Theorem 31 Let be a complete -metric space and let ≼ be a partial preorder on X. Let be an n-tuple of mappings from into itself verifying if and if . Let be a mapping verifying the mixed monotone property on X. Assume that there exist such that
for which are ⊑-comparable. Suppose either F is continuous or is regular. If there exist verifying for all i, then F has, at least, one ϒ-fixed point.
Notice that (6) and (7) are very different contractivity conditions. For instance, (6) would be simpler if the image of all are sets with a few points.
Proof Define and let for all i. If , then for all i is equivalent to . By recurrence, define for all i and all m, and we have that . This means that the sequence is ⊑-monotone non-decreasing. Since is complete, it is only necessary to prove that is -Cauchy in order to deduce that it is -convergent. By item 3 of Lemma 24, it will be sufficient to prove that each sequence is G-Cauchy. Firstly, notice that means that
Hence
Furthermore, for all m,
Therefore, for all i and all m,
Since ψ is non-decreasing, for all i and all m,
Applying Lemma 8 using
for all i and all m, we deduce that
Next, we prove that every sequence is G-Cauchy reasoning by contradiction. Suppose that are not G-Cauchy () and are G-Cauchy, being . From Proposition 2, for all , there exist and subsequences and such that, for all ,
Now, let and . Since are G-Cauchy, for all , there exists such that if , then . Define . Therefore, we have proved that there exists such that if then
Next, let be such that . Let be such that and define . Consider the numbers until finding the first positive integer verifying
Now let be such that and define . Consider the numbers until finding the first positive integer verifying
Repeating this process, we can find sequences such that, for all ,
Note that by (10), for all , so
for all and all . Next, for all k, let be an index such that
Notice that, applying (G5) twice and (11), for all k and all j,
Applying Proposition 2 to guarantee that the following points are ⊑-comparable, the contractivity condition (7) assures us for all k
Consider the sequence
If this sequence has a subsequence that converges to zero, then we can take limit when in (13) using this subsequence, so that we would have , which is impossible since . Therefore, the sequence (14) has no subsequence converging to zero. In this case, taking in Lemma 6, there exist and such that for all . It follows that, for all , . Thus, by (13) and (12),
Taking limit in (15) as and taking into account (9), we deduce that , which is impossible. The previous reasoning proves that every sequence is G-Cauchy.
Corollary 25 guarantees that the sequence is -Cauchy. Since is complete (again by Corollary 25), there exists such that , that is, if then
Suppose that F is G-continuous. In this case, item 3 of Lemma 29 implies that is -continuous, so and . By the unicity of the -limit, , which means that Z is a ϒ-fixed point of F.
Suppose that is regular. In this case, by Corollary 25, is also regular. Then, taking into account that is a ⊑-monotone non-decreasing sequence such that , we deduce that for all m. From Proposition 2, since , then and are ⊑-comparable for all i and all m. Notice that for all i and all m,
It follows from condition (7) and (8) that, for all i,
By (16) we deduce that
which means that
Since , we conclude that , that is, Z is a ϒ-fixed point of F. □
If we take in Theorem 26, then we get the following results.
Corollary 32 Let be a complete -metric space and let ≼ be a partial preorder on X. Let be an n-tuple of mappings from into itself verifying if and if . Let be a mapping verifying the mixed monotone property on X. Assume that there exists such that
for which are ⊑-comparable. Suppose either F is continuous or is regular. If there exist verifying for all i, then F has, at least, one ϒ-fixed point.
If we take for all , with , in Corollary 32, then we derive the following result.
Corollary 33 Let be a complete -metric space and let ≼ be a partial preorder on X. Let be an n-tuple of mappings from into itself verifying if and if . Let be a mapping verifying the mixed monotone property on X. Assume that there exists such that
for which are ⊑-comparable. Suppose either F is continuous or is regular. If there exist verifying for all i, then F has, at least, one ϒ-fixed point.
Example 34 Let and let G be the G-metric on X given, for all , by . Then is complete and G generates the discrete topology on X. Consider on X the following partial order:
Define by
Then the following statements hold.
-
1.
F is a G-continuous mapping.
-
2.
If verify , then either or . In particular, and F has the mixed monotone property on X.
-
3.
If are ⊑-comparable, then . In particular, (17) holds for .
For simplicity, henceforth, suppose that n is even and let A (respectively, B) be the set of all odd (respectively, even) numbers in .
-
4.
For a mapping , we use the notation and consider
Then if i is odd and if i is even. Let .
-
5.
Take if i is odd and if i is even. Then for all i.
Therefore, we can apply Corollary 33 to conclude that F has, at least, one ϒ-fixed point. To finish, we prove the previous statements.
If , then there exists such that for all . Since X is discrete, then for all . This proves that is the discrete topology on X.
-
1.
If are n sequences such that for all i, then there exists such that for all and all i. Then and F is G-continuous.
-
2.
If verify , the either (in this case, there is nothing to prove) or . Then either or . In particular,
Hence F has the mixed monotone property on X.
-
3.
Suppose that are ⊑-comparable, and we claim that . Indeed, assume, for instance, that for all i. By item 2, for all i, either or . Then
If for all i, the proof is similar. Next, we prove that (17) holds using . If , then . Therefore
Suppose that are ⊑-comparable. It follows that
It is clear that (17) holds if the previous number is 0. On the contrary, suppose that
Then or (both cases are similar). Assume, for instance, that . Then there exists such that . In particular
Therefore
This means that
Therefore, in this case, (17) also holds.
-
4.
It is evident.
-
5.
Since for all i, then for all i. If i is odd, then . If i is even, then , so .
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The second author has been partially supported by Junta de Andalucía and by project FQM-268 of the Andalusian CICYE.
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Roldán, A., Karapınar, E. Some multidimensional fixed point theorems on partially preordered -metric spaces under -contractivity conditions. Fixed Point Theory Appl 2013, 158 (2013). https://doi.org/10.1186/1687-1812-2013-158
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DOI: https://doi.org/10.1186/1687-1812-2013-158