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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 287492, 13 pages

http://dx.doi.org/10.1155/2014/287492
Research Article

Discussion on “Multidimensional Coincidence Points” via Recent Publications

1Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia

2Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

3University of Jaén, Campus las Lagunillas s/n, 23071 Jaén, Spain

Received 17 March 2014; Accepted 23 March 2014; Published 8 May 2014

Academic Editor: Jen-Chih Yao

Copyright © 2014 Saleh A. Al-Mezel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show that some definitions of multidimensional coincidence points are not compatible with the mixed monotone property. Thus, some theorems reported in the recent publications (Dalal et al., 2014 and Imdad et al., 2013) have gaps. We clarify these gaps and we present a new theorem to correct the mentioned results. Furthermore, we show how multidimensional results can be seen as simple consequences of our unidimensional coincidence point theorem.

1. Introduction and Preliminaries

In the sequel, will be a nonempty set and will represent a partial order on . Given with , let denote by the product space of identical copies of .

In [1], Guo and Lakshmikantham introduced the notion of coupled fixed point and, thus, they initiated the investigation of multidimensional fixed point theory.

Definition 1 (Guo and Lakshmikantham [1]). Let be a given mapping. We say that is a coupled fixed point of if

Following this initial paper [1], in 2006, Bhaskar and Lakshmikantham [2] obtained some coupled fixed point theorems for mapping (where is a partially ordered metric space) by defining the notion of mixed monotone mapping.

Definition 2 (see [2]). Let be a partially ordered set. A mapping . is said to have the mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in ; that is, for any ,

After that, Lakshmikantham and Ćirić [3] proved coupled fixed/coincidence point theorems for mappings and by introducing the concept of the mixed -monotone property. Inspired by these papers [2, 3], Berinde and Borcut defined tripled fixed points and established some tripled fixed point theorems.

Definition 3 (Berinde and Borcut [4]). Let be a given mapping. We say that is a tripled fixed point of if

Definition 4 (see [4]). Let be a partially ordered set and . We say that has the mixed monotone property if is monotone nondecreasing in and and it is monotone nonincreasing in ; that is, for any

As a natural extension, Karapınar [5] studied the quadruple case (see also [6, 7]).

Definition 5 (see [5]). An element is called a quadruple fixed point of if

Definition 6 (see [5]). Let be a partially ordered set and . We say that has the mixed monotone property if is monotone nondecreasing in and and it is monotone nonincreasing in and ; that is, for any

When a mapping is involved, we have the notion of coincidence point. We will only recall the corresponding definitions in the quadruple case since they are similar in other dimensions.

Definition 7 (see [6]). An element is called a quadrupled coincident point of the mappings and if

Definition 8 (see [6]). Let be a partially ordered set and let and be two mappings. We say has the mixed -monotone property if is -nondecreasing in and and is -nonincreasing in and ; that is, for any ,

It is very natural to extend the definition of 2-dimensional fixed point (coupled fixed point), 3-dimensional fixed point (tripled fixed point), and so on to multidimensional fixed point ( -tuple fixed point) (see, e.g., [819]). In this paper, we give some remarks on the notion of -tuple fixed point given in several papers, such as Imdad et al. [9], Dalal et al. [8], and Ertürk and Karakaya [20, 21]. Notice that this paper can be considered as a continuation of Karapınar and Roldán [10, 22]. We note also that authors preferred to say “ -tuplet fixed point” [20, 21] or “ -tuplet fixed point” [8, 9] instead of “ -tuple fixed point”.

Definition 9 (see [8, 9, 20]). An element is called an -tuple fixed point of the mapping if

Definition 10 (see [8, 9, 20]). Let be a partially ordered set and let be a mapping. We say has the mixed monotone property if is nondecreasing in odd arguments and is nonincreasing in its even arguments; that is, for any ,

Definition 11 (see [8, 9, 20]). An element is called an -tuple coincidence point of the mappings and if

Definition 12 (see [8, 9, 20]). Let be a partially ordered set and let and be two mappings. We say has the mixed -monotone property if is -nondecreasing in odd arguments and is -nonincreasing in its even arguments; that is, for any ,

Using these preliminaries, the following result was announced in [9]. Notice that in that paper, the authors used the notation to refer to the product space .

Theorem 13 (Imdad et al. [9], Theorem 13). Let be a partially ordered set equipped with a metric such that is a complete metric space. Assume that there is a function with and for all . Further, suppose that and are two maps such that has the mixed -monotone property satisfying the following conditions:(i) , (ii) is continuous and monotonically increasing,(iii) is a commutating pair,(iv) for all with , , ,…, .

Also, suppose that either(a) is continuous or(b) has the following properties: if a nondecreasing sequence , then for all ; if a nonincreasing sequence , then for all .

If there exist such that then and have a -tupled coincidence point; that is, there exists such that

Based on this theorem, Dalal et al. [8] extended the previous result to compatible mappings in the following sense.

Definition 14 (Dalal et al. [8]). Let be a metric space provided with a partial order and let and be two mappings. We will say that is a compatible pair if whenever are sequences in such that for some .

Theorem 15 (Dalal et al. [8], Theorem 3.2). Let be a partially ordered set equipped with a metric such that is a complete metric space. Assume that there is a function with and for all . Further, let and be two maps such that has the mixed -monotone property satisfying the following conditions:(i) , (ii) is continuous and monotonically increasing,(iii)the pair is compatible,(iv) for all with , , ,…, .

Also, suppose that either(a) is continuous or(b) has the following properties: if a nondecreasing sequence , then for all ; if a nonincreasing sequence , then for all . If there exists such that then and have a -tupled coincidence point; that is, there exists such that

2. Some Remarks

Firstly we notice that, in the case , Definitions 9 and 11, do not extend the notion of tripled coincidence point in the sense of Berinde and Borcut [4]. Therefore, their results are not extensions of well-known results in the tripled case. This fact shows that the odd case is not well-posed by Definitions 9 and 11 or, more precisely, the mixed monotone property is not useful to ensure the existence of coincidence points. In this sense, we have the following result.

Theorem 16. Theorem   in [9] is not valid if is odd.

Proof. It is sufficient to examine the case to indicate the mentioned invalidity. It is evident that the illustrative proof for the case can be analogously extended to the case in which is odd. We follow the lines of the proof of Theorem  3.1 in [8]. Let be the initial points. We construct three recursive sequences , , and in the following way: Due to the assumption, we derive that Then, the authors concluded that these sequences verify, for all , Now, we will show that it is impossible to prove that because the mixed -monotone property leads to contrary inequalities. Indeed, we derive the following inequalities: Furthermore, By combining the inequalities above, we conclude that Notice that in the third component the inequality is on the contrary Then, we find that Consequently, we cannot get the inequality , since other possibilities yield to another cases in which points are not comparable.

By using the same argument above, we also conclude that Corollaries and   in [9] are not valid. Similarly, we may prove the following result.

Corollary 17. Theorem  3.1 in [8] is not valid if is odd.

In Theorem 16, we investigate the case in which is odd. But we must emphasize that, when is even, the main results of Dalal et al. [8] are also very weak. To prove it, we show the following example inspired by [23].

Example 18. Let be the set of all real numbers provided with its usual order and the Euclidean metric for all . Let and be the mappings given by It is easy to check that the contractivity condition of Theorem 16 is not satisfied. Indeed, consider , , , and . Then, we have that Thus, it is impossible to find (as it was defined in [8]) such that However, it is clear that is the only common -tuple fixed point of and .

3. Corrected Versions of the Mentioned Theorems

For the sake of completeness and to conclude this paper, in this section, we state a corrected version of Theorem  3.1 in [8], which immediately leads to a corrected version of Theorem  13 in [9]. For this purpose, we recollect here some notations, definitions, and results from the literature (that can also be found in [10, 1416]).

First at all, instead of Definitions 9 and 11, we recall here the concept of multidimensional fixed/coincidence point introduced by Roldán et al. in [13] (see also [1416]), which is an extension of Berzig and Samet’s notion given in [12].

Throughout this section, fix such that and let and be two mappings. Fix a nontrivial partition of ; that is, and are nonempty subsets of such that and . We will denote Henceforth, let be mappings from into itself and let be the -tuple .

Definition 19 (Roldán et al. [13, 16]). A point is called a -coincidence point of the mappings and if If is the identity mapping on , then is called a -fixed point of the mapping .

It is clear that the previous definition extends the notions of coupled, tripled, and quadruple fixed/coincidence points. In fact, if we represent a mapping throughout its ordered image, that is, , then(i)Gnana-Bhaskar and Lakshmikantham’s coupled fixed points occur when , , and ;(ii)Berinde and Borcut’s tripled fixed points are associated with , , , and ;(iii)Karapınar’s quadruple fixed points are considered when , , , , and ;(iv)Berzig and Samet’s multidimensional fixed points are given when and .

For more details see [13].

A partial order on can be extended to a partial order on in the following way. If is a partially ordered space, and , we will use the following notation: Consider on the product space the following partial order: for , We say that two points and are comparable if or .

Using this partial order, the mixed -monotone property is as follows.

Definition 20 (see [13]). Let be a partially ordered space. We say that has the mixed -monotone property (with respect to ) if is -monotone nondecreasing in arguments of and -monotone nonincreasing in arguments of ; that is, for all and all ,

Remark 21 (see [10]). In order to ensure the existence of -coincidence/fixed points, it is very important to assume that the mixed -monotone property is compatible with the permutation of the variables; that is, the mappings of should verify

Remark 22 (see [10]). Notice that, in fact, when is even, Definitions 11 and 12 can be seen as particular cases of the previous definitions when is the set of all odd numbers and is the family of all even numbers in and the mappings are appropriate permutations of the variables.

The following definitions are usual in the field of fixed point theory.

Definition 23. An ordered metric space is a metric space provided with a partial order .

Definition 24 (see [2]). An ordered metric space is said to be nondecreasing-regular (resp., nonincreasing-regular) if we have that (resp., ) for all when is any sequence verifying and (resp., ) for all . And is said to be regular if it is both nondecreasing-regular and nonincreasing-regular.

Definition 25. Let be a partially ordered set and let be two mappings. We will say that is monotone -nondecreasing if for all such that .

Remark 26. If is -nondecreasing and , then . It follows from

Lemma 27 (see [16]). Let be a metric space and define , for all , by Then is metric on . And is complete if, and only if, is complete.

Lemma 28 (see [16]). Let be an ordered metric space and let and be two mappings. Let be an -tuple of mappings from into itself verifying if and if  . Define ,   for all , by (1)If has the mixed -monotone property, then is monotone -nondecreasing.(2)If is -continuous, then is also -continuous.(3)If is -continuous, then is -continuous.(4)A point is a -fixed point of if, and only if, is a fixed point of .(5)A point is a -coincidence point of and if, and only if, is a coincidence point of and .(6)If is regular, then is also regular.

The commutativity and compatibility of the mappings are defined as follows.

Definition 29. We will say that two mappings are commuting if for all . We will say that and are commuting if for all .

The following notion was introduced in order to avoid the necessity of commutativity.

Definition 30 (see [2426]). Let be an ordered metric space. Two mappings are said to be O-compatible if provided that is a sequence in such that is -monotone and

The natural extension to an arbitrary number of variables is the following one.

Definition 31. Let be an ordered metric space and let and be two mappings. Let be an -tuple of mappings from into itself verifying if and if . We will say that is a -compatible pair if whenever are sequences in such that are -monotone and

Notice that the previous definition is different from Definition 14 because we impose that the sequences are -monotone.

Lemma 32. If and are -compatible, then and are -compatible.

Inspired by Boyd and Wong’s theorem [27], Mukherjea [28] introduced the following kind of control functions: The following property is well-known.

Lemma 33. Let and let be a sequence. If and for all , then .

Using this kind of control functions, we present the following theorem.

Theorem 34. Let be an ordered metric space and let be two mappings such that the following properties are fulfilled;(i) ;(ii) is monotone -nondecreasing;(iii)there exists such that ;(iv)there exists verifying Also assume that, at least, one of the following conditions holds:(a) is complete, and are continuous, and the pair is -compatible;(b) is complete and and are continuous and commuting;(c) is complete and is nondecreasing-regular;(d) is complete, is closed, and is nondecreasing-regular;(e) is complete, is continuous and monotone -nondecreasing, the pair is -compatible, and is nondecreasing-regular.Then and have, at least, a coincidence point.

Proof. We divide the proof into four steps.

Step  1. We claim that there exists a sequence such that is -nondecreasing and for all . Starting from given in and taking into account that , there exists such that . Then . Since is -nondecreasing, . Now , so there exists such that . Then . Since is -nondecreasing, . Repeating this argument, there exists a sequence such that

Now, let us define for all .

Step  2. We claim that for all . Since for all , it follows from that

Step  3. We claim that . We consider two possibilities.(i)Suppose that there is such that . Then . Remark 26 guarantees that . By induction, the same reasoning proves that if there is such that , then for all and, in this case, it is clear that .(ii)Suppose that for all . In this case, by Lemma 33.

Step  4. We claim that is a Cauchy sequence. Let us show that is Cauchy reasoning by contradiction. Suppose that is not Cauchy. Then there exist and partial subsequences and verifying , , and for all ( is the least integer number, greater that , such that ). Since , we have . By (e), and using Step , Using the contractivity condition , Moreover Taking limit as in (53) and using , Step 3, and (51), we get the contradiction This contradiction proves that, in any case, is a Cauchy sequence. Now, we prove that and have a coincidence point distinguishing between cases (a)–(e).

Case . is complete, and are continuous, and the pair is -compatible. As is complete, there exists such that . Since for all , we also have that . As and are continuous, then and . Taking into account that the pair is -compatible, we deduce that . In such a case, we conclude that that is, is a coincidence point of and .

Case . is complete and and are continuous and commuting. It is obvious because implies .

Case . is complete and is nondecreasing-regular. As is a Cauchy sequence in the complete space , there is such that . Let be any point such that . In this case, . We are also going to show that , so we will conclude that (and is a coincidence point of and ).

Indeed, as is regular and is -nondecreasing and converging to , we deduce that for all . Applying the contractivity condition (iv),

We are going to show that (i)If , then because .(ii)Suppose that there is some such that . Remark 26 guarantees that . This proves that if there is some such that , then , so (57) also holds.

In any case, (57) holds and this implies that converges to . This completes this case.

Case . is complete, is closed, and is nondecreasing-regular. It follows from the fact that a closed subset of a complete metric space is also complete. Then, is complete and Case is applicable.

Case . is complete, is continuous and monotone -nondecreasing, the pair is -compatible, and is nondecreasing-regular. As is complete, there exists such that . As for all , we also have that . As is continuous, . Furthermore, as the pair is -compatible, then As , the previous property means that . We are going to show that and this finishes the proof.

Indeed, since is -nondecreasing, converges to , and is nondecreasing-regular, we have that for all . Moreover, as is monotone -nondecreasing, we deduce that for all . Applying the contractivity condition (iv), We claim that (i)If , then because .(ii)Suppose that there is some such that . Remark 26 guarantees that . This proves that if there is some such that , then , so (60) also holds.

In any case, (60) holds and this implies that converges to . This completes the proof.

Inspired by Berinde’s approach [23], we deduce the following result which removes the weakness of Theorem   in [8].

Corollary 35. Let be an ordered metric space, let and be two mappings, and let be an -tuple of mappings from into itself verifying  if and if . Suppose that the following properties are fulfilled:(i) ;(ii) has the mixed -monotone property;(iii)there exists such that for all ;(iv)there exists verifying for all such that for all .

Also assume that at least one of the following conditions holds;(a) is complete, and are continuous, and the pair is -compatible;(b) is complete and and are continuous and commuting;(c) is complete and is regular;(d) is complete, is closed, and is regular;(e) is complete, is continuous and monotone -nondecreasing, the pair is -compatible, and is regular.

Then and have, at least, a -coincidence point.

Proof. Notice that the contractivity condition (61) means that for all such that . Therefore, it is only necessary to apply Theorem 34 to the mappings defined in Lemma 28.

We now reconsider Example 18.

Example 36. Let be the set of all real numbers provided with its usual order and the Euclidean metric for all . Let