Some Inevitable Remarks on “Tripled Fixed Point Theorems for Mixed Monotone Kannan Type Contractive Mappings”

Abstract

We advise that the proof of Theorem 12 given by Borcut et al. (2014) is not correct, and it cannot be corrected using the same technique. Furthermore, we present some similar results as an approximation to the opened question if that statement is valid.

1. Introduction

The definition of coupled fixed point was firstly given by Guo and Lakshmikantham in [1]. This concept, in the context of metric space, was reconsidered by Gnana Bhaskar and Lakshmikantham [2] in 2006 and by Lakshmikantham and Ciric [3] in the coincidence case. Later, Karapınar investigated this notion in the context of cone metric space. After that, Berinde and Borcut [4] presented the notion of tripled fixed point obtaining similar results, and the same authors extended their work to the coincidence case in [5] (see also, e.g., [6–9]).

A coupled fixed point of is a point such that and . In order to ensure existence and uniqueness of coupled fixed points, Bhaskar and Lakshmikantham introduced the concept of mapping having the mixed monotone property. Henceforth, let be a partial order on . The mapping is said to have the mixed monotone property (with respect to ) if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any ,

Inspired by the previous notions, Berinde and Borcut defined the concepts of tripled fixed point and mixed monotone property as follows. A tripled fixed point of is a point such that , , and . The mapping is said to have the mixed monotone property (with respect to ) if is monotone nondecreasing in and , and it is monotone nonincreasing in ; that is, for any , The second equation that defines a tripled fixed point, that is, , uses the point twice in the arguments of . This fact is necessary to ensure the existence of tripled fixed points of a nonlinear contration because, in such a case, the mixed monotone property is applicable.

Very recently, as a continuation of their pioneering works in the tripled case, Borcut et al. announced in [10] the following result.

Theorem 1 (Borcut et al. [10], Theorem 12). Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Assume that there exists a such that for all , , . Also suppose that either(a) is continuous, or(b) has the following properties:(i)if a nondecreasing sequence , then for all ;(ii)if a nonincreasing sequence , then for all .
If there exist such that then has a tripled fixed point in ; that is, there exist such that

This note is to advise that the proof given by the authors of the previous result is not correct, and it cannot be corrected using the same technique. Furthermore, we present some similar results as an approximation to the opened question if the previous theorem is valid.

2. A Review of the Incorrect Proof

Let us review the lines of their proof. Based on , the authors defined, recursively, for all , and they proved that and were monotone nondecreasing sequences and was a monotone nonincreasing sequence in . Then, they defined, for all , Using the contractivity condition (3), they proved that, taking into account that , , and , Based on this inequality, the authors immediately announced that (see [10, page 4, inequalities and ]). However, these last two inequalities are false. In fact, we can only prove that However, comparing (9) with (11), we notice that does not necessarily coincide with and, similarly, is not necessarily equal to . Therefore, inequality (9) cannot be ensured.

Exactly in the same way, it can be possible to see that (10) cannot be proved using the contractivity condition (3). In such a case, the proof given by the authors, which decisively used inequalities (9) and (10), is false.

3. Some Berinde and Borcut’s Type Tripled Fixed Point Theorems

For the moment, the question about whether Theorem 1 is valid is opened. The following results are some approximations to this problem, using contractivity conditions that are inspired in (3). The main aim of this section is to show some results in this line of research using a well-known result by Ćirić [11]. Our technique is based on some very recent works which showed that most of coupled/tripled/quadrupled fixed point results can be reduced to their corresponding unidimensional theorems in different frameworks (see, for instance, [12–18]). Before that, let us introduce some notation and basic results.

Given a binary relation on , let us define If is a partial order on , then is also a partial order on .

Given a metric on , let us define , for all , by Then and are metrics on . In addition to this, if is complete, then and are also complete.

Given a mapping , let us denote by the mapping Notice that a tripled fixed point of is nothing but a fixed point of . If is -continuous, then is -continuous. Furthermore, if has the mixed monotone property with respect to , then is nondecreasing with respect to (see [12]). We also recall the following result.

Definition 2. Let be a metric on and let be a partial order on . We will say that is regular if it verifies the following two properties:(i)if a nondecreasing sequence , then for all ;(ii)if a nonincreasing sequence , then for all .

Lemma 3. If is regular, then is also regular.

The first version of the following theorem was given by Ćirić in 1972 (see [11]) in the case of metric spaces that were not necessarily partially ordered. A partially ordered version can be found, for example, in [19]. Our main results will be consequences of the next result.

Theorem 4 (see e.g., [19]). Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a nondecreasing mapping and let be such that for all such that . Also assume that is continuous or is regular. If there exist such that , then has a fixed point.

In the following result, we found some terms that play an important role in the contractivity condition (3).

Theorem 5. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that

Proof. As has the mixed monotone property on with respect to , it follows that is -nondecreasing. Let us define, for all , Assume that , , and ; that is, . In this case, the contractivity condition (16) guarantees that Taking into account that , , and , we also find the same upper bound: Furthermore, as , , and , Joining the last three inequalities, we deduce that, for all such that , As , Theorem 4 guarantees that has a fixed point; that is, has a tripled fixed point.

Theorem 6. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that

Proof. Following the lines of the previous proof, consider provided with the metric . Assume that , , and ; that is, . In this case, the contractivity condition (24) guarantees that and the same upper bound is valid for . Moreover, as , , and ,
Therefore Theorem 4 guarantees that has a fixed point; that is, has a tripled fixed point.

The following particularization is also inspired by some Berinde and Borcut’s results.

Corollary 7. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that and for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that

Proof. Let us define . If are such that , then so the previous theorem is applicable.

The following result presents a contractivity condition more similar to (3) than (16). It follows from the previous result using .

Corollary 8. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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 Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors offer thanks to anonymous referees for their remarkable comments, suggestion, and ideas which helped to improve this paper. Antonio-F. Roldán-López-de-Hierro has been partially supported by Junta de Andalucía by Project FQM-268 of the Andalusian CICYE.