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Mathematical Problems in Engineering
Volume 2012, Article ID 409872, 17 pages
http://dx.doi.org/10.1155/2012/409872
Research Article

New Meir-Keeler Type Tripled Fixed-Point Theorems on Ordered Partial Metric Spaces

1Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1 4011, Tunisia
2Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 3 February 2012; Accepted 27 March 2012

Academic Editor: Zheng-Guang Wu

Copyright © 2012 Hassen Aydi and Erdal Karapınar. 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 some new Meir-Keeler type tripled fixed-point theorems on a partially ordered complete partial metric space. Also, as application, some results of integral type are given.

1. Introduction and Preliminaries

In the last century, the theory of fixed points has appeared as a crucial technique in the study of nonlinear phenomena. Particularly, the tools in fixed-point theory have an application in such diverse fields as biology, chemistry, physics, economics, computer sciences, and engineering.

Recently, fixed-point theorems are considered on partial metric spaces on which self-distance of some points may not be zero. This phenomenon was discovered by Matthews [1] when he considered the tools of metric spaces in the field of semantics and domain theory in computer science (see, e.g., [2, 3]). After the initial results of Mathews, other papers have been released on partial metric spaces (see e.g., [420]).

Another important development is reported in fixed-point theory via ordered metric spaces. Fixed-point theorems in ordered sets were discussed by Ran and Reurings [21]. Subsequently, many results in this direction were given (see, e.g., [2231]).

In this paper, we combine two recent trends, partial metric spaces and ordered sets, and discuss the existence and uniqueness of some new Meir-Keeler type tripled fixed-point theorems in the context of partially ordered partial metric spaces.

Let be a nonempty set. 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 [1, 5].

Definition 1.1 (see, e.g., [1, 5, 15]). 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, .

Lemma 1.2 (see, e.g., [1, 5, 15]). 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 1.3 (see, e.g., [4, 15, 16]). Let be a partial metric space. Then, (A) if , then ,(B) if , then .

Remark 1.4. If , may not be 0.

Lemma 1.5 (see, e.g., [4, 15, 16]). Let as in a partial metric space where . Then, for every .

is called a partially ordered partial metric space if is a partially ordered set and is a partial metric space. Further, if is a complete partial metric space, then is called a partially ordered complete partial metric space. Hereafter, we assume that and we use the notation Also, take the mapping such that where and .

Let be a partially ordered set. We consider the following partial order (also denoted by ≤) on the product space : where . Moreover, we say that is equal to if and only if , and . In the sequel, we need the following definitions.

Definition 1.6 (see [32]). Let be a partially ordered set and a given mapping. We say that has the mixed monotone property if is monotone nondecreasing in and , and it is monotone nonincreasing in , that is, for any,

Definition 1.7 (see [32]). An element is called a tripled fixed point of if

Berinde and Borcut [32] proved the following theorem.

Theorem 1.8. Let be a partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property on . Assume that there exist constants such that for which for all , and . Assume that 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 there exist such that

Recently, Theorem 1.8 is extended to cone metric spaces by Rao and Kishore [33]. On the other hand, very recently, Aydi et al. [34] introduced the following concepts.

Definition 1.9 (see [34]). Let be a partially ordered set and . We say that has the mixed strict monotone property if, for any,

Definition 1.10 (see [34]). Let be a partially ordered metric space. A mapping is said to be a generalized Meir-Keeler type contraction if, for any , there exists a such that for all with , and .

In the following, we consider the partial case of Definition 1.10 and we introduce the following.

Definition 1.11. Let be a partially ordered partial metric space. A mapping is said to be a generalized -Meir-Keeler type contraction if, for any , there exists a such that for all with , and .

Remark 1.12. It is immediate to show that if is a generalized -Meir-Keeler type contraction, then for all with or .

Proposition 1.13. Let be a partially ordered partial metric space and a given mapping. If (1.8) is satisfied, then is a generalized -Meir-Keeler type function.

Proof. Assume that (1.8) is satisfied. For all , one can check that (1.13) is satisfied with .

In the sequel, we use the following notations given in [34]. Let be such that, for, Let be such that We consider sequences , , and such that for .

Our first auxiliary result is as follows.

Proposition 1.14. Let be a partially ordered partial metric space, and letbe a given mapping such that the following hypotheses hold:(i) has the mixed strict monotone property,(ii) is a generalized -Meir-Keeler type function,(iii) such that , and .

Then,

Proof. Let and . We show that with .
Due to the fact that has the mixed strict monotone property, together with the assumption that , and , we obtain Analogously, we have Thus, (1.19) holds for . By using the same arguments, we show that (1.19) holds also for . In fact, Similarly, we find Inductively, we get that (1.19) holds.
By Remark 1.12, together with (1.19), we have Let . Combining (1.24)–(1.26), we get If we denote , then, by definition of the partial metric and (1.27), we have Consequently, the sequence is decreasing. Hence, converges, say to . Clearly, if , we have finished. Suppose, on the contrary, . Thus, there exists such that In particular, for , we have that is equal to It follows from (1.19) and the hypothesis (ii) that which is equivalent to Moreover, we have Combining (1.33) and (1.34), we have Thus, which is a contradiction with respect to (1.29), and so . We conclude that

Remark 1.15. The previous proposition remains true if, in (iii), we change the assumption with the following

2. Existence of Tripled Fixed Point

The following theorem is our first main result.

Theorem 2.1. Let be a partially ordered complete 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 .
Assume that satisfies the following hypotheses:(i) has the mixed strict monotone property,(ii) is a generalized -Meir-Keeler type function,(iii)there exist such that Then, has a tripled fixed point, that is, there exist such that Also, .

Proof. Let be as in (iii). We construct sequences , , and according to (1.17).
We claim that, for all , we have Indeed, we will use a mathematical induction to prove (2.3). Clearly, we have Suppose now that the inequalities in (2.3) hold for some . By the mixed strict monotone property of , together with (1.17), we have Thus, (2.3) holds for all .
Putting and and by Proposition 1.14, we get which is equivalent to Take an arbitrary . It follows from (2.7) that there exists such that Without loss of the generality, assume that and define the following set: We claim that Take . Then, by (2.8) and the triangle inequality (which still holds for partial metrics), we have We consider the following two cases.Case 1 (). By Remark 1.12 and the definition of , the inequality (2.11) turns into Case 2 (). That is, Since , then, by (ii), we have Hence, combining (2.14) and (2.11), we get On the other hand, using (i), one can easily check that Hence, we conclude that (2.10) holds. By (2.8), we have that , and so, by (2.10) we get Then, for all , we have By definition of , we have Consequently, by definition of the metric , , so we get Therefore, , , and are Cauchy sequences in the metric space . Since is a complete partial metric space, then, by Lemma 1.2, is also a complete metric space. Hence, there exists a point such that Again, by Lemma 1.2 and (2.19), we obtain We will prove that To this aim, take an arbitrary . Since then there exist such that by (2.22) for all , , . Now, taking and using Remark 1.12 with the assumption by (2.25), we get Analogously, we get that which yield that

Remark 2.2. Theorem 2.1 remains true if we replace (iv) with one of the following statements.
There exist such that

3. Uniqueness of Tripled Fixed Point

In this section, we will prove the uniqueness of the tripled fixed point.

Theorem 3.1. In addition to hypotheses of Theorem 2.1, assume that, for all , , there exists that is comparable to and . Then, has a unique tripled fixed point.

Proof. The set of tripled fixed points of is not empty due to Theorem 2.1. We suppose that are two tripled fixed points of . We distinguish the following two cases.Case 1. is comparable to with respect to the ordering in , where Without loss of the generality, we may assume that By this, definition of , Lemma 1.3, and Remark 1.12, we have which is a contradiction and therefore must be .Case 2. is not comparable to . By assumption, there exists which is comparable to both and . Without loss of the generality, we may assume that From Proposition 1.14 and (3.4), we have By triangle inequality, we derive By Lemma 1.3, we get .

4. Results of Integral Type

Motivated by Suzuki [35] and on the same lines of [31, Theorem  3.1], one can prove the following result.

Theorem 4.1. Let be a partially ordered complete partial metric space, and let be a given mapping. 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 generalized -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 be a mapping satisfying the following hypotheses:(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, has a tripled fixed point.

To end this paper, we give the following corollary.

Corollary 4.3. Let be a partially ordered complete partial metric space be a mapping satisfying the following hypotheses:
(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, has a tripled fixed point.

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

Remark 4.4. By taking , we retrieve the analogous of Theorem 1.8 of Berinde and Borcut on ordered partial metric spaces (with ). In fact, assume that (1.8) holds for , that is, for all . From this inequality, we get that which corresponds to (4.4) with . Then, we may apply Corollary 4.3.

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