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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 327273, 20 pages
http://dx.doi.org/10.1155/2012/327273
Research Article

Coupled Fixed Points for Meir-Keeler Contractions in Ordered Partial Metric Spaces

1Department of Mathematics and Computer Sciences, Çankaya University, 06530 Ankara, Turkey
2Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1, 4011 H. Sousse, Tunisia
3Department of Mathematics, Atılım University, İncek, 06836 Ankara, Turkey

Received 18 February 2012; Revised 19 April 2012; Accepted 2 May 2012

Academic Editor: Rafael Martinez-Guerra

Copyright © 2012 Thabet Abdeljawad 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

In this paper, we prove the existence and uniqueness of a new Meir-Keeler type coupled fixed point theorem for two mappings and on a partially ordered partial metric space. We present an application to illustrate our obtained results. Further, we remark that the metric case of our results proved recently in Gordji et al. (2012) have gaps. Therefore, our results revise and generalize some of those presented in Gordji et al. (2012).

1. Introduction and Preliminaries

Fixed point theory is an important tool in the study of nonlinear analysis as it is considered to be the key connection between pure and applied mathematics with wide applications in economics, physical sciences, such as biology, chemistry, physics, differential equations, and almost all engineering fields (see, e.g., [113]). From the engineering point of view there are numerous problems in adaptive systems where convergence, optimal performance, and stability are key issues. In this direction many case studies with engineering applications can be described by contraction mappings and their fixed point iterations, such as linear and nonlinear filters, image restoration and image retrieval, and in many other areas where this theory helps to describe and/or understand the phenomenon. Indeed, the relaxation in linear systems, and relaxation and stability in neural networks can be analyzed in this light, where examples for a posteriori and normalized learning algorithms for adaptive filters for monophonic and stereophonic echo cancelation can be presented [14, 15].

Also it is worth mentioning that Matthews introduced the notion of partial metric space, which provides an area with great potential for the development of fixed point theory, as well as tools of conducting studies on denotational semantics of data-flow networks [16].

As a result it is evident that the importance of fixed point theory cannot be ruled out. Banach fixed point theorem [17] is the cornerstone of this topic. The result of Banach has drawn considerable interest of many authors. There are very different approaches in the study of generalization of a Banach fixed point theorem.

One of the interesting generalizations was announced by Matthews [16]. The author introduced the notion of partial metric spaces and proved the analog of Banach fixed point theorem. Roughly speaking, a partial metric space is a generalization of a metric spaces in which self distance of some points may not be zero. Matthews [16] discovered this phenomena when he tried to overcome problems of applying metric space techniques in the subfield of computer science: semantics and domain theory (see, e.g., [18, 19]). After the pioneer result of Mathews, remarkably good results have been reported on partial metric spaces (see, e.g., [2040]).

On the other hand, considering the existence and uniqueness of a fixed point in partially ordered sets initiated a new trend in fixed point theory. The first result in this direction was given by Turinici [41], where he extended Banach contraction principle in partially ordered sets. Ran and Reurings [42] presented some applications of Turinici's theorem to matrix equations. After this intriguing paper, so many exceptionally good results have been revealed in this direction (see, e.g., [4350]). Worth mentioning, Gnana Bhaskar and Lakshmikantham [44] introduced the notion of a coupled fixed point in the class of partially ordered metric spaces. Motivated by the above history, we devote this paper to prove the existence and uniqueness of coupled fixed points for a new Meir-Keeler type mappings in ordered partial metric spaces.

First, we recall basic definitions and crucial results. Hereafter, we assume that and we use the notation

Definition 1.1 (see [44]). Let be a partially ordered set and . is said to have the mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in , that is, for any ,

Definition 1.2 (see [44]). An element is said to be a coupled fixed point of the mapping if

The following two results of Bhaskar and Lakshmikantham in [44] were proved in the context of cone metric spaces in [51].

Theorem 1.3 (see [44]). Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on . Assume that there exists with If there exists such that and , then there exist such that and .

Theorem 1.4 (see [44]). 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 has the following properties: (i)if a nondecreasing sequence , then , for all , (ii)if a nonincreasing sequence , then , for all . Assume that there exists a with If there exists such that and , then there exist such that and .

Inspired by Definition 1.1, the following concept of a -mixed monotone mapping was introduced by Lakshmikantham and Ćirić [47].

Definition 1.5 (see [47]). Let be partially ordered set and and . is said to have mixed -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in , that is, for any ,

It is clear that Definition 1.5 reduces to Definition 1.1 when is the identity.

Definition 1.6 (see [47]). An element is called a coupled coincidence point of mappings and if and is called a coupled common fixed of and , if The mappings and are said to commute if for all .

Very recently, Gordji et al. [31] replaced mixed  g-monotone  property with a mixed  strict  g-monotone  property and improved the results in [47].

Definition 1.7 (see [31]). Let be a partially ordered set and and . is said to have the mixed strict -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in , that is, for any ,

If we replace with identity map in (1.10), we get the definition of mixed strict monotone property of .

A partial metric is a function satisfying the following conditions: (P1)If , then , (P2), (P3), (P4), for all . Then is called a partial metric space. If is a partial metric on , then the function given by is a metric on . Each partial metric on generates a topology on with a base of the family of open -balls , where for all and . Similarly, closed -ball is defined as . For more details see for example [16, 21].

Definition 1.8 (see [16, 21, 33]). Let be a partial metric space. (i)A sequence in converges to whenever . (ii)A sequence in is called Cauchy whenever exists (and finite). (iii) is said to be complete if every Cauchy sequence in converges, with respect to , to a point , that is, . (iv)A mapping is said to be continuous at if for each there exists such that .

Lemma 1.9 (see [16, 21, 33]). Let be a partial metric space. (a)A sequence is Cauchy if and only if is a Cauchy sequence in the metric space , (b) is complete if and only if the metric space is complete. Moreover,

Lemma 1.10 (see [20]). Let be a partial metric space. Then (A)If then . (B)If , then .

Remark 1.11. If , may not be .

The following two lemmas can be derived from the triangle inequality (P4).

Lemma 1.12 (see [20]). Let as in a partial metric space , where . Then for every .

Lemma 1.13 (see [36]). Let and . If then .

Remark 1.14. Limit of a sequence in a partial metric space is not unique.

Example 1.15. Consider with . Then is a partial metric space. Clearly, is not a metric. Observe that the sequence converges both for example to and , so no uniqueness of the limit.

We give the partial case of a definition given in [31].

Definition 1.16 (see [31]). Let be a partially ordered partial metric space. Let and . The mapping is said to be a -Meir-Keeler type contraction if for any there exists a such that for all with , .

If we replace with the identity in (1.13) and a metric on , the is called a Meir-Keeler type contraction.

Definition 1.17. Let be a partially ordered partial metric space. Let and . The mapping is said to be a strict -Meir-Keeler type contraction if there exists such that for any there exists a such that for all with , .

If we replace with the identity in (1.14) and if a metric on , the is called a strict Meir-Keeler type contraction. Further, it can be shown easily that every strict Meir-Keeler (resp., strict -Meir-Keeler) type contraction is a Meir-Keeler (resp., -Meir-Keeler) type contraction.

Let be a partial metric space. Note that the mappings defined by forms a partial metric on where and .

The following fact can be derived easily from Definition 1.16.

Lemma 1.18. Let be a partially ordered partial metric space. Let and . If is a -Meir-Keeler type contraction, then one has for all with , or , .

Proof. Without loss of generality, suppose that and where . It is clear that . Set . Since is a -Meir-Keeler type contraction, then, for this , there exits such that for all with and . The result follows by choosing , , , , that is,

Remark 1.19. Let be a partially ordered partial metric space. Let and . If is a strict -Meir-Keeler type contraction, then we have for all with , or , .

Proof. The proof is similar to Lemma 1.18 above.

2. Existence of Coupled Fixed Points

The following theorem is our first main result.

Theorem 2.1. Let be a partially ordered partial metric space. Suppose that has the following properties: (a)if is a sequence such that for each and , then for each , (b)if is a sequence such that for each and , then for each . Let and be mappings such that and is a complete subspace of . Suppose that satisfies the following conditions: (i) has the mixed strict -monotone property, (ii) is a -Meir-Keeler type contraction, (iii)there exist such that Then and have a coupled coincidence point, that is, there exist such that

Proof. Let be such that and . We construct the sequence and in the following way. Due to the assumption , we are able to choose such that and . By repeating the same argument, we can choose such that and . Inductively, we observe that We claim that, for all We will use the mathematical induction to show (2.4). By assumption (iii), we have Assume that the inequalities in (2.4) hold for some . Regarding the mixed -strict monotone property of , we have By repeating the same arguments, we observe that Combining the above inequalities, together with (2.3), we get So, (2.4) holds for all . Set Taking Lemma 1.18 and (2.4) into account, we get If we add the previous two inequalities side by side, we obtain that . Hence, is monotone decreasing sequence in . Since the sequence is bounded below, there exists such that .
We prove . Suppose on the contrary that . Thus, there is a positive integer such that for any , we have where and is chosen by (ii). In particular, for , we have Regarding the assumption (iii) together with (2.12) and (2.4), we have which is equivalent to Similarly, we have Summing the two above inequalities which contradicts (2.11) for . Thus, . That is, Consequently, we have By condition (P3), we have so letting , we get Analogously, we have We claim that the sequences and are Cauchy in .
Take an arbitrary . It follows from (2.17) that there exists such that Without loss of the generality, assume that and define the following set Take . We claim that Take . Then, by (2.22) and the triangle inequality (which still holds for partial metrics) we have We distinguish two cases.
First Case. .
By Lemma 1.18 and the definition of , the inequality (2.25) turns into
Second Case. .
In this case, we have Since and , by (ii), we get Also, we have Since and , by (ii), we get Thus, combining (2.25), (2.28) and (2.30), we obtain On the other hand, using (i), it is obvious that We conclude that . Since , so that is, (2.24) holds. By (2.22), we have . This implies with (2.24) that Then, for all , we have . This implies that for all , we have Thus, the sequences and are Cauchy in . By Lemma 1.9, and are also Cauchy in . Again by Lemma 1.9, ) is complete. Thus, there exist such that by using (2.20) and (2.21), we arrive at Since the sequences and are monotone increasing and monotone decreasing, respectively, by properties (a) and (b), we conclude that for each . Therefore, having in mind that is a -Meir-Keeler type contraction, by (2.37) and Lemma 1.18, we get From (2.36), by Lemma 1.12, we obtain so . Analogously we get .

Remark 2.2. We remak that Theorem 2.1 has been proved recently in [31] in the category of partially ordered metric spaces. However, they proceed the proof without using the assumptions (a) and (b) stated in our Theorem. They claimed that and by using the fact that the sequences and are increasing and decreasing, respectively. In our belief, this step is not true and cannot be achieved without using the assumptions (a) and (b). Actually, this may not be true if the partial ordering, for example, is obtained via nonstrongly minihedral cones.

Corollary 2.3. Let be a complete ordered partial metric space. Suppose that has the following properties: (a)if is a sequence such that for each and , then for each , (b)if is a sequence such that for each and , then for each .Let be a given mapping. Suppose that satisfies the following conditions: (i) has the mixed strict monotone property, (ii) is a Meir-Keeler type contraction, (iii)there exist such that Then, has a coupled fixed point, that is, there exist such that

Proof. It follows by taking , the identity mapping on , in Theorem 2.1.

3. Uniqueness of Coupled Fixed Points

Let be a partially ordered set. We endow by the following order (denoted ) Moreover, and are called -comparable if either or . In case , we shortly say that and are comparable and denote by . In this section, we will prove the uniqueness of the coupled fixed point.

Theorem 3.1. In addition to the hypotheses of Theorem 2.1, assume that for all non -comparable points , , there exists such that is comparable to both and . Further, assume that and commute and is a strict Meir-Keeler type contraction. Then, and have a unique coupled common fixed point, that is, there exists such that

Proof. The set of coupled coincidence points of and is not empty due to Theorem 2.1. If is the only coupled coincidence point of and , then commutativity of and implies that Hence, is a coupled coincidence point of and and by uniqueness we conclude that
Now suppose that are two coupled coincidence points of and . We show that and . To this end we distinguish the following two cases.
First Case. is -comparable to with respect to the ordering in , where Without loss of the generality, we may assume that By definition of and Lemma 1.18 we have which is a contradiction. Therefore, we have . Hence
Second Case. is not -comparable to .
By assumption, there exists such that is comparable to both and . Then, we have Setting , , , , and , as in the proof of Theorem 2.1, we get Since is comparable with , we have and . By using that has the mixed strict monotone property, we observe that and for all . Thus, by Remark 1.19, we get that
Inductively, we derive that The right hand side of above inequality tends to zero as . Hence, Analogously, we get that By the triangle inequality, we have Combining all observation above, we get that and . Therefore, In both cases above, we have shown that (3.16) holds. Now, let and . By the commutativity of and with the fact that and , we have Thus, is a coupled coincidence point of and . Setting and in (3.17). Then, by (3.16) we have From (3.17) we get that Hence, the pair is the coupled common fixed point of and .
Finally, we prove the uniqueness of the coupled common fixed point of and . Actually, if is another coupled common fixed point of and , then follows from (3.16).

Remark 3.2. We remark that Theorem 3.1 above has been recently proved in [31] without assuming that the mapping is a strict -Meir-Keeler contraction. This leads to a gap in the proof of Theorem  2.6 there.

Corollary 3.3. Suppose that all the hypotheses of Corollary 2.3 hold, and further, for all , , there exists that is comparable to and . Further, assume that is a strict Meir-Keeler type contraction. Then, has a unique coupled fixed point.

4. Applications

Motivated by Suzuki [52] and on the same lines of Theorem  3.1 of [53], one can prove the following result.

Theorem 4.1. Let be a partially ordered partial metric space. Let and be given mappings such that . Assume that there exists a function from into itself satisfying the following:(I) and for every ,(II) is nondecreasing and right continuous,(III)for every , there exists such that for all and .
Then, is a -Meir-Keeler type function.

The following result is an immediate consequence of Theorems 2.1 and 4.1.

Corollary 4.2. Let be a partially ordered complete partial metric space. Given and such that , is a complete subspace and the following hypotheses hold:(i) has the mixed -strict monotone property,(ii)for every , there exists such that for all and , where is a locally integrable function satisfying for all ,(iii)there exist such that Assume that the hypotheses (a) and (b) given in Theorem 2.1 hold. Then, and have a coupled coincidence point.

To end this paper, we give the following corollary.

Corollary 4.3. Let be a partially ordered partial metric space. Given and such that , is a complete subspace and the following hypotheses hold:(i) has the mixed -strict monotone property,(ii)for all and , where and is a locally integrable function from into itself satisfying for all ,(iii)there exist such that Assume that the hypotheses (a) and (b) of Theorem 2.1 hold. Then, and have a coupled coincidence point.

Proof. For all , we take and we apply Corollary 4.2.

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