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Chinese Journal of Mathematics
Volume 2013 (2013), Article ID 859531, 8 pages
http://dx.doi.org/10.1155/2013/859531
Research Article

A Generalization of Prešić Type Mappings in 0-Complete Ordered Partial Metric Spaces

1Department of Applied Mathematics, Shri Vaishnav Institute of Technology and Science, Gram Baroli, Sanwer Road, Indore 453331, India
2Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

Received 25 July 2013; Accepted 13 August 2013

Academic Editors: G. Liu and S. A. Marano

Copyright © 2013 Satish Shukla and Stojan Radenović. 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

The purpose of this paper is to prove some coincidence and common fixed point results for mappings satisfying Prešić type contraction condition in 0-complete ordered partial metric spaces. The results proved in this paper generalize and extend the results of Prešić and Matthews in 0-complete ordered partial metric spaces. Some examples are included which show that the generalization is proper.

1. Introduction

Banach contraction mapping principle is one of the most interesting and useful tools in applied mathematics. In recent years many generalizations of Banach contraction mapping principle have appeared. In 1965 Prešić [1, 2] extended Banach contraction mapping principle to mappings defined on product spaces and proved the following theorem.

Theorem 1. Let be a complete metric space, a positive integer, and a mapping satisfying the following contractive type condition: for every , where are nonnegative constants such that . Then there exists a unique point such that . Moreover if are arbitrary points in and for , then the sequence is convergent and .

Note that condition (1) in the case reduces to the well-known Banach contraction mapping principle. So, Theorem 1 is a generalization of the Banach fixed point theorem. Some generalization of Theorem 1 can be seen in [111].

On the other hand in 1994 Matthews [12] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. In partial metric space the usual metric was replaced by partial metric, with a property that the self-distance of any point may not be zero. Result of Matthews is generalized by several authors in different directions (see, e.g., [1329]). Romaguera [30] introduced the notion of -Cauchy sequence and -complete partial metric spaces and proved some characterizations of partial metric spaces in terms of completeness and -completeness. Some results on -complete partial metric spaces can be seen in [3032].

The existence of fixed point in partially ordered sets was investigated by Ran and Reurings [33] and then by Nieto and Rodríguez-López [34, 35]. Fixed point results in ordered partial metric spaces were obtained by several authors (see, e.g., [14, 1619, 27, 29]). Very recently, in [7] (see also [36]) authors introduced the ordered Prešić type contraction and generalized the result of Prešić and proved some fixed point theorems for such mappings. In this paper, we generalize and extend the result of Prešić [1, 2] in -complete ordered partial metric spaces. A generalization of result of Prešić in -complete partial metric spaces is also established. Some examples are included which show that the generalization is proper.

First we recall some definitions and properties of partial metric space [12, 30, 32, 37, 38].

Definition 2. A partial metric on a nonempty set is a function ( stands for nonnegative reals) such that for all :(P1), (P2), (P3), (P4).
A partial metric space is a pair such that is a nonempty set and is a partial metric on .

It is clear that if , then from (P1) and (P2) . But if , may not be . Also every metric space is a partial metric space, with zero self-distance.

Example 3. If is defined by , for all , then is a partial metric space.

Each partial metric on generates a topology on which has a base the family of open -balls , where for all and .

Theorem 4 (see [12]). For each partial metric the pair , where for all , is a metric space.

Here is called induced metric space and is induced metric. In further discussion until unless specified will represent induced metric space.

Let be a partial metric space. (1)A sequence in converges to a point if and only if .(2)A sequence in is called Cauchy sequence if there exists (and is finite) .(3) is said to be complete if every Cauchy sequence in converges with respect to to a point such that .(4)A sequence in is called -Cauchy sequence if . The space is said to be -complete if every -Cauchy sequence in converges with respect to to a point such that .

Lemma 5 (see [12, 30, 32, 37]). Let be a partial metric space and any sequence in .(i) is a Cauchy sequence in if and only if it is a Cauchy sequence in metric space .(ii) is complete if and only if the metric space is complete. Furthermore, if and only if .(iii)Every -Cauchy sequence in is Cauchy in .(iv)If is complete, then it is -complete.

The converse assertions of (iii) and (iv) do not hold. Indeed the partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an easy example of a -complete partial metric space which is not complete. Also, it is easy to see that every closed subset of a -complete partial metric space is -complete.

Definition 6. Let be a nonempty set, a positive integer, and a mapping. If , then is called a fixed point of .

Definition 7. Let be a nonempty set, a positive integer, and and mappings. (a)An element is said to be a coincidence point of and if . (b)If , then is called a point of coincidence of and . (c)If , then is called a common fixed point of and . (d)Mappings and are said to be commuting if for all . (e)Mappings and are said to be weakly compatible if they commute at their coincidence points.

Lemma 8 (see [8]). Let be a nonempty set, a positive integer, and , two weakly compatible mappings. If and have a unique point of coincidence , then is the unique common fixed point of and .

Remark that the previous definition in the case reduces to the usual definitions of commuting and weakly compatible mappings in the sense of [39] (for details, see Introduction from [39]).

Definition 9. Let a nonempty set be equipped with a partial order “” such that is a partial metric space; then is called an ordered partial metric space. A sequence in is said to be nondecreasing with respect to “” if . Let be a positive integer and a mapping; then is said to be nondecreasing with respect to “” if for any finite nondecreasing sequence we have . Let be a mapping. is said to be -nondecreasing with respect to “” if for any finite nondecreasing sequence we have .

Remark 10. For previous definitions reduce to usual definitions of fixed point and nondecreasing mapping in partial metric space.

Definition 11. Let be a nonempty set equipped with partial order “”, and let be a mapping. A nonempty subset of is said to be well ordered if every two elements of are comparable. Elements are called -comparable if and are comparable. is called -well ordered if for all , and are -comparable; that is, and are comparable.

Example 12. Let , “” a partial order relation on defined by ,. Let and be defined by ,,,. Then it is clear that is not well ordered but it is -well ordered.

Let be an ordered partial metric space, a positive integer. Let be a mapping; is called ordered Prešić contraction if for all with , where are nonnegative constant such that .

If (2) is satisfied for all , then is called Prešić contraction.

Note that in ordered partial metric spaces a Prešić contraction is necessarily an ordered Prešić contraction, but converse may not be true (see Example 21 of this paper).

Remark 13. Since every metric space is a partial metric space and there exist mappings which are Prešić contraction in partial metric spaces but not in metric spaces (see Example 20 of this paper), therefore the class of Prešić contraction in partial metric spaces is wider than that in metric spaces.
The following lemma will be useful in sequel.

Lemma 14. Let be any ordered partial metric space, a positive integer. Let , be two mappings satisfying for all with where are nonnegative constant such that . If and have a point of coincidence , then .

Proof. Suppose that is a point of coincidence of and with coincidence point , that is, . As , using (3) we obtain as , we obtain .

Now we can state our main results.

2. Main Results

Theorem 15. Let be any -complete ordered partial metric space and a positive integer. Let , be two mappings such that and is a closed subspace of . Suppose the following conditions hold: (I)for all with , where are nonnegative constant such that ; (II) there exist such that ; (III) is -nondecreasing; (IV) if a nondecreasing sequence converges to , then for all and .
Then and have a point of coincidence. If in addition and are weakly compatible, then and have a common fixed point with . Moreover, the set of common fixed points of and is -well ordered if and only if and have a unique common fixed point.

Proof. Starting with given we define a sequence as follows: let . As and , define . Then and is -nondecreasing, so , that is, . Continuing this procedure, we obtain that is,
Thus is a nondecreasing sequence with respect to “.”
For simplicity set , where .
By mathematical induction we will show that
According to the definition of it is clear that (8) is true for . Let the following inequalities be the induction hypothesis.
As , so using (5) we obtain and the inductive proof of (8) is complete.
Let with ; then
As , previous inequality implies that
Thus is a -Cauchy sequence in . Since is -complete as it is closed subspace, it follows that converges in . Therefore where for some .
We will show that is coincidence point of and and .
Then for any integer , we have By assumption (IV) it follows that for all and , so using (5), previous inequality implies that Using (13) and the fact that , previous inequality implies that
Thus is point of coincidence, and is coincidence point of and .
Suppose and are weakly compatible, so so is a point of coincidence of and , and by Lemma 14, we have . Again since we obtain from (5) that As ,, and it follows from previous inequality that as , we obtain Thus is common fixed point of and .
Suppose that the set of common fixed points is -well ordered. We will show that common fixed point is unique. Assume on the contrary that is another common fixed point of and , that is, and . As and are -comparable let, for example, . From (5), it follows that
As and from Lemma 14, , it follows from previous inequality that a contradiction. Therefore common fixed point is unique. For converse, if common fixed point of and is unique, then the set of common fixed points of and is singleton therefore -well ordered.

Remark 16. For previous theorem is a generalization and extension of result of Matthews [12] in ordered partial metric spaces.

Taking , that is, identity mapping of , we get the following fixed point result for ordered Prešić contraction in partial metric spaces.

Corollary 17. Let be any -complete ordered partial metric space and a positive integer. Let be a mapping and the following conditions hold: (I) is an ordered Prešić contraction; (II)there exist such that ; (III) is nondecreasing; (IV)if a nondecreasing sequence converges to , then for all .
Then has a fixed point with . Moreover, the set of fixed points of is well ordered if and only if has a unique fixed point.

Following theorem generalizes and extends the result of Prešić in partial metric spaces.

Theorem 18. Let be any -complete partial metric space and a positive integer. Let , be two mappings such that , is a closed subspace of , and for all , where are nonnegative constant such that . Then and have a point of coincidence. Furthermore, if and are weakly compatible, then and have a unique common fixed point with .

Proof. We note that the inequality (23) is true for all ; therefore the proof of theorem follows from similar process as used in the proof of Theorem 15.

Taking that is, identity mapping in the previous theorem, we obtain the following corollary.

Corollary 19 (Prešić type). Let be any -complete partial metric space and a positive integer. Let be a mapping satisfying for all , where are nonnegative constant such that . Then has a unique fixed point with .

Following is an example which illustrates that a Prešić type contraction in partial metric space need not to be a Prešić type contraction in metric space.

Example 20. Let and defined by for all . Then is a -complete partial metric space. For , suppose be defined by Then is Prešić type contraction in partial metric space , that is, it satisfies (24) with , while it is not a Prešić type contraction in usual metric space where for all (note that is also the induced metric).

Proof. For and condition (24) becomes for all , where is a nonnegative constant such that . Note that if , then (26) is satisfied trivially.
If and , then (26) is valid for . Indeed since so result follows.
If any two of are equal to , for example, if and , then (26) is valid. In this case
Hence, if , then .
Similarly in all possible cases (26) is satisfied for . Note that all other conditions of Corollary 19 are satisfied and has a unique fixed point , that is, , with .
Again for , will be Prešić contraction in if for all , where , are nonnegative constant such that . We will show that (29) is not satisfied for certain points in . Let ; then but ; therefore for these values of , and (29) does not hold. Thus is not a Prešić contraction in , and so we cannot apply the result of Prešić to conclude the existence of fixed point of .

Following example illustrates that an ordered Prešić contraction need not to be a Prešić contraction in ordered partial metric spaces, and the fixed point of ordered Prešić contraction may not be unique (when set of fixed points of is not well ordered).

Example 21. Let and let order relation “” be defined by suppose be defined by
Then is an ordered complete partial metric space. For , suppose be a mapping defined by
Then(a) is an ordered Prešić contraction that, satisfies (2) with ,(b) is not a Prešić contraction; that is, does not satisfy (2) for all .

Proof. (a) For and , (2) becomes for all with , where is nonnegative constant such that . Note that if , then (34) is satisfied trivially.
Now we have to check the validity of inequality (34) only for and .
Let with , that is, ; then (34) is valid for . Indeed since so result follows.
Similarly in all possible cases (34) is satisfied for .
Note that all other conditions of Corollary 17 are satisfied (except that the set of fixed points of is well ordered) and has two fixed points and , that is, ,, with . Note that set of fixed points of is not well ordered as .
(b) For , (2) becomes
We will show that (36) is not satisfied for all , with , in particular, let and ; then but ; therefore for these values of ,, and (36) does not hold. Thus is not Prešić contraction in .

Conflict of Interests

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

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