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Discussion of coupled and tripled coincidence point theorems for φ-contractive mappings without the mixed g-monotone property
Fixed Point Theory and Applications volume 2014, Article number: 92 (2014)
Abstract
After the appearance of Ran and Reuring’s theorem and Nieto and Rodríguez-López’s theorem, the field of fixed point theory applied to partially ordered metric spaces has attracted much attention. Coupled, tripled, quadrupled and multidimensional fixed point results has been presented in recent times. One of the most important hypotheses of these theorems was the mixed monotone property. The notion of invariant set was introduced in order to avoid the condition of mixed monotone property, and many statements have been proved using these hypotheses. In this paper we show that the invariant condition, together with transitivity, lets us to prove in many occasions similar theorems to which were introduced using the mixed monotone property.
MSC:46T99, 47H10, 47H09, 54H25.
1 Introduction
One of the core recent research topics in Fixed Point Theory is multidimensional fixed point results in the context of various abstract space. This trend was initiated by the well-known paper of Gnana-Bhaskar and Lakshmikantham [1]. Following this pioneering work, several authors reported various results in the setting of partially ordered metric spaces. Recently, the notion of coupled fixed point was extended to the higher dimensions by defining tripled, quadrupled and, hence, multidimensional fixed points [2–5]. In [6], Roldán et al. proved that most of the multidimensional fixed point results can be derived from the existing fixed point theorems in the context of partially preordered metric spaces with the additional hypothesis of the mixed monotone property. Meanwhile, in the literature, several multidimensional fixed point theorems have appeared in which the authors omitted the notion of mixed property by adding a weaker conditions such as F-closed, F-invariant, etc.
In this paper we will show that the notion of transitive F-closed (or F-invariant) set is equivalent to the concept of preordered set and, then some recent multidimensional results using F-invariant sets can be reduced to well-known results on partially ordered metric spaces.
2 Preliminaries
Henceforth, let X be a nonempty set. Given a positive integer n, let be the product space . Let be the set of all nonnegative integers. We will use n, m and k to denote nonnegative integers. Unless otherwise stated, ‘for all n’ will mean ‘for all ’.
Definition 1 (Roldán et al. [7])
A preorder (or a quasiorder) ≼ on X is a binary relation on X that is reflexive (i.e., for all ) and transitive (if verify and , then ). In such a case, we say that is a preordered space (or a preordered set). If a preorder ≼ is also antisymmetric ( and implies ), then ≼ is called a partial order.
Throughout this manuscript, let be a metric space and let ≼ be a preorder (or a partial order) on X. In the sequel, and will denote mappings.
Definition 2 A point is
-
a coupled coincidence point of F and g if , and ;
-
a tripled coincidence point of F and g if , , and ;
-
a quadrupled coincidence point of F and g if , , , and .
If g is the identity mapping on X, then a point verifying the previous conditions is a coupled (respectively, tripled, quadrupled) fixed point of F due to Gnana-Bhaskar and Lakshmikantham [1] (respectively, Berinde and Borcut [2], Karapınar [3]).
Definition 3 If is a preordered space and are two mappings, we will say that T is a -nondecreasing mapping if for all such that . If g is the identity mapping on X, T is ≼-nondecreasing.
In [8], the author called g-isotone to -nondecreasing mappings, especially in the case in which X is a product space .
Definition 4 A fixed point of a self-mapping is a point such that . A coincidence point of two mappings is a point such that . A common fixed point of is a point such that .
Definition 5 We will say that T and g are commuting if for all , and we will say that F and g are commuting if for all .
Remark 6 If are commuting and is a coincidence point of T and g, then is also a coincidence point of T and g.
The following notion was introduced in order to avoid the necessity of commutativity.
Let be a metric space provided with a partial order ≼. Two mappings are said to be O-compatible if
provided that is a sequence in X such that is ≼-monotone and
In 2003, Ran and Reuring proved the following version of the Banach theorem applicable to metric spaces endowed with a partial order.
Theorem 8 (Ran and Reurings [12])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is ≼-nondecreasing.
-
(c)
T is continuous.
-
(d)
There exists such that .
-
(e)
There exists a constant such that for all with .
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
Later, Nieto and Rodríguez-López [13] slightly modified the hypothesis of the previous result swapping condition (c) by the fact that is nondecreasing-regular in the following sense.
Definition 9 Let be an ordered set endowed with a metric d. We will say that is nondecreasing-regular (respectively, nonincreasing-regular) if any ≼-nondecreasing (respectively, ≼-nonincreasing) sequence d-convergent to , we have (respectively, ) for all m. is regular if it is both nondecreasing-regular and nonincreasing-regular.
Inspired by Boyd and Wong’s theorem [14], Mukherjea [15] introduced the following kind of control functions:
and he proved a version of the following result in which the space is not necessarily endowed with a partial order (but the contractivity condition holds over all pairs of points of the space).
Theorem 10 Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(a)
is complete.
-
(b)
T is ≼-nondecreasing.
-
(c)
Either T is continuous or is nondecreasing-regular.
-
(d)
There exists such that .
-
(e)
There exists such that for all with .
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain uniqueness of the fixed point.
An interesting version of the previous result in the coupled case for compatible mappings is the following one.
Theorem 11 (Choudhury and Kundu [9], Theorem 3.1)
Let be a partially ordered set and let there be a metric d on X such that is a complete metric space. Let be such that and for all . Let and be two mappings such that F has the mixed g-monotone property and satisfy
Let , g be continuous and monotone increasing and F and g be compatible mappings. Also suppose
-
(a)
F is continuous or
-
(b)
X has the following properties:
-
(i)
if a nondecreasing sequence , then for all ;
-
(ii)
if a nonincreasing sequence , then for all .
If there exist such that and , then there exist such that and , that is, F and g have a coupled coincidence point in X.
A partial order ≼ on X can be extended to a partial order ⊑ on defining, for all ,
Another interesting generalization of Theorem 10 was given by Wang in [8] using this extended partial order on .
Theorem 12 (Wang [8], Theorem 3.4)
Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Let and be a G-isotone mapping for which there exists such that for all , with ,
where is defined, for all , by
Suppose and also suppose either
-
(a)
T is continuous, G is continuous and commutes with T, or
-
(b)
is regular and is closed.
If there exists such that and are ⊑-comparable, then T and G have a coincidence point.
Some other generalizations of the previous result can be found in Romaguera [16] (to partial metric spaces, but not necessarily provided with a partial order).
In order to guarantee the existence and uniqueness of a solution of periodic boundary value problems, Gnana Bhaskar and Lakshmikantham (and, subsequently, Lakshmikantham and Ćirić; see [17]) proved, in 2006, existence and uniqueness of a coupled fixed point (a notion introduced by Guo and Laksmikantham [18]) in the setting of partially ordered metric spaces by introducing the notion of mixed monotone property.
In order to ensure the existence of coupled fixed points, Gnana Bhaskar and Lakshmikantham introduced the following condition.
Definition 13 (Gnana Bhaskar and Lakshmikantham [1])
Let be a partially ordered set and . We say that F has the mixed monotone property if is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any ,
Many result were proved to ensure the existence of coupled fixed point. One of the common properties of all these results is the fact that the mapping must verify the mixed monotone property. Searching for a generalization of this kind of theorems, Samet and Vetro [19] succeeded in proving some results in which the mapping F did not necessarily have the mixed monotone property.
Definition 14 (Samet and Vetro [19])
Let be a metric space and be a given mapping. Let M be a nonempty subset of . We say that M is an F-invariant subset of if, for all ,
-
(i)
;
-
(ii)
.
The following theorem is the main result in [19].
Theorem 15 (Samet and Vetro [19])
Let be a complete metric space, be a continuous mapping and M be a nonempty subset of . We assume that
-
(i)
M is F-invariant;
-
(ii)
there exists such that ;
-
(iii)
for all , we have
where α, β, θ, γ, δ are nonnegative constants such that .
Then F has a coupled fixed point, i.e., there exists such that and .
Later, Sintunaravat et al. [20] introduced the notion of transitive property so as to extend the Lakshmikantham and Ćirić’s theorem (see [17]).
Definition 16 (Sintunaravat et al. [20])
Let be a metric space and M be a subset of . We say that M satisfies the transitive property if, for all ,
Then they proved the following result.
Theorem 17 (Sintunaravat et al. [20])
Let be a complete metric space and M be a nonempty subset of . Assume that there is a function with and for each , and also suppose that is a mapping such that
for all . Suppose that either
-
(a)
F is continuous or
-
(b)
for any two sequences , with ,
for all , then for all .
If there exists such that and M is an F-invariant set which satisfies the transitive property, then there exist such that and , that is, F has a coupled fixed point.
In recent times, it has been proved that many coupled, tripled, and quadrupled results can be reduced to the unidimensional case, that is, to Theorems 8 and 10 (see, for instance, Samet et al. [21], Agarwal et al. [22] and Roldán et al. [6], Ding et al. [23], Karapınar [24] and Karapınar and Roldán [25]). Furthermore, in some cases, it is not necessary to consider a partial order, but a preorder (see Roldán et al. [7]).
Recently, in 2013, Batra and Vashistha [26] introduced the concept of -invariant set which is a generalization of an F-invariant set introduced by Samet and Vetro [19] and proved the existence of coupled common fixed point theorems for nonlinear contractions under c-distance in cone metric spaces having an -invariant subset.
More recently, Charoensawan [27], based on Batra and Vashistha’s results, introduced the tripled case as follows.
Definition 18 (Charoensawan [27])
Let be a metric space and be a given mapping. Let M be a nonempty subset of . We say that M is an F-invariant subset of if, and only if, for all ,
The following concept is an extension of Definition 16.
Definition 19 (Charoensawan [27])
Let be a metric space and M be a subset of . We say that M satisfies the transitive property if, and only if, for all ,
Definition 20 (Charoensawan [27])
Let be a metric space and , be given mappings. Let M be a nonempty subset of . We say that M is an -invariant subset of if, and only if, for all ,
Notice that in the previous definitions, it is not necessary to consider neither a metric nor a partial order on X. Then this author proved the following result.
Theorem 21 (Charoensawan [27], Theorem 3.7)
Let be a complete metric space and M be a nonempty subset of . Assume that there is a function with and for each , and also suppose that and are two continuous functions such that
for all with or . Suppose that , g commutes with F.
If there exists such that
and M is an -invariant set which satisfies the transitive property, then there exist such that
Not knowing them and independently from Charoensawan’s results, Kutbi et al. [28] used a bidimensional extension of F-invariant subset as follows.
Definition 22 (Kutbi et al. [28])
We say that M is an F-closed subset of if, for all ,
Then they succeeded in proving some results using this property rather than the mixed monotone property and the concept of F-invariant set. For instance, the following one.
Theorem 23 (Kutbi et al. [28])
Let be a complete metric space, let be a continuous mapping and let M be a subset of . Assume the following.
-
(i)
M is F-closed;
-
(ii)
there exists such that ;
-
(iii)
there exists such that for all , we have
Then F has a coupled fixed point.
The following two lemmas can be found in the literature, but we recall them here for the sake of completeness.
Lemma 24 Let be a mapping and let be a sequence.
-
1.
If and for all m, then .
-
2.
If and is a sequence verifying for all m and , then .
Lemma 25 If is a sequence on a metric space that is not Cauchy, then there exist and two subsequences and such that, for all :
Furthermore, if , then
In this paper we observe that if is F-invariant and has the transitive property, we could induce a preorder on such that Theorem 17 can be seen as an easy consequence of Theorem 10. Actually, we will present an unidimensional result that can be particularized, following well-known techniques, to the multidimensional case.
3 Main results
Before showing our main results, some remarks must be done. Firstly, consider the family
This family of control functions was employed by Sintunaravat et al. in Theorem 17 and by Charoensawan in Theorem 21. However, notice that it is not as general as Wang’s family Φ because the value is not necessarily determined if . In this sense, .
Secondly, notice that Charoensawan’s notion of F-invariant set is similar to Kutbi et al.’s notion of F-closed set, but it is different from Samet and Vetro’s original concept because property (i) in Definition 14 is not imposed. Then, coherently with Definition 22, we prefer calling these subsets employing the term F-closed.
Definition 26 Let be two mappings and let be a subset. We will say that M is
-
-closed if for all such that ;
-
-compatible if for all such that .
Definition 27 We will say that a subset is transitive if implies that .
Definition 28 Let be a metric space and let be a subset. We will say that is regular if for all sequence such that and for all m, we have for all m.
We introduce a notion of continuity weaker than the usual concept.
Definition 29 Let be a metric space, let be a subset and let . A mapping is said to be M-continuous at x if for all sequence such that and for all m, we have . T is M-continuous if it is M-continuous at each .
Remark 30 Every continuous mapping is also M-continuous, whatever M.
In order to avoid the commutativity condition of the mappings T and g, and inspired by Definition 7, we present the following notion of -compatibility.
Definition 31 Let be a metric space and let be a subset. Two mappings are said to be -compatible if
provided that is a sequence in X such that for all and
Remark 32 If T and g are commuting, then they are also -compatible, whatever M.
The main result of this paper is the following one.
Theorem 33 Let be a complete metric space, let be two mappings such that and let be a -compatible, -closed, transitive subset. Assume that there exists such that
Also assume that, at least, one of the following conditions holds.
-
(a)
T and g are M-continuous and -compatible;
-
(b)
T and g are continuous and commuting;
-
(c)
is regular and gX is closed.
If there exists a point such that , then T and g have, at least, a coincidence point.
Proof Starting from such that , we have , so there exists such that . Therefore . As M if -closed, . Again, since , there exists such that . Therefore . As M if -closed, . Repeating this process, there exists a sequence such that
If there exists such that , then , so is a coincidence point of T and g. In this case, the proof is finished. On the contrary, assume that
As for all m, the contractivity condition (5) implies that
Applying (7) and item 1 of Lemma 24 to , we deduce that
Next, we show that is a Cauchy sequence reasoning by contradiction. Suppose that is not a Cauchy sequence. Taking into account (8), a well-known reasoning guarantees that there exist and two partial subsequences and such that
Since M is transitive, it follows from (6) that for all k. Let us apply the contractivity condition (5) to and , and we get, for all k,
By (9) and (10), is a sequence of real numbers, stricly greater than , that converges to . In particular, since ,
Letting in (11) and using (10) and (12), we deduce that
which is impossible. This contradiction proves that is a Cauchy sequence.
As is complete, there is such that . Next we distinguish between hypotheses (a), (b), and (c).
Case (a). Assume that T and g are M-continuous and -compatible. Since and for all , we have and . Furthermore, as T and g are -compatible, we have . Therefore,
which means that x is a coincidence point of T and g.
Case (b). By Remarks 30 and 32, if T and g are continuous and commuting, then they are also M-continuous and -compatible, so item (a) is applicable.
Case (c). Now, suppose that is regular and gX is closed. As and gX is closed, then , that is, there is such that . Taking into account that is regular, then for all m. Applying the contractivity condition (5), we have
Now we distinguish whether or not. If , item 2 of Lemma 24 (applied to and for all m) ensures us that , that is, converges to Tz. In this case, , so z is a coincidence point between T and g. Next, suppose that and we are going to show that
-
If , then since .
-
Suppose that there is some such that . Then and . Taking into account that M is -compatible, we deduce that . Therefore and (13) also holds for .
In any case, (13) holds for all . Therefore and, by the unicity of the limit, . □
If one takes in Theorem 33, where and for all , and remove the transitivity condition, we get the following statement.
Corollary 34 Let be a complete metric space, let be two mappings such that and let be a -compatible, -closed set. Assume that there exists such that
Also assume that, at least, one of the following conditions holds.
-
(a)
T and g are M-continuous and -compatible;
-
(b)
T and g are continuous and commuting;
-
(c)
is regular and gX is closed.
If there exists a point such that , then T and g have, at least, a coincidence point.
Proof We use the same structure as in the proof of Theorem 33. By following its lines, we derive
and, therefore,
By standard techniques, one can easily deduce that is a Cauchy sequence (notice that it is not necessary the transitivity condition because we do not need to apply the contractivity condition to ). Since is complete, there is such that . The rest is the same as in the proof of Theorem 33. □
We particularize the previous theorem, obtaining the following version of Theorem 12, in which a partial order is not necessary.
Corollary 35 Let be a complete metric space and let ≼ be a transitive relation on X. Let be two mappings such that and T is -nondecreasing. Suppose that there exists such that
Also suppose that
Assume that either
-
(a)
T and g are continuous and commuting, or
-
(b)
is regular and gX is closed.
If there exists a point such that , then T and g have, at least, a coincidence point.
Proof Consider the subset . Then the following properties hold.
-
M is nonempty because .
-
M is transitive because ≼ is so.
-
If are such that , then . Since T is -nondecreasing, we have , that is, . Therefore, M is -closed.
-
Let be such that and . If , then , so . On the contrary, if ≼ is antisymmetric, since T is -nondecreasing, we have
In any case, M is -compatible.
-
It is clear that is regular when is regular.
Then M verifies all conditions of Theorem 33 and it guarantees that T and g have, at least, a coincidence point. □
In the previous result, the condition does not mean that necessarily . For instance, if we define for all , then we have and .
Corollary 36 Theorem 12 immediately follows from Theorem 33.
Proof The result follows from Corollary 35 when we observe that if is regular, then is also regular (where is defined in Theorem 12 and ⊑ is given by (1)). □
As we have pointed out in Preliminaries, recently, some authors have showed that many coupled, tripled, quadrupled and multidimensional fixed point results can be reduced to the unidimensional case. Next we show how coupled and tripled coincidence point results (for instance, Theorems 17 and 21) can be directly deduced from the ‘unidimensional’ Theorem 33.
We introduce the following notation. Let be a partially ordered space and let . Given two mappings and , define
Clearly, a coincidence point between F and g is nothing but a coincidence point between and .
A nonempty subset M of is -closed if M is -closed, that is,
-
: ;
-
: .
Corollary 37 Theorem 17 immediately follows from Theorem 33.
Proof It is only necessary to consider g as the identity mapping on X and the metric on given by
□
Define
-
: ;
-
: .
Definition 38 We will say that is -compatible if is -compatible (that is, for all such that ).
Lemma 39 Let , let and be two mappings and let be a nonempty subset of . Then the following properties hold.
-
1.
is reflexive whatever .
-
2.
satisfies the transitive property if, and only if, is a preorder (reflexive and transitive) on .
Next, assume that is -compatible.
-
3.
is -closed if, and only if, the mapping is -nondecreasing.
-
4.
If is F-invariant, then the mapping is -nondecreasing.
Proof We include the proof assuming that because the case is exactly the same.
-
(1)
It is obvious. (2) Suppose that satisfies the transitive property and let and . If or , then it is apparent that . In other case, and . Since satisfies the transitive property, then , which means that . Thus, is transitive, that is, a preorder on . The converse is similar.
-
(3)
Suppose that is -closed. Let be such that , that is, , which means that either or . If , then because is -compatible. On the other case, if , then because is -closed. This is equivalent to , that is, . Therefore, is -nondecreasing.
Assume that is -nondecreasing and let such that . In this case, . As is -nondecreasing, we have , which means that either or .
-
(4)
It follows from the fact that M is also F-closed. □
Corollary 40 Theorem 21 immediately follows from Theorem 33.
Proof It follows from Lemma 39 and Corollary 35 considering the subset and taking into account the following facts:
-
if and if is given, then , where for all ;
-
when d is a complete metric on X, the mapping , defined by
is a complete metric on .
□
In the same way, the following result can be proved using the -compatibility involved in Theorem 33.
Corollary 41 Theorem 11 immediately follows from Theorem 33.
Remark 42
-
1.
Notice that the main results in [28] do not assume that M verifies the transitive property. Therefore, they cannot be directly deduced from Theorem 33. However, they are consequences of Corollary 34.
-
2.
Finally, we point out that, following well-known techniques (which can be found in [7], Section 6), the unidimensional Theorem 33 can be particularized to the multidimensional case in which X is replaced by , and generating a large list of coupled, tripled and quadrupled possible results.
Example 43 Let provided with the Euclidean metric and let define the subset , the mappings and the function by
Since all conditions of Theorem 33 hold, then T and g has a coincidence point. In this case, T and g are continuous, and the only coincidence point between T and g is .
Example 44 Let be provided with the Euclidean metric and let us define the subset , the mappings , and the function by
If we take , we have . Moreover, . If , we distinguish three cases.
-
If , then
-
If and , then
-
If , then .
Notice that T and g are not continuous on X, but is regular and gX is closed. Since all conditions of Theorem 33 hold, then T and g has a coincidence point. In this case, all points in are coincidence points between T and g.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The third author, therefore, acknowledges with gratitude DSR for financial support. The second author has been partially supported by Junta de Andalucía by Project FQM-268 of the Andalusian CICYE.
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Karapınar, E., Roldán, A., Shahzad, N. et al. Discussion of coupled and tripled coincidence point theorems for φ-contractive mappings without the mixed g-monotone property. Fixed Point Theory Appl 2014, 92 (2014). https://doi.org/10.1186/1687-1812-2014-92
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DOI: https://doi.org/10.1186/1687-1812-2014-92