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Erdal Karapınar, Antonio Roldán, Juan Martínez-Moreno, Concepcion Roldán, "Meir-Keeler Type Multidimensional Fixed Point Theorems in Partially Ordered Metric Spaces", Abstract and Applied Analysis, vol. 2013, Article ID 406026, 9 pages, 2013. https://doi.org/10.1155/2013/406026
Meir-Keeler Type Multidimensional Fixed Point Theorems in Partially Ordered Metric Spaces
We study the existence and uniqueness of a fixed point of the multidimensional operators which satisfy Meir-Keeler type contraction condition. Our results extend, improve, and generalize the results mentioned above and the recent results on these topics in the literature.
Fixed point theory plays a crucial role in nonlinear functional analysis. In particular, fixed point results are used to prove the existence (and also uniqueness) when solving various type of equations. On the other hand, fixed point theory has a wide application potential in almost all positive sciences, such as Economics, Computer Science, Biology, Chemistry, and Engineering. One of the initial results in this direction (given by S. Banach), which is known as Banach fixed point theorem or Banach contraction mapping principle  is as follows. Every contraction in a complete metric space has a unique fixed point. In fact, this principle not only guarantees the existence and uniqueness of a fixed point, but it also shows how to get the desired fixed point. Since then, this celebrated principle has attracted the attention of a number of authors (e.g., see [1–39]). Due to its importance in nonlinear functional analysis, Banach contraction mapping principle has been generalized in many ways with regards to different abstract spaces. One of the most interesting results on generalization was reported by Guo and Lakshmikantham  in 1987. In their paper, the authors introduced the notion of coupled fixed point and proved some related theorems for certain type mappings. After this pioneering work, Gnana Bhaskar and Lakshmikantham  reconsidered coupled fixed point in the context of partially ordered sets by defining the notion of mixed monotone mapping. In this outstanding paper, the authors proved the existence and uniqueness of coupled fixed points for mixed monotone mappings and they also discussed the existence and uniqueness of solution for a periodic boundary value problem. Following these initial papers, a significant number of papers on coupled fixed point theorems have been reported (e.g., see [6, 11, 13, 19, 22, 23, 29, 31–33, 36, 38, 40]).
Following this trend, Berinde and Borcut  extended the notion of coupled fixed point to tripled fixed point. Inspired by this interesting paper, Karapınar  improved this idea by defining quadruple fixed point (see also [25–28]). Very recently, Roldán et al.  generalized this idea by introducing the notion of -fixed point, that is to say, the multidimensional fixed point.
Another remarkable generalization of Banach contraction mapping principle was given by Meir and Keeler . In the literature of this topic, Meir-Keeler type contraction has been studied densely by many selected mathematicians (e.g., see [2–4, 9, 20, 21, 36, 39]).
In this paper, we prove the existence and uniqueness of fixed point of multidimensional Meir-Keeler contraction in a complete partially ordered metric space. Our results improve, extend, and generalize the existence results on the topic in fixed point theory.
Preliminaries and notation about coincidence points can also be found in . Let be a positive integer. Henceforth, will denote a nonempty set, and will denote the product space . Throughout this paper, and will denote nonnegative integers and . Unless otherwise stated, “for all ” will mean “for all ” and “for all ” will mean “for all .”
A metric on is a mapping satisfying, for all , , , From these properties, we can easily deduce that and for all . The last requirement is called the triangle inequality. If is a metric on , we say that is a metric space (for short, an MS).
Definition 1 (see ). A triple is called a partially ordered metric space if is an MS and is a partial order on .
Definition 2 (see ). An ordered MS is said to have the sequential -monotone property if it verifies the following. (i) If is a nondecreasing sequence and , then for all . (ii) If is a nonincreasing sequence and , then for all .
If is the identity mapping, then is said to have the sequential monotone property.
Henceforth, fix a partition of ; that is, and . We will denote that If is a partially ordered space, , and , we will use the following notation:
Let and be two mappings.
Definition 3 (see ). We say that and are commuting if for all .
Definition 4 (see ). Let be a partially ordered space. We say that has the mixed -monotone property (w.r.t. ) if is -monotone nondecreasing in arguments of and -monotone nonincreasing in arguments of ; that is, for all , , and all ,
Henceforth, let be mappings from into itself, and let be the -tuple .
Definition 5 (see ). 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 .
Remark 6. If and are commuting and is a -coincidence point of and , then also is a -coincidence point of and .
With regards to coincidence points, it is possible to consider the following simplification. If is a permutation of and we reorder (5), then we deduce that every coincidence point may be seen as a coincidence point associated to the identity mapping on .
Lemma 7. Let be a permutation of , and let and . Then, a point is a -coincidence point of the mappings and if and only if is a -coincidence point of the mappings and .
Therefore, in the sequel, without loss of generality, we will only consider -coincidence points where , that is, that verify for all .
If one represents a mapping throughout its ordered image, that is, , then(i)Gnana-Bhaskar and Lakshmikantham's election in is and ;(ii) Berinde and Borcut's election in is , and ;(iii) Karapnar's election in is , , , and .
For more details, see . We will use the following result about real sequences in the proof of our main theorem.
Lemma 8. If is a sequence in an MS that is not Cauchy, then there exist and two subsequences and such that, for all , , , and .
Meir and Keeler generalized the Banach contraction mapping principle in the following way.
Definition 9 (Meir and Keeler ). A Meir-Keeler mapping is a mapping on an MS such that for all , there exists verifying that if and , then .
Lim characterized this kind of mappings in terms of a contractivity condition using the following class of functions.
Definition 10 (Lim ). A function will be called an L-function if (a) , (b) for all , and (c) for all , there exists such that for all .
Theorem 11 (Lim ). Let be an MS, and let . Then is a Meir-Keeler mapping if and only if there exists an (nondecreasing, right-continuous) L-map such that
Using a result of Chu and Diaz , Meir and Keeler  proved that every Meir-Keeler mapping on a complete MS has a unique fixed point. Since then, many authors have developed this notion in different ways (e.g., see [2–4, 9, 20, 21, 36, 39]). For instance, in , Samet introduces the concept of generalized Meir-Keeler type function as follows.
Definition 12 (see ). Let be a partially ordered metric space and a given mapping. We say that is a generalized Meir-Keeler type function if for all , there exists such that
Then, the author  proved some coupled fixed point theorems via generalized Meir Keeler type mappings. In this paper, we extend the notion of generalized Meir-Keeler type mappings in various ways and get some fixed point results by the help of these notions.
3. Multidimensional Meir-Keeler-Type Mappings
Henceforth, let be a partially ordered MS and let and be two mappings.
Definition 13. We will say that is a (multidimensional) -Meir-Keeler type mapping, ((MK) mapping) if it verifies the following two properties.
(MK1) If verify for all , then .
(MK2) For all , there exists such that if verify for all and
If is the identity mapping on , we will say that is a (-dimensional) Meir-Keeler type mapping.
On the one hand, notice that, in a wide sense, property (MK1) may be interpreted as property (MK2) for . On the other hand, we observe that our definition may not be compared with the original one due to Meir and Keeler since we assume that has a partial order. In any case, if , has a partial order and is the identity mapping on , and we can only establish that if is a Meir-Keeler mapping in the sense of Definition 9, then is a Meir-Keeler-type mapping in the sense of Definition 13, but the converse does not hold.
Remark 14. If is an injective mapping on , then all mappings verify (MK1).
Lemma 15. Let be a mapping on a partially ordered MS , and let be such that for all . (1)If verifies (MK2), then either for all or (2)If is a -Meir-Keeler type mapping, then and the equality is achieved if and only if for all .
Proof. If the condition “ for all ” does not hold, then . Hence, . If is a -Meir-Keeler-type mapping, the case “ for all ” means that the equality is achieved.
This global contractivity condition (10) is not strong enough to ensure that has a fixed point. For instance, if , then for all has no fixed point. In order to characterize this kind of mappings in different ways, we recall some definitions and results.
Definition 16. The -modulus of uniform continuity of is, for all ,
Remark 17. The identity mapping on a set will be denoted by . If is a mapping, then will be defined by for all . If is a metric space, then , given by for all , , is a metric on . A partial order on may be induced on by if and only if for all (notice that this partial order depends on the partition of ). Then, also is a partially ordered MS. Furthermore, given any , will denote the mapping defined by for all . It is obvious that for all .
Theorem 18. Let be a partially ordered MS, and let and be two mappings. Then, the following statements are equivalent.
(MK) is a -Meir-Keeler-type mapping.
(MK3) For all , there exists such that
(MK4) for all .
(MK5) and verify (MK1), and there exists an (nondecreasing, right-continuous) L-function such that for all verifying for all and .
(MK6) For all , the mapping is a -Meir-Keeler-type mapping on .
(MK7) There exists such that the mapping is a -Meir-Keeler-type mapping on .
Proof. [(MK)⇒(MK3)]: Fix , and let given by (MK2). Let be such that for all , and let . If , then for all , and so by (MK1). In another case, . If , then by (MK2). Now, suppose that . Then, , where is also given by (MK2), and . Hence, (MK3) holds.
[(MK3)⇒(MK4)]: Given , let verifying (MK3). Then, , and so .
[(MK4)⇒(MK)]: On the one hand, if for all , then for all , and so verify (MK1). On the other hand, let , and define . Therefore, . Since is a supremum, there exists such that if for all and , then . In particular, if for all and , then , .
[(MK)⇔(MK5)]: It is possible to follow step by step the proof of Proposition 1 in  with slight changes.
[(MK)⇔(MK6)⇔(MK7)]: It is apparent taking into account (12).
The following result is a particular case taking for all .
Corollary 19. Let be a partially ordered metric space, and let and be two mappings. Assume that there exists such that for all verifying for all . Then, is a -Meir-Keeler-type mapping.
Lemma 20. Every generalized Meir Keeler type function in the sense of Samet is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13 taking as the identity mapping on the MS.
Proof. Suppose that is a generalized Meir Keeler type function in the sense of Samet. Fix and let verifying (7). Let such that , , and . We have to prove that . If and , there is nothing to prove. Next, suppose that . Let
If , then and , which is false. Then, . On the other hand,
If , then by (7). Finally, if , taking in (7), we have that (where is taken as in (7)), and so . This proves that is a 2-dimensional Meir-Keeler type mapping associated to .
Remark 21. Converse of Lemma 20 does not hold. For instance, let be provided with its usual metric and partial order . Take and consider for all . Then, is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13 (taking as the identity mapping on ), but, if , it is not a generalized Meir Keeler type function in the sense of Samet.
Indeed, we firstly prove that is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13 (taking as the identity mapping on ). Let . Consider any (i.e., ) and define . Consider such that and verifying . In particular, . Then, It follows that is a 2-dimensional Meir-Keeler-type mapping in the sense of Definition 13. Next, we claim that if , then is not a generalized Meir Keeler type function in the sense of Samet. Let . If was a generalized Meir Keeler type function in the sense of Samet, it would be verifying (7). Take , , and . Then, , and However, since .
4. Main Results
In the following result, we show sufficient conditions to ensure the existence of -coincidence points, where .
Theorem 22. Let be a complete MS, and let a partial order on . Let be an -tuple of mappings from into itself verifying if and if . Let and be two mappings such that is a -Meir-Keeler-type mapping and has the mixed -monotone property on , , and is continuous and commuting with . Suppose that either is continuous or has the sequential -monotone property. If there exist verifying for all , then and have, at least, one -coincidence point.
Proof. The proof is divided in six steps. We follow the strategy of Theorem 9 in .
Step 1. There exist sequences , such that , for all and all .
Step 2. for all and all .
Define for all . Firstly, suppose that there exists such that . Then, for all , so is a -coincidence point of and and we have finished. Therefore, we may reduce to the case in which for all ; that is,
Step 3. We claim that for all (i.e., , ). Indeed, as for all and all , then condition (MK2), Lemma 15, and (20) imply that, for all and all , Taking maximum on , we deduce that the sequence is nonincreasing and lower bounded. Therefore, it is convergent; that is, there exists such that (and for all ). We claim that . On the contrary, assume that . Let be a positive number associated to by (MK2). Since there exists such that if , then . By (MK2), it follows that, for all , Taking maximum on , we deduce that But this is impossible. Then, we have just proved that . Therefore, , which means that As for all and all , then for all .
Step 4. Every sequence is Cauchy. Suppose that are not Cauchy and are Cauchy, being . By Lemma 8, for all , there exist and subsequences and such that Let and . Since are Cauchy, there exists such that if , , then for all .
Let such that , , and define , . As for some , there exists such that , . Thus, we can consider the numbers , until finding the first positive integer verifying
Now, let such that , and define , . Since , we can consider the numbers until finding the first positive integer verifying Repeating this process, we can find sequences such that, for all , Since , we know that , , for all . Therefore, for all , Let verifying (MK3) using , and let such that if , then for all . Then, for all and all , Applying (MK3), it follows, for all and all , that but this contradicts (30) since . This contradiction shows us that every sequence is Cauchy.
Existence of a fixed point is derived by standard techniques. Indeed, since is complete, there exist such that for all . As is continuous and commutes with ,
Step 5. Suppose that is continuous. In this case, since for all and is continuous, for all . Then, for all ; that is, is a -coincidence point of and .
Step 6. Suppose that has the sequential -monotone property. In this case, by step 2, we know that for all and all. This means that the sequence is monotone. As , we deduce that for all and all. This condition implies that, for all and all , (the first case occurs when and the second one when ). Fix , and we claim that . Indeed, let arbitrary, and let verifying (MK3). Since for all , there exists such that if , then for all . Applying (MK3) and (35), Therefore, for all . In conclusion, is a -coincidence point of and .
5. Uniqueness of -Coincidence Points
For the uniqueness of a fixed point, we need the following notion. Consider on the product space the following partial order: for , We say that two points and are comparable if or .
By following the lines of Theorem 11 in , we prove the uniqueness of the coincidence point.
Theorem 23. Under the hypothesis of Theorem 22, assume that for all -coincidence points , of and , there exists such that is comparable, at the same time, to and to .
Then, and have a unique -coincidence point such that for all .
It is natural to say that is injective on the set of all -coincidence points of and when for all implies for all when are two -coincidence points of and . For example, this is true that is injective on .
Corollary 24. In addition to the hypotheses of Theorem 23, suppose that is injective on the set of all -coincidence points of and . Then, and have a unique -coincidence point.
Corollary 25. In addition to the hypotheses of Theorem 23, suppose that is comparable to for all . Then, .
In particular, there exists a unique such that , which verifies .
Proof. Let and we are going to show that by contradiction. Assume that and let such that . As is comparable to , then either for all or for all . Since for all , we know that either for all or for all . Now, we have to distinguish between two cases.
If for all (i.e., for all ), then which is impossible since . Now, suppose that , . In this case, item 1 of Lemma 15 guarantees that which also is impossible. This contradiction proves that ; that is, for all .
Remark 26. Notice that a mixed strict monotone mapping in the sense of [36, Definition 2.1] is always a mixed monotone mapping in our sense (where and is the identity mapping on ). Then, Theorems 2.1, 2.2, 2.3, and 2.4 in  (and, by extension, theorems by Gnana Bhaskar and Lakshmikantham ) are consequence of our main results.
Example 27. Let and be usual metric on . Consider the mapping and . It is clear that is monotone nonincreasing in odd arguments and is monotone nondecreasing in even arguments. All conditions of Theorems 22 and 23 are satisfied. It is clear that is the unique fixed point.
Example 28. Let be provided with its usual partial order and its usual metric . Let , and let real numbers such that there exist verifying . Let , and consider and , for all . Then, is monotone nondecreasing in those arguments for which and monotone nonincreasing in those arguments for which . Furthermore, taking , Corollary 19 shows that is a -Meir-Keeler-type mapping. Actually, all conditions of Theorems 22 and 23 are satisfied. Indeed, it is clear that is the unique fixed point of .
Proof of Theorem 23
Proof. From Theorem 22, the set of -coincidence points of and is nonempty. The proof is divided in two steps.
Step 1. We claim that if are two -coincidence points of and , then