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Abstract and Applied Analysis
Volume 2012, Article ID 192581, 18 pages
http://dx.doi.org/10.1155/2012/192581
Research Article

Coupled Coincidence Points of Mappings in Ordered Partial Metric Spaces

1Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia
2Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Belgrade, Serbia
3University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11020 Beograd, Serbia

Received 31 October 2011; Accepted 1 January 2012

Academic Editor: Ferhan M. Atici

Copyright © 2012 Zorana Golubović 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

New coupled coincidence point and coupled fixed point results in ordered partial metric spaces under the contractive conditions of Geraghty, Rakotch, and Branciari types are obtained. Examples show that these results are distinct from the known ones.

1. Introduction

In recent years many authors have worked on domain theory in order to equip semantics domain with a notion of distance. In particular, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. Subsequently, several authors (see, e.g., [29]) have proved fixed point theorems in partial metric spaces.

Many generalizations of Banach's contractive condition have been introduced in order to obtain more general fixed point results in metric spaces and their generalizations. We mention here conditions introduced by Geraghty [10], Rakotch [11], and Branciari [12].

The notion of a coupled fixed point was introduced and studied by Bhaskar and Lakshmikantham in [13]. In subsequent papers several authors proved various coupled and common coupled fixed point theorems in partially ordered metric spaces (e.g., [1417]). These results were applied for investigation of solutions of differential and integral equations. In a recent paper [18], Berinde presented a method of reducing coupled fixed point results in ordered metric spaces to the respective results for mappings with one variable.

In this paper, we further develop the method of Berinde and obtain new coupled coincidence and coupled fixed point results in ordered partial metric spaces, under the contractive conditions of Geraghty, Rakotch, and Branciari types. Examples show that these results are distinct from the known ones. In particular, they show that using the order and/or the partial metric enables conclusions which cannot be obtained in the classical case.

2. Notation and Preliminary Results

2.1. Partial Metric Spaces

The following definitions and details can be seen in [19].

Definition 2.1. A partial metric on a nonempty set is a function such that, for all ,,,,. The pair is called a partial metric space.

It is clear that if , then from (P1) and (P2) . But if , may not be 0.

Each partial metric on generates a topology on which has as a base the family of open -balls , , where for all and . A sequence in converges to a point , with respect to , if . This will be denoted as , , or .

If is a partial metric on , then the function given by is metric on . Furthermore, if and only if

A basic example of a partial metric space is the pair , where for all . The corresponding metric is Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1, 19].

Remark 2.2. Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function need not be continuous in the sense that and implies .

Definition 2.3. Let be a partial metric space. Then,a sequence in is called a Cauchy sequence if exists (and is finite);the space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

Lemma 2.4. Let be a partial metric space. is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .The space is complete if and only if the metric space is complete.

Definition 2.5. Let be a nonempty set. Then is called an ordered partial metric space if (i) is a partially ordered set, and (ii) is a partial metric space.
We will say that the space satisfies the ordered-regular condition (abr. (ORC)) if the following holds: if is a nondecreasing sequence in with respect to such that as , then for all .

Definition 2.6 (see [13, 14]). Let be a partially ordered set, , and . is said to have -mixed monotone property if the following two conditions are satisfied: If (the identity map), we say that has the mixed monotone property. A point is said to be a coupled coincidence point of and if and and their common coupled fixed point if and .

Definition 2.7 (see [20]). Let be a metric space, and let and . The pair is said to be compatible if whenever and are sequences in such that and for some .

2.2. Some Auxiliary Results

Lemma 2.8. (i) Let be an ordered partial metric space. If relation is defined on by and is given by then is an ordered partial metric space. The space is complete iff is complete.
(ii) If and , and has the -mixed monotone property, then the mapping given by is -nondecreasing with respect to , that is, where .
(iii) If is continuous in (i.e., with respect to ), then is continuous in (i.e., with respect to ). If is continuous from to (i.e., and imply ), then is continuous in .

Proof. (i) Relation is obviously a partial order on . To prove that is a partial metric on , only conditions (P1) and (P4) are nontrivial.
(P1) If , then obviously holds. Conversely, let , that is, We know by (P2) that and . Adding up, we obtain that , and since in fact equality holds, we conclude that and . Similarly, we get that and . Hence, and applying property (P1) of partial metric , we get that and , that is, .
(P4) Let . Then
(ii, iii) The proofs of these assertions are straightforward.

Remark 2.9. Let be the metric associated with the partial metric as in (2.1). It is easy to see that, with notation as in the previous lemma, is the associated metric to the partial metric on . We note, however, that when we speak about continuity of mappings, we always assume continuity in the sense of the partial metric , that is, in the sense of the respective topology . This should not be confused with the approach given by O'Neill in [3] where both - and -continuity were assumed.

It is easy to see that (using notation as in the previous lemma), the mappings and are -compatible (in the sense of Definition 2.7) if and only if the mappings and are -compatible in the usual sense (i.e., , whenever is a sequence in such that ).

Assertions similar to the following lemma (see, e.g., [21]) were used (and proved) in the course of proofs of several fixed point results in various papers.

Lemma 2.10. Let be a metric space, and let be a sequence in such that is decreasing and If is not a Cauchy sequence, then there exist and two sequences and of positive integers such that the following four sequences tend to when :

As a corollary (applying Lemma 2.10 to the associated metric of a partial metric , and using Lemma 2.4) we obtain the following.

Lemma 2.11. Let be a partial metric space, and let be a sequence in such that is decreasing and If is not a Cauchy sequence in , then there exist and two sequences and of positive integers such that the following four sequences tend to when :

3. Coupled Coincidence and Fixed Points under Geraghty-Type Conditions

Let denote the class of real functions satisfying the condition An example of a function in may be given by for and . In an attempt to generalize the Banach contraction principle, Geraghty proved in 1973 the following.

Theorem 3.1 (see [10]). Let be a complete metric space, and let be a self-map. Suppose that there exists such that holds for all . Then has a unique fixed point and for each the Picard sequence converges to when .

Subsequently, several authors proved such results, including the very recent paper of Ðukić et al. [22].

We begin with the following auxiliary result.

Lemma 3.2. Let be an ordered partial metric space which is complete. Let be self-maps such that is continuous, , and one of these two subsets of is closed. Suppose that is -nondecreasing (with respect to ) and there exists with or . Assume also that there exists such that holds for all such that and are comparable. Assume that either 1°: is continuous and the pair is - compatible or 2°: satisfies (ORC). Then, and have a coincidence point in .

Proof. The proof follows the lines of proof of [22, Theorems 3.1 and 3.5].
Take with, say, , and using that is -nondecreasing and that form the sequence satisfying , , and Since and are comparable, we can apply the contractive condition to obtain Consider the following two cases: for some ; for each .Case 1. Under this assumption, we get that and it follows that . By induction, we obtain that for all and so for all . Hence, is a Cauchy sequence, converging to , and is a coincidence point of and .Case 2. We will prove first that in this case the sequence is strictly decreasing and tends to 0 as .
For each we have that Hence, is strictly decreasing and bounded from below, thus converging to some . Suppose that . Then, it follows from (3.7) that wherefrom, passing to the limit when , we get that . Using property (3.1) of the function , we conclude that , that is, , a contradiction. Hence, is proved.
In order to prove that is a Cauchy sequence in , suppose the contrary. As was already proved, as , and so, using (P2), as . Hence, using (2.1), we get that as . Using Lemma 2.11, we obtain that there exist and two sequences and of positive integers such that the following four sequences tend to when : Putting in the contractive condition and , it follows that Hence, and . Since , it follows that , which is in contradiction with .
Thus, is a Cauchy sequence, both in and in . Hence, it converges (in and in ) to a point (we suppose to be closed, that is, complete; the case when is closed is treated similarly) such that Also, it follows easily that We will prove that and have a coincidence point.
(i) Suppose that is continuous and that is a -compatible pair. We have that It follows that and is a coincidence point of and .
(ii) If satisfies (ORC), since is an increasing sequence tending to , we have that for each . So we can apply (P4) and the contractive condition to obtain Letting we get . Hence, we obtain that and is a coincidence point.

Now, we are in the position to prove the main result of this section.

Theorem 3.3. Let be an ordered partial metric space which is complete, and let and be such that has the -mixed monotone property. Suppose that there exists such that holds for all satisfying ( and ) or ( and ); is continuous, , and one of these two subsets of is closed;there exist such that ( and ) or ( and ); is continuous and is compatible in the sense of Definition 2.7, or satisfies (ORC). Then there exist such that that is, and have a coupled coincidence point.

Proof. Let relation , partial metric , and mappings , on be defined as in Lemma 2.8. Then is an ordered partial metric space which is complete and is a -nondecreasing self-map on . Moreover,there exists such that holds for all -comparable ; is continuous, , and one of these two subsets of is closed;there exists such that and are comparable; is continuous and the pair is -compatible, or satisfies (ORC).Thus, all the conditions of Lemma 3.2 are satisfied (with and ). Hence, there exists such that , that is, there exist such that and . Therefore, and have a coupled coincidence point.

Putting (the identity map) in Theorem 3.3, we obtain the following.

Corollary 3.4. Let be an ordered partial metric space which is complete, and let have the mixed monotone property. Suppose thatthere exists such that holds for all satisfying ( and ) or ( and );there exist such that ( and ) or ( and ); is continuous, or satisfies (ORC).Then there exist such that that is, has a coupled fixed point.

If is a standard metric, this reduces to [15, Corollary 2.3]. The following example shows how Corollary 3.4 can be used.

Example 3.5. Let be ordered by the standard relation ≤. Consider the partial metric on given by . Let be given as Finally, take given as for and . We will show that conditions of Corollary 3.4 hold true.

Take arbitrary satisfying and (the other possible case is treated symmetrically), and denote and . Consider the six possible cases: 1°: , 2°: , 3°: , 4°: , 5°: , and 6°: . It is easy to check that in all these cases and that . Since holds for each (as far as it holds for ), we obtain that . The conditions of Corollary 3.4 are satisfied and has a coupled fixed point (which is ).

4. Coupled Coincidence and Fixed Points under Rakotch-Type Conditions

Let denote the class of real functions satisfying the condition Rakotch proved in 1962 the following.

Theorem 4.1 (see [11]). Let be a complete metric space, and let be a self-map. Suppose that there exists such that holds for all . Then has a unique fixed point .

We will prove the respective result for the existence of a coupled coincidence point in the frame of partial metric spaces. We begin with the following auxiliary result, which may be of interest on its own.

Lemma 4.2. Let be an ordered partial metric space which is complete. Let be self-maps such that is continuous, , and one of these two subsets of is closed. Suppose that is -nondecreasing (with respect to ) and there exists with or . Assume also that there exists such that holds for all such that and are comparable. Assume that either 1°: is continuous and the pair is -compatible or 2°: satisfies (ORC). Then, and have a coincidence point in .

Proof. The proof is similar to the proof of Lemma 3.2, so we note only the basic step.
With replaced by , (3.7) shows that is strictly decreasing, thus converging to some . Suppose that . Then, it follows that a contradiction. Hence, is proved. The rest of the proof is identical to that of Lemma 3.2.

The following example shows that the existence of order may be crucial.

Example 4.3. Let be endowed with the partial metric defined by The order is given by Then it is easy to check that is an ordered partial metric space which is complete and satisfies (ORC). Consider the mappings given as Take the function given by . It is easy to check that the contractive condition (4.3) holds. Indeed, if and are comparable, say , then either or ( and ). In the first case, and ; hence, trivially holds (whichever function is chosen). In the second case, and and . All the conditions of Lemma 4.2 are fulfilled and mappings and have a coincidence point ().

On the other hand, consider the same problem, but without order. Then the contractive condition does not hold and the conclusion about the coincidence point cannot be obtained in this way. Indeed, take any with . Then, and . Hence, if , then and .

Now, we are in the position to give

Theorem 4.4. Let be an ordered partial metric space which is complete, and let and be such that has the -mixed monotone property. Suppose thatthere exists such that holds for all satisfying ( and ) or ( and ) and conditions (ii), (iii), (iv), respectively, (iv′) of Theorem 3.3 hold true. Then there exist such that that is, and have a coupled coincidence point.

Proof. The proof is analogous to the proof of Theorem 3.3, and so is omitted.

Putting (the identity map) in Theorem 4.4, we obtain the following.

Corollary 4.5. Let be an ordered partial metric space which is complete, and let have the mixed monotone property. Suppose thatthere exists such that holds for all satisfying ( and ) or ( and ) and conditions (ii), (iii), respectively, (iii′) of Corollary 3.4 hold true. Then there exist such that that is, has a coupled fixed point.

5. Coupled Coincidence and Fixed Points under Integral Conditions

Denote by the set of functions that are Lebesgue integrable and summable (having finite integral) on each compact subset of , and satisfying condition for each . Branciari [12] was the first to use integral-type contractive conditions in order to obtain fixed point results. He proved the following.

Theorem 5.1 (see [12]). Let be a complete metric space, and let be a self-map satisfying for some and some . Then has a unique fixed point and for each the Picard sequence converges to when .

Subsequently, several authors proved such results, including the very recent paper of Liu et al. [23].

We begin with the following.

Lemma 5.2. Let be an ordered partial metric space which is complete. Let be self-maps such that is continuous, , and one of these two subsets of is closed. Suppose that is -nondecreasing (with respect to ) and there exists with or . Assume also that there exists and such that holds for all such that and are comparable. Assume that either 1°: is continuous and the pair is -compatible or satisfies (ORC). Then, and have a coincidence point in .

Proof. Take with, say, , and using that is -nondecreasing and that form the sequence satisfying , , and Since and are comparable, we can apply the contractive condition (5.2) to obtain If for some , then it easily follows that and is a point of coincidence of and . Suppose that for each (and, hence, the strict inequality holds in (5.4)). We will prove that is a nonincreasing sequence. Suppose, to the contrary, that for some . Then, since , we get that a contradiction.
Denote by the limit of the nonincreasing sequence of positive numbers. Suppose that . Then, using Lemmas 2.1 and 2.2 from [23], the contractive condition (5.2), and property (4.1) of function , we get that a contradiction. We conclude that .
Now we prove that is a Cauchy sequence in (and in ). If this were not the case, as in the proof of Lemma 3.2, using Lemma 2.11, we would get that there exist and two sequences and of positive integers such that the following four sequences tend to when : Putting in the contractive condition and , it follows that Passing to the upper limit as and using properties of functions , as well as [23, Lemma 2.1], we get a contradiction Thus, is a Cauchy sequence, converging to some (which we suppose to be closed in ) such that . We will prove that and have a coincidence point.
(i) Suppose that is continuous and is -compatible. As in the proof of Lemma 3.2, we get that and is a coincidence point of and .
(ii) If satisfies (ORC), since is an increasing sequence tending to , we have that for each . So we can apply (P4) and the contractive condition to obtain Passing to the upper limit as , using properties of functions , , as well as [23, Lemma 2.1], we get that . Hence, we obtain that .

Putting and in Lemma 5.2, we obtain an ordered partial metric extension of Branciari's Theorem 5.1. The following example shows that this extension is proper.

Example 5.3. Let be endowed with the standard metric and order, and consider defined by , and , . Take given by . Then condition (5.1) does not hold. Indeed, would imply that for all . But taking and would give that , a contradiction.

On the other hand, consider the same problem in an (ordered) partial metric space, with the partial metric given by . Then, condition (5.2) reduces to which is, for , equivalent to and holds true (for each ) if . Obviously, has a (unique) fixed point .

The following theorem is obtained from Lemma 5.2 in a similar way as Theorems 3.3 and 4.4 from respective lemmas.

Theorem 5.4. Let be an ordered partial metric space which is complete, and let and be such that has the -mixed monotone property. Suppose thatthere exists and such that holds for all satisfying ( and ) or ( and ); and conditions (ii), (iii), (iv), respectively, (iv′) of Theorem 3.3 hold true. Then there exist such that that is, and have a coupled coincidence point.

Putting in Theorem 5.4, we obtain the following.

Corollary 5.5. Let be an ordered partial metric space which is complete, and let have the mixed monotone property. Suppose thatthere exists and such that holds for all satisfying ( and ) or ( and ) and conditions (ii), (iii), respectively, (iii′) of Corollary 3.4 hold true. Then there exist such that that is, has a coupled fixed point.

Acknowledgments

The authors are thankful to the referees for very useful suggestions that helped to improve the paper. They are also thankful to the Ministry of Science and Technological Development of Serbia.

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