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.

Hamed H. Alsulami 1 and Erdal Karapinar 1,2 and Antonio Francisco Roldan Lopez de Hierro 3

Academic Editor:Sehie Park

1, Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
2, Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
3, University of Jaen, Campus las Lagunillas, s/n, 23071 Jaen, Spain

Received 16 March 2014; Accepted 3 July 2014; 5 January 2015

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.

1. Introduction

Following Matthews [1], a partial metric on a nonempty set [figure omitted; refer to PDF] is a mapping [figure omitted; refer to PDF] verifying, for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] In this case, [figure omitted; refer to PDF] is called a partial metric space . Although the authors of [2] used the notation [figure omitted; refer to PDF] for a partial metric space, we prefer using [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , the mapping [figure omitted; refer to PDF] , given by [figure omitted; refer to PDF] is a metric on [figure omitted; refer to PDF] .

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

Definition 1 (Mutlu et al. [2], Definition 9).

Assume that [figure omitted; refer to PDF] is a partially ordered set and [figure omitted; refer to PDF] . [figure omitted; refer to PDF] and [figure omitted; refer to PDF] mappings have the following properties: [figure omitted; refer to PDF]

Theorem 2 (Mutlu et al. [2], Theorem 10).

Suppose that [figure omitted; refer to PDF] is a partially ordered set and [figure omitted; refer to PDF] is a partial metric on [figure omitted; refer to PDF] with [figure omitted; refer to PDF] being a complete partial metric space. Assume that [figure omitted; refer to PDF] are satisfied by Definition 1 and also are continuous mappings possessing the mixed monotone property on [figure omitted; refer to PDF] . Let there be a nonincreasing function [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and also having [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , with [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . If there exists [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] at the time, [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

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, [figure omitted; refer to PDF] will be a nonempty set and [figure omitted; refer to PDF] will represent the product space [figure omitted; refer to PDF] of 2 identical copies of [figure omitted; refer to PDF] .

Definition 3.

A binary relation [figure omitted; refer to PDF] on [figure omitted; refer to PDF] is a nonempty subset [figure omitted; refer to PDF] . One will write [figure omitted; refer to PDF] (or [figure omitted; refer to PDF] ) if [figure omitted; refer to PDF] . A binary relation [figure omitted; refer to PDF] on [figure omitted; refer to PDF] is reflexive if [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , and it is transitive if [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . A reflexive and transitive relation on [figure omitted; refer to PDF] is a preorder (or a quasiorder) on [figure omitted; refer to PDF] . If a preorder [figure omitted; refer to PDF] is also antisymmetric ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] imply [figure omitted; refer to PDF] ), then [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] be a given mapping. We say that [figure omitted; refer to PDF] is a coupled fixed point of [figure omitted; refer to PDF] if [figure omitted; refer to PDF]

Definition 5.

Given two mappings [figure omitted; refer to PDF] , we say that [figure omitted; refer to PDF] is a common coupled fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] if [figure omitted; refer to PDF]

Henceforth, we will use the notation [figure omitted; refer to PDF] Functions in [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are. If they are arbitrary, then, for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

: Therefore, [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are monotone.

(4) The authors did not clarify if [figure omitted; refer to PDF] is either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . In any case, the test function [figure omitted; refer to PDF] can take arbitrary real values. It is clear that the contractivity condition (4) implies that [figure omitted; refer to PDF] takes nonnegative values in different points, but it does not cover all possibilities. In particular, the function [figure omitted; refer to PDF] is not declared at [figure omitted; refer to PDF] . Then, [figure omitted; refer to PDF] can take any real value (its image is not restricted to [figure omitted; refer to PDF] ).

(5) The previous remark is important because if we take [figure omitted; refer to PDF] in (4), we deduce that [figure omitted; refer to PDF]

: which, in the metric case, let bound the distance [figure omitted; refer to PDF] by [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , which is a very strong restriction on the mappings [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

(6) In [2], page 3, equation (15), the authors announced [figure omitted; refer to PDF]

: However, it is not clear why [figure omitted; refer to PDF] . Even if we would be able to prove that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] (which was not proved), the condition [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . However, the authors affirmed in [2], page 3, equation (16), that "Similarly, we can obtain [figure omitted; refer to PDF]

: Let us see where the mistake is. Theorem 2 only assumes that the inequality [figure omitted; refer to PDF]

: occurs provided that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; that is, the first argument of [figure omitted; refer to PDF] must be [figure omitted; refer to PDF] -lower than the first argument of [figure omitted; refer to PDF] . As the authors defined [figure omitted; refer to PDF]

: then [figure omitted; refer to PDF]

: In this case, it was not proved that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In fact, the contrary inequalities were announced; that is, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . If both inequalities hold, then [figure omitted; refer to PDF] , which means that [figure omitted; refer to PDF] is a coupled fixed point of [figure omitted; refer to PDF] . However, the proof must analyse the case in which [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

(8) Similarly, the contractivity condition (4) cannot be applied to study the term [figure omitted; refer to PDF] , because [figure omitted; refer to PDF]

: but, in this case, the inequalities [figure omitted; refer to PDF] and [figure omitted; refer to PDF] cannot be proved in the case [figure omitted; refer to PDF] since the contrary inequalities are supposed.

(9) When the authors tried to prove that the sequences [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are Cauchy, as usual, they reasoned by contradiction. They announced that if [figure omitted; refer to PDF] is not Cauchy, then there exist [figure omitted; refer to PDF] and two partial subsequences [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: and if [figure omitted; refer to PDF] is the smallest index verifying this property, then [figure omitted; refer to PDF]

: (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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , can be deduced from (16), in which only [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF]

: Taking into account that [figure omitted; refer to PDF] is a metric on [figure omitted; refer to PDF] , if this property was true, then the sequences [figure omitted; refer to PDF] and [figure omitted; refer to PDF] would be constant for all [figure omitted; refer to PDF] which, in general, is false. In fact, it is well known that if there is some [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] 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 ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] must be constant and equal). Therefore, we are going to show an example in which other hypotheses hold, where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are not constant, but [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have no common coupled fixed point.

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

Let [figure omitted; refer to PDF] be such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . As [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] On the other hand, [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] Taking into account that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] must not be employed in the contractivity condition, and items 7-8 suggest that it is very difficult to use two different mappings [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the contractivity condition as we cannot compare, at the same time, the terms [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are not involved in the second member of the contractivity condition, it is almost impossible to control the term [figure omitted; refer to PDF] when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] be a binary relation on [figure omitted; refer to PDF] .

(i) Two points [figure omitted; refer to PDF] are called [figure omitted; refer to PDF] -comparable if [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .

(ii) A subset [figure omitted; refer to PDF] is said to be [figure omitted; refer to PDF] -well ordered if every two points of [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -comparable.

(iii): A mapping [figure omitted; refer to PDF] is called [figure omitted; refer to PDF] -nondecreasing if [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] .

Definition 7.

One will say that [figure omitted; refer to PDF] is a partially ordered partial metric space (sometimes, it is also known as ordered partial metric space ) if [figure omitted; refer to PDF] is a partial metric on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a partial order on [figure omitted; refer to PDF] .

Definition 8 (Nashine et al. [4]).

Let [figure omitted; refer to PDF] be a partially ordered set. A pair of mappings [figure omitted; refer to PDF] is said to be weakly increasing if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . The mapping [figure omitted; refer to PDF] is said to be [figure omitted; refer to PDF] -weakly isotone increasing if for all [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] .

Very recently, Nashine et al. [4] proved the following result.

Theorem 9 (Nashine et al. [4], Theorem 3.6).

Let [figure omitted; refer to PDF] be a complete partially ordered partial metric space. Let [figure omitted; refer to PDF] be two mappings such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] -comparable [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a continuous function with [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . We suppose the following:

(i) [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -weakly isotone increasing,

(ii) [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous.

Then, the set [figure omitted; refer to PDF] of common fixed points of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is nonempty, and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Moreover, the set [figure omitted; refer to PDF] is well ordered if and only if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] be a complete partially ordered partial metric space and let [figure omitted; refer to PDF] be two continuous mappings such that, for all [figure omitted; refer to PDF] verifying [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] And [figure omitted; refer to PDF] is a continuous function with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] . Also assume that for all [figure omitted; refer to PDF] , we have the fact that [figure omitted; refer to PDF] Then, the set [figure omitted; refer to PDF] of common coupled fixed points of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is nonempty, and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

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

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

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 [figure omitted; refer to PDF] and/or by replacing [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . One can also get several corollaries by replacing [figure omitted; refer to PDF] with the various combinations of the terms in [figure omitted; refer to PDF] . 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]).

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 Roldan Lopez de Hierro has been partially supported by Junta de Andalucia by project FQM-268 of the Andalusian CICYE.

Conflict of Interests

The authors declare that they have no competing interests.

References

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