Abstract and Applied Analysis

Volume 2012 (2012), Article ID 451239, 12 pages

doi:10.1155/2012/451239

Research Article

Nonlinear Contractions in 0-Complete Partial Metric Spaces

Abd Ghafur Bin Ahmad,¹ Zaid Mohammed Fadail,¹ Vesna Ćojbašić Rajić,² and Stojan Radenović³

¹School of Mathematical Sciences, Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor Darul Ehsan, Malaysia

²Faculty of Economics, University of Belgrade, Kamenička 6, 11000 Beograd, Serbia

³Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd, Serbia

Received 20 March 2012; Accepted 23 May 2012

Academic Editor: Gaston Mandata N'Guerekata

Copyright © 2012 Abd Ghafur Bin Ahmad 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.

Using the setting of 0-complete partial metric spaces, some common fixed point results of maps that satisfy the nonlinear contractive conditions are obtained. These results generalize and improve the existing fixed point results in the literature in the sense that weaker conditions are used. An example shows how our result can be used when the corresponding result in standard metric cannot.

1. Introduction and Preliminaries

Matthews [1] generalized the concept of a metric space by introducing partial metric spaces. Based on the notion of partial metric spaces, Matthews [1, 2], Oltra and Valero [3], Ilić et al. [4, 5] obtained some fixed point theorems for mappings satisfying different contractive conditions. Recently, Abdeljawad et al. [6] proved one fixed point result for generalized contraction principle with control functions on partial metric spaces. For some new results on partial metric and cone metric spaces, see [1–27].

The aim of this paper is to continue the study of common fixed points of mappings but now in 0-complete partial metric spaces, under nonlinear generalized contractive conditions.

Consistent with Matthews [1, 2], O'Neill [21, 22], and Oltra et al. [23], the following definitions and results will be needed throughout this paper.

Definition 1.1. A partial metric on a nonempty set is a function 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 .

For a partial metric on , the function given by

is a (usual) metric on . Each partial metric on generates a topology on with a base of the family of open balls , where for all and .

Definition 1.2 (see [1, 20]). (i) A sequence in a partial metric space converges to if and only if .

(ii) a sequence in a partial metric space is called 0-Cauchy if .(iii) a partial metric space is said to be 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that . In this case, is a 0-complete partial metric on .

Remark 1.3. (1) (see [20]) Clearly, a limit of a sequence in a partial metric space does not need to be unique. Moreover, the function does not need to be continuous in the sense that and implies . For example, if and for , then for for each and so, for example, and when .

(2) (see [6]) However, if , then for all .

Assertions similar to the following lemma were used (and proved) in the course of proofs of several fixed point results in various papers [8, 9, 20, 28].

Lemma 1.4. Let be a partial metric space and let be a sequence in such that is nonincreasing and that

If is not a 0-Cauchy sequence in , then there exist and two sequences and of positive integers such that and the following four sequences tend to when

Definition 1.5 (see [29]). Let and be self-maps of a set . If for some , then is called a coincidence point of and , and is called a point of coincidence of and . The pair of self-maps is weakly compatible if they commute at their coincidence points.

The following lemma is Proposition 1.4 of [29].

Lemma 1.6. Let and be weakly compatible self-maps of a set . If and have a unique point of coincidence , then is the unique common fixed point of and .

Definition 1.7 (see [30]). The following two classes of mappings are defined as is a nondecreasing right continuous function such that for all , and is a nondecreasing function such that for all .

It is clear for the function

that but .

Lemma 1.8 (see [19]). If , then for all and .

2. Common Fixed Point Results

In this section, we obtain some common fixed point results defined on 0-complete partial metric spaces.

Theorem 2.1. Let be a partial metric space and be mappings on with . Assume that

for all , where . If or is a complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. First, we prove that and have a unique point of coincidence (if it exists). If with and with , we assume . Applying (2.1) we have

This implies a contradiction, so we conclude that .

Now, consider the sequence defined by and . Consider the two possible cases:

(i) for some . In this case ( a point of coincidence). Because of Lemma 1.6 (unique point of coincidence and weakly compatible mappings) we have a unique common fixed point.(iii) for every .

Applying (2.1) with and , we have,

If is a maximum, then we have

which is a contradiction. So, we conclude that the maximum is . Now, we have the following:

It follows from (2.5) that the sequence is monotone decreasing. According to the properties of function , it follows

when . Therefore, .

Next, we prove that is a 0-Cauchy sequence in the space (). It is sufficient to show that is a 0-Cauchy sequence. Assume the opposite. Then using Lemma 1.4 we get that there exist and two sequences and of positive integers and sequences

all tend to when . Applying condition (2.1) to elements and and putting for each , we get that

When , we have

Using properties of , we obtain a contradiction , since .

This shows that is a 0-Cauchy sequence in the space and hence is a 0-Cauchy sequence in . If we suppose that is a 0-complete subspace of , then there exists such that

If is a 0-complete subspace of with , (2.10) also holds.

Let and . We show that is a point of coincidence of and . Suppose that . We have

As , we have and by Remark 1.3 (2) . Since , there exists such that for every we have and . Now, for , we obtain

or

As we have a contradiction:

This implies , that is, . Hence, and have a (unique) point of coincidence. From Lemma 1.6 follows that this is a unique common fixed point of and .

Remark 2.2. Assumption about right continuity of the function is only used in the proof that is a 0-Cauchy sequence.

In the following theorem we consider a weaker assumption for the function . As a compensation we assume a bit stronger contractive condition.

Theorem 2.3. Let be a partial metric space and be mappings on with . Assume that

for all , where . If or is a complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. We can prove that and have a unique point of coincidence in a similar way like in Theorem 2.1. If we consider the sequence defined by and , we used Theorem 2.1 to show .

Here, we only prove that is a 0-Cauchy sequence in the space by using induction. Let us denote for each , We have , as . For there exists such that for every we have . Assuming that for some and it holds that and we need to prove that .

We have

Then (2.16) becomes . This shows that is a 0-Cauchy sequence in . Further, proof is similar as in Theorem 2.1, so we omit it.

Corollary 2.4. Let be a partial metric space and let be a map such that

for all , where . If is a 0-complete subspace of , then has a unique fixed point.

Proof. It follows from Theorem 2.1 by taking (the identity map).

Corollary 2.5. Let be a partial metric space and let be a map such that

for all , where . If is a 0-complete subspace of , then has a unique fixed point.

Proof. It follows from Theorem 2.3 by taking (the identity map).

Corollary 2.6. Let be a partial metric space and let be mappings on with . Assume that

for all , where . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique fixed point.

Proof. It follows from Theorem 2.1 by taking , where .

Corollary 2.7. Let be a partial metric space and let be mappings on with . Assume that

for all , where and . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique fixed point.

Proof. It follows from Corollary 2.6 by noting that

Corollary 2.8. Let be a partial metric space and let be mappings on with . Assume that

for all , where . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique fixed point.

Proof. It follows from Corollary 2.7.

From Corollary 2.8 follows the theorem proved by Jungck [31], where we consider a 0-complete partial metric space instead of a complete metric space, under the assumption that and are commuting mappings.

Corollary 2.9. Let be a 0-complete partial metric space and be a map such that

for all , where . If is a closed subspace of , then has a unique fixed point.

Proof. It follows from Theorem 2.3 and Corollary 2.5. by taking (the identity map).

Under a weaker assumption for the function , from Corollary 2.9 follows the theorem proved by Boyd and Wong [32] where we consider a partial metric instead of a standard metric.

From Corollary 2.7 we deduce the following corollaries, which are extensions of some well-known theorems.

Corollary 2.10. Let be a 0-complete partial metric space and let be a map such that

for all , where . Then has a unique fixed point.

This corollary is an extension of Banach contraction theorem on a 0-complete partial metric space. This corollary is already mentioned in Matthews [2] for a complete partial metric space, but is true for a 0-complete partial metric spaces.

Corollary 2.11. Let be a 0-complete partial metric space and let be a map such that

for all , where . Then has a unique fixed point.

This corollary is extension of Kannan theorem [33] (4) on a 0-complete partial metric spaces.

Corollary 2.12. Let be a 0-complete partial metric space and let be a map such that

for all , where a and . Then has a unique fixed point.

This corollary is extension of Reich theorem [33] (8) on a 0-complete partial metric spaces.

Corollary 2.13. Let be a 0-complete partial metric space and let be a map such that

for all , where . Then has a unique fixed point.

This corollary is extension of Chatterjea theorem [33] (11) on a 0-complete partial metric spaces.

Corollary 2.14. Let be a 0-complete partial metric space and let be a map such that at least one of the following is true

for all , where . Then has a unique fixed point.

This corollary is extension of Zamfirescu theorem [33] (19) on a 0-complete partial metric spaces.

We demonstrate the use of Theorem 2.1 with the help of the following example.

Example 2.15. Let , where denotes the set of rational numbers and is given by . Then is a 0-complete partial metric space which is not complete. Suppose that are such that for all and

Then . Without loss of generality assume that . Then and . From (2.1) follows

and so . If , then we have (this is true because ). If then we have , which is true. It follows that has a unique fixed point.

On the other hand, consider the same problem in the standard metric and take . Then, from (2.1) follows and

Hence does not hold and the existence of a unique fixed point cannot be obtained.

Remark 2.16. Note that Theorem 2.1 improves [12, Theorem 1 and Corollary 1], [25, Theorem 3, Corollaries 1 and 2 and Theorem 4], and [13, Corollary 2.3] since our assumptions are weaker than the assumptions from [12, 13, 25] in several places.

Finally, it is worth to notice that all new results in recently papers [7, 10, 15, 20, 27] are true if partial metric space is 0-complete instead complete.

Acknowledgments

The authors are thankful to the referees for their remarks which helped to improve the presentation of the paper. The authors (first and second) would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant OUP-UKM-FST-2012. The third and the fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.

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