Imperative Remarks for “On Common Coupled Fixed Point Theorems for Comparable Mappings in Ordered Partially Metric Spaces” and an Answer to the Question: How to Smooth It Away

Abstract

We show that the main result in the work by Mutlu et al. is not true. We explain point by point some of its main mistakes and we propose an alternative version to smooth away the defects of it.

1. Introduction

Following Matthews [1], a partial metric on a nonempty set is a mapping verifying, for all , In this case, is called a partial metric space. Although the authors of [2] used the notation for a partial metric space, we prefer using in order to avoid confusion with the metric case. Every metric space is a partial metric space, but the converse is false. For a partial metric on , the mapping , given by is a metric on .

In [2], the authors introduced the following definition and announced the following theorem.

Definition 1 (Mutlu et al. [2], Definition 9). Assume that is a partially ordered set and . and mappings have the following properties:

Theorem 2 (Mutlu et al. [2], Theorem 10). Suppose that is a partially ordered set and is a partial metric on with being a complete partial metric space. Assume that are satisfied by Definition 1 and also are continuous mappings possessing the mixed monotone property on . Let there be a nonincreasing function such that and for all and also having and , with for . If there exists with and at the time, with and .

In this paper, we show that Definition 1 is not clear. Therefore, Theorem 2 is not well posed. Furthermore, its proof has many mistakes. We illustrate that it fails with an example. Finally, we propose a correct version of Theorem 2.

2. Preliminaries

To better understand our main claims, let us introduce the following definitions and notation. In the sequel, will be a nonempty set and will represent the product space of 2 identical copies of .

Definition 3. A binary relation on is a nonempty subset . One will write (or ) if . A binary relation on is reflexive if for all , and it is transitive if for all such that and . A reflexive and transitive relation on is a preorder (or a quasiorder) on . If a preorder is also antisymmetric ( and imply ), then is called a partial order.
In [3], Guo and Lakshmikantham introduced the notion of coupled fixed point and, thus, they initiated the investigation of multidimensional fixed point theory.

Definition 4 (Guo and Lakshmikantham [3]). Let be a given mapping. We say that is a coupled fixed point of if

Definition 5. Given two mappings , we say that is a common coupled fixed point of and if
Henceforth, we will use the notation Functions in are called comparison functions.

3. Main Remarks about Theorem 2

In the following lines, we must do the following commentaries in order to advise researchers about proving new results based on Theorem 2.(1)First of all, we point out that Definition 1 is not clear because it does not explain how the sequences and are. If they are arbitrary, then, for all ,   Therefore, , so for all . Hence, both mappings are constant, and the result is not interesting at all.(2)As a consequence, Theorem 2 was incorrectly proved. Precisely, its proof collects very different mistakes.(3)Although Theorem 2 assumes that and have the mixed monotone property, this condition was not used through its proof. We suppose that it is not necessary. Only Definition 1 is employed to prove that the iterative sequences and are monotone.(4)The authors did not clarify if is either or . In any case, the test function can take arbitrary real values. It is clear that the contractivity condition (4) implies that takes nonnegative values in different points, but it does not cover all possibilities. In particular, the function is not declared at . Then, can take any real value (its image is not restricted to ).(5)The previous remark is important because if we take in (4), we deduce that  which, in the metric case, let bound the distance by . If , then for all , which is a very strong restriction on the mappings and .(6)In [2], page 3, equation (15), the authors announced  However, it is not clear why . Even if we would be able to prove that and (which was not proved), the condition is not guaranteed in a partial metric space. Precisely, this is the characteristic property of partial metric spaces. Therefore, the second inequality in (10) is false.(7)With respect to the previous remark, it is also necessary to point out that the contractivity condition (4) does not permit us to upper bound, for instance, the term . However, the authors affirmed in [2], page 3, equation (16), that “Similarly, we can obtain  Let us see where the mistake is. Theorem 2 only assumes that the inequality

 occurs provided that and ; that is, the first argument of must be -lower than the first argument of . As the authors defined  then  In this case, it was not proved that and . In fact, the contrary inequalities were announced; that is, and . If both inequalities hold, then , which means that is a coupled fixed point of . However, the proof must analyse the case in which for all .(8)Similarly, the contractivity condition (4) cannot be applied to study the term , because  but, in this case, the inequalities and cannot be proved in the case since the contrary inequalities are supposed.(9)When the authors tried to prove that the sequences and are Cauchy, as usual, they reasoned by contradiction. They announced that if is not Cauchy, then there exist and two partial subsequences and such that  and if is the smallest index verifying this property, then  (see [2], page 3, equations (26) and (27)). However, the authors did not justify neither why we can suppose that the subindices are even nor why (17), involving the partial subsequences and , can be deduced from (16), in which only and have a role. In [4], the authors justified the unidimensional case but did not study the coupled case.(10)Other important mistakes can be found in [2], page 4, equation (39), where the author announced that

 Taking into account that is a metric on , if this property was true, then the sequences and would be constant for all which, in general, is false. In fact, it is well known that if there is some such that , then is the common coupled fixed point.(11)Finally, we point out that Theorem 7 in [2] is incorrectly enunciated.

4. An Example

It is not clear how we can show a counterexample of Theorem 2 because Definition 1 is not well posed. Item 1 of Section 3 shows that, in general, it is a very restrictive hypothesis ( and must be constant and equal). Therefore, we are going to show an example in which other hypotheses hold, where and are not constant, but and have no common coupled fixed point.

Let provided with its usual partial order and let for all . Then, is a complete partial metric space. Let us define and by Then, and have the mixed monotone property, both mappings are continuous, and . Letting and , we have the fact that and . However, the condition is impossible when , so and cannot have a common coupled fixed point. It only remains to prove that the contractivity condition (4) holds.

Let be such that and . As , then , so On the other hand, Therefore, Taking into account that we conclude that inequality (22) holds.

5. A Correct Version

Taking into account the commentaries given in Section 3, we propose a correct version of Theorem 2. Item 6 shows that the terms must not be employed in the contractivity condition, and items 7-8 suggest that it is very difficult to use two different mappings and in the contractivity condition as we cannot compare, at the same time, the terms , , and . If and are not involved in the second member of the contractivity condition, it is almost impossible to control the term when and can be even and odd.

In recent times, many coupled/tripled/quadrupled/multidimensional fixed point theorems in various abstract metric spaces have come to be simple consequences of their corresponding unidimensional results (see, e.g., [5–10] and the references therein). Following this line of research, we present here a correct version of Theorem 2 for three reasons mainly: (1) for the sake of completeness; (2) to describe how coupled results in partial metric spaces can be deduced from the unidimensional case; (3) to show some possible hypotheses to ensure the existence of common coupled fixed points when we work with two different mappings. Before doing it, we need to introduce the following preliminaries.

Definition 6. Let be a binary relation on .(i)Two points are called -comparable if or .(ii)A subset is said to be -well ordered if every two points of are -comparable.(iii)A mapping is called -nondecreasing if implies .

Definition 7. One will say that is a partially ordered partial metric space (sometimes, it is also known as ordered partial metric space) if is a partial metric on and is a partial order on .

Definition 8 (Nashine et al. [4]). Let be a partially ordered set. A pair of mappings is said to be weakly increasing if and for all . The mapping is said to be -weakly isotone increasing if for all , we have .
Very recently, Nashine et al. [4] proved the following result.

Theorem 9 (Nashine et al. [4], Theorem 3.6). Let be a complete partially ordered partial metric space. Let be two mappings such that for all -comparable , where and is a continuous function with for each , . We suppose the following:(i) is -weakly isotone increasing,(ii) and are continuous.
Then, the set of common fixed points of and is nonempty, and for . Moreover, the set is well ordered if and only if and have one, and only one, common fixed point.

Based on this result, we present a coupled version that can be interpreted as a correct version of Mutlu et al.’s theorem.

Theorem 10. Let be a complete partially ordered partial metric space and let be two continuous mappings such that, for all verifying or , where And is a continuous function with and for each . Also assume that for all , we have the fact that Then, the set of common coupled fixed points of and is nonempty, and for all .

To prove it, we use the following notation and basic facts. Let be a partial metric on and define by Then, is a partial metric space. Now, let be a binary relation on and define the relation on by Then, is also a binary relation on with the following property: if is a partial order on , then is a partial order on .

Given two mappings , let us denote by the mappings If is -continuous, then is -continuous. Using the notation given in (25), the contractivity condition (26) can be rewritten as (24) in the sense that for all such that or (i.e., -comparable points of ). Furthermore, inequalities (28) are equivalent to that is, is -weakly isotone increasing in the partially ordered set . Applying Theorem 9, the set of common fixed points of and is nonempty, and , for . Notice that a common fixed point of and is nothing but a common coupled fixed point of and . This means that the set of common coupled fixed points of and is nonempty and for all common coupled fixed points of and .

6. Conclusions

We first note that we can suggest further corrected forms for the paper [2]. We prefer Theorem 10 since it is the best possible corrected result inspired from the very defective main result in [2], that is, Theorem 2.

Secondly, we can list several consequences of Theorem 10, for instance, by taking and/or by replacing with . One can also get several corollaries by replacing with the various combinations of the terms in . Furthermore, it is easy to state the analog of Theorem 10 in the context of “complete partial metric space,” instead of “complete partially ordered partial metric space.” Regarding the skeleton of the paper, we avoid listing all these results that can be easily derived by the reader.

Finally, independently from the structure of the abstract space (e.g., metric space, partial metric space, G-metric space, b-metric space, etc.), we underline the fact that multidimensional fixed point theorems and, in particular, coupled fixed point theorems can be derived from the existing corresponding results in the literature (see, e.g., [5–8]).

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

This research was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper. Antonio Francisco Roldán Lόpez de Hierro has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE.