Abstract
Recently, some (common) multidimensional fixed theorems in partially ordered complete metric spaces have appeared as a generalization of existing (usual) fixed point results. Unexpectedly, we realized that most of such (common) coupled fixed theorems are either weaker or equivalent to existing fixed point results in the literature. In particular, we prove that the results included in the very recent paper (Charoensawan and Thangthong in Fixed Point Theory Appl. 2014:245, 2014) can be considered as a consequence of existing fixed point theorems on the topic in the literature.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
Multidimensional fixed point theory was initiated in 2006 by Gnana Bhaskar and Lakshmikantham [1]. In fact, the authors [1] investigated the existence and uniqueness of a coupled fixed point of certain operators in the context of a partially ordered set to solve a periodic boundary value problem. Since then, multidimensional fixed point theorems have been investigated heavily by several authors; see, e.g., [1–29] and related references therein.
In this short note, we underline the fact that most of the multidimensional fixed point theorems can be derived from the existing (uni-dimensional) fixed point results in the literature. In particular, we shall show that the result in the recent report [6] can be considered in this frame.
For the sake of completeness, we recollect some basic definitions, notations and results on the topic in the literature. Throughout the paper, let X be a nonempty set. Given a positive integer n, let be the product space . Let be the set of all nonnegative integers. In the sequel, n, m and k will be used to denote nonnegative integers. Unless otherwise stated, ‘for all n’ will mean ‘for all ’.
Definition 1.1 (Roldán and Karapınar [22])
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 imply ), 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 1.2 A point is:
-
a coupled coincidence point of F and g if ,
-
a tripled coincidence point of F and g if ,
-
a quadrupled coincidence point of F and g if ,
Notice that when we take g as the identity mapping on X, then a point verifying the related conditions above is a coupled (respectively, tripled, quadrupled) fixed point of F due to Gnana Bhaskar and Lakshmikantham [1] (respectively, Berinde and Borcut [9], Karapınar [13]).
Definition 1.3 (Al-Mezel et al. [21])
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 [28], -nondecreasing mappings were called g-isotone mappings (in particular, when X is a product space ).
Definition 1.4 A fixed point of a self-mapping is a point such that . A coincidence point between two mappings is a point such that . A common fixed point of is a point such that .
Definition 1.5 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 1.1 If are commuting and is a coincidence point of T and g, then is also a coincidence point of T and g.
In 2003, Ran and Reurings characterized the Banach contraction mapping principle in the context of partially ordered metric space.
Theorem 1.1 (Ran and Reurings [20])
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.
After Ran and Reurings’ result, fixed point theorems have been investigated heavily. One of the interesting results in this direction was reported by Nieto and Rodríguez-López in [19], who slightly modified the hypothesis of the previous result swapping condition (c) with the fact that is nondecreasing-regular as follows.
Definition 1.6 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 is d-convergent to , we have that (respectively, ) for all m. And is regular if it is both nondecreasing-regular and nonincreasing-regular.
Inspired by Boyd and Wong’s theorem [10], Mukherjea [18] introduced the following kind of control functions:
and 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 1.2 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.
A partial order ≼ on X can be extended to a partial order ⊑ on defining, for all ,
An interesting generalization of the previous result was given by Wang in [28] using this extended partial order on .
Theorem 1.3 (Wang [28], Theorem 3.4)
Let be a partially ordered set and suppose that 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.
For further generalizations of the previous result, we refer readers to papers of Romaguera [25] and in Al-Mezel et al. [21].
Gnana Bhaskar and Lakshmikantham introduced the following condition in order to guarantee the existence of coupled fixed points
Definition 1.7 (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 ,
On the other hand, Samet and Vetro [26] succeeded in proving some results in which the mapping F did not necessarily have the mixed monotone property.
Definition 1.8 (Samet and Vetro [26])
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 [26].
Theorem 1.4 (Samet and Vetro [26])
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 .
Furthermore, Sintunavarat et al. [27] introduced the notion of transitive property to reconsider the Lakshmikantham and Ćirić’s theorem (see [17]) without the mixed monotone property.
Definition 1.9 (Sintunavarat et al. [27])
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 1.5 (Sintunavarat et al. [27])
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)
if 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.
Recently, Charoensawan [11], based on Batra and Vashistha’s results, introduced the tripled case as follows.
Definition 1.10 (Charoensawan [11])
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 1.9.
Definition 1.11 (Charoensawan [11])
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 1.12 (Charoensawan [11])
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 ,
In the previous definitions, it is not necessary to consider either a metric or a partial order on X.
Theorem 1.6 (Charoensawan [11], 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
Meanwhile, Kutbi et al. [16] used a bidimensional extension of an F-invariant subset as follows.
Definition 1.13 (Kutbi et al. [16])
We say that M is an F-closed subset of if, for all ,
The following one is the main result of Kutbi et al. [16].
Theorem 1.7 (Kutbi et al. [16])
Let be a complete metric space, let be a continuous mapping, and let M be a subset of . Assume that:
-
(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.
2 Main results
In this section we shall indicate our main result. Before stating the main theorem, we give necessary remarks. First of all, we consider the following family:
Notice that this family of control functions was employed by Sintunavarat et al. in Theorem 1.5 and by Charoensawan in Theorem 1.6. Here, we should mention that it is not as general as Wang’s family Φ since the value is not necessarily determined if . Thus, we have in this sense.
Secondly, we pay attention to the following fact: 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 1.8 is not imposed. Then, coherently with Definition 1.13, we prefer calling these subsets employing the term F-closed.
Definition 2.1 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 2.2 ([29])
We will say that a subset is transitive if implies that .
Definition 2.3 ([29])
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 that for all m.
Definition 2.4 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 2.1 If T and g are commuting, then they are also -compatible, whatever M.
The main result in [29], using the previous notions, is the following one.
Theorem 2.1 (Karapınar et al. [29])
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.
The following one is the main result of [6].
Theorem 2.2 (Charoensawan and Thangthong [6], Theorem 3.1)
Let be a partially ordered set and M be a nonempty subset of , and let d be a metric on X such that is a complete metric space. Assume that are two generalized compatible mappings such that G is continuous, and for any , there exist such that , , and . Suppose that there exists such that the following holds:
for all with
Also suppose that either
-
(a)
F is continuous or
-
(b)
for any three sequences , and with
and
for all implies
If there exist such that
and M is an -closed, then there exist such that , , and , that is, F and G have a tripled point of coincidence.
The following remarks must be done in order to clarify some facts stated in [6] to the reader.
-
In the previous theorem, the authors assumed that is a partially ordered set. Clearly, it is a superfluous hypothesis.
-
We understand that ‘’ is an erratum and that it must be replaced by ‘’.
-
In [6], Example 3.2 is invalid since does not necessarily belong to when are arbitrary.
Let . It is easy to show that the mappings , defined by
for all , are metrics on Y.
Now, given a mapping , let us define the mapping by
It is simple to show the following properties.
Lemma 2.1 (see, e.g., [7])
The following properties hold:
-
(1)
is complete if and only if (and ) is complete;
-
(2)
F has the mixed monotone property if and only if is monotone nondecreasing with respect to ⪯;
-
(3)
is a tripled fixed point of F if and only if is a fixed point of .
-
(4)
is a tripled coincidence point of F and G if and only if is a coincidence point of and .
-
(5)
is a tripled common fixed point of F and G if and only if is a common fixed point of and .
As a consequence of the previous facts, next we show that Theorem 2.2 is not a true extension: indeed, it can be seen as a simple corollary of Theorem 2.1.
Theorem 2.3 Theorem 2.2 follows from Theorem 2.1.
Proof Notice that condition (3) is equivalent to
for all such that (notice that ). By Lemma 2.1, all conditions of Theorem 2.1 are satisfied. □
3 Final remarks
In this section, we underline that the common/coincidence point theorem in [6] can be concluded as a fixed point theorem. For this purpose, we first recall the following crucial lemma.
Lemma 3.1 ([30])
Let X be a nonempty set and be a function. Then there exists a subset such that and is one-to-one.
Theorem 3.1 Let be a complete metric space, let be a mapping, and let be a T-closed, transitive subset. Assume that there exists such that
Assume that either
-
(a)
T is continuous, or
-
(b)
is regular.
If there exists a point such that , then T and g have, at least, a fixed point.
We skip the proof of this theorem since it can be considered as a special case of Theorem 2.1. Indeed, if we take g as the identity map on X, we conclude the result. On the other hand, by the following lemma, we shall show that Theorem 2.1 can be derived from Theorem 3.1.
Theorem 3.2 Theorem 2.1 is a consequence of Theorem 3.1.
Proof By Lemma 3.1, there exists such that and is one-to-one. Define a map by . Since g is one-to-one on , we conclude that h is well defined. Note that
for all . Since is complete, by using Theorem 3.1, there exists such that . Hence, T and g have a point of coincidence. It is clear that T and g have a unique common fixed point whenever T and g are weakly compatible. □
From Theorem 2.3 and Theorem 3.2 we conclude the following result.
Theorem 3.3 Theorem 2.2 is a consequence of Theorem 3.1.
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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-López-de-Hierro, AF. A note on ‘-Closed set and tripled point of coincidence theorems for generalized compatibility in partially metric spaces’. J Inequal Appl 2014, 522 (2014). https://doi.org/10.1186/1029-242X-2014-522
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DOI: https://doi.org/10.1186/1029-242X-2014-522