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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 715456, 6 pages
http://dx.doi.org/10.1155/2013/715456
Research Article

Further Remarks on Fixed-Point Theorems in the Context of Partial Metric Spaces

1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 27 June 2013; Accepted 29 July 2013

Academic Editor: Chi-Ming Chen

Copyright © 2013 Mohamed Jleli 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 fixed-point theorems on metric spaces are established, and analogous results on partial metric spaces are deduced. This work can be considered as a continuation of the paper Samet et al. (2013).

1. Introduction

In 1994, Matthews [1] introduced the concept of partial metric spaces as a part of the study of denotational semantics of dataflow networks and showed that the Banach’s contraction principle can be generalized to the partial metric context for applications in program verification. Later on, many authors studied fixed-point theorems on partial metric spaces (see, e.g., [29] and references therein).

We start by recalling some basic definitions and properties of partial metric spaces (see [1, 5] for more details).

Definition 1. A partial metric on a nonempty set is a function such that for all , we have(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 . A basic example of a partial metric space is the pair , where for all . Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1].

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

Definition 2. Let be a partial metric space and a sequence in . Then converges to a point if and only if . We may write this as ; is called a Cauchy sequence if exists and is finite; is said to be complete if every Cauchy sequence in converges, with respect to , to a point , such that .
If is a partial metric on , then the function given by is a metric on .

Lemma 3. Let be a partial metric space. Then(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space ;(b)a partial metric space is complete if and only if the metric space is complete; furthermore, if and only if

In [1], Matthews extended the Banach contraction principle to the setting of partial metric spaces.

Theorem 4 (see [1]). Let be a complete partial metric space and a given mapping. Suppose that there exists a constant such that Then has a unique fixed point . Moreover, we have .

Very recently, in [10, 11], the authors proved that a large class of fixed-point theorems in partial metric spaces, including Matthews result, are immediate consequences of fixed-point theorems in metric spaces. More precisely, in [11], the authors established the following result.

Theorem 5 (see [11]). Let be a complete metric space, , and a lower semicontinuous function. Suppose that there exists such that for all . Then has a unique fixed point . Moreover, one has .

Now, taking in Theorem 4, we obtain that for all . Applying Theorem 5, we obtain immediately the result of Matthews. Another technique suggested by Haghi et al. [10] let be a metric on defined by It is not difficult to show that if is complete, then is complete. Now, Matthews’ contraction is equivalent to for all . So, the result of Matthews can be deduced immediately from the Banach contraction principle. Observe that in Theorem 5, if we consider the mapping defined by then is a metric on and is a complete metric space. Moreover, the contractive condition in Theorem 5 is equivalent to for all . So, Theorem 5 can also be deduced from the Banach contraction principle.

Define the function by Then, the contractive condition in Theorem 5 is equivalent to for all . In this paper, we establish new fixed-point theorems in metric spaces involving a function , where the mapping is not a metric on . Some fixed-point theorems in partial metric spaces are deduced from our main results in metric spaces.

2. Main Result

We denote by the set of functions satisfying the following conditions:(i) for all ,(ii),(iii) is continuous.

As examples, the following functions belong to :(a),(b),(c),(d),(e).

We denote by the set of functions satisfying the following conditions:(j) is nondecreasing,(jj) for each , where is the th iterate of .

It is not difficult to show that if , then for every . These functions are known in the literature as -comparison functions.

Definition 6. Let be a metric space, , , and . We say that the pair satisfies an -generalized -contraction if there exists such that for every .

Our main result is giving by the following theorem.

Theorem 7. Let be a complete metric space, , and . Suppose that the following conditions hold: (1) is lower semicontinuous,(2)there exist and such that the pair satisfies an -generalized -contraction.
Then has a unique fixed point . Moreover, one has .

Proof. Let . Define the sequence by for all positive integer . Let be a positive integer such that . Using condition with and , we obtain that Without restriction of the generality, we can suppose that for every . Suppose that In this case, we have Note that . Indeed, if , from condition (i), we have , which implies that , that is a contradiction with the assumption for every . Since for every , we get that which is a contradiction. Then, we deduce that for every . Using the property (i) of and the monotony property of , we obtain that for every . Fix , and let be a positive integer (given by (jj)) such that Let ; using the triangular inequality and (20), we obtain Thus we proved that is a Cauchy sequence in the metric space . Since is complete, there exists some such that as . We shall prove that is a fixed point of . At first, observe that from condition (jj) and (20), we have which implies (since is lower semicontinuous) that Using condition with and , we get that for every . Suppose that . Using the continuity of , (23), and (24), we have Then, there exists a positive integer such that for every . Thus, we have Letting in the above inequality, we get that which is a contradiction. Thus, we proved that . Now, suppose that is another fixed point of . We can observe that Indeed, applying (14) with , we get that which implies (30). Now, applying (14) with and , we obtain Using (24) and (30), we obtain which implies that ; that is, . Thus, we proved that has a unique fixed point with .

Example 8. Let be the mapping defined by We endow with the standard metric . Let for all , for all , and for all . Clearly, is continuous, , and . Moreover, we have for all . By Theorem 7, has a unique fixed point () with . Note that in this example, the Banach contraction principle cannot be used since the mapping is not continuous.

3. Particular Cases

In this section, new fixed point results are deduced from our main result given by Theorem 7.

Corollary 9. Let be a complete metric space, , and . Suppose that the following conditions hold: (1) is lower semicontinuous,(2) there exists such that for all .

Then has a unique fixed point . Moreover, we have .

Proof. It follows from Theorem 7 with .

Corollary 10. Let be a complete metric space, , and . Suppose that the following conditions hold:(1) is lower semicontinuous,(2) there exists such that for all .

Then has a unique fixed point . Moreover, we have .

Proof. It follows from Theorem 7 with .

Corollary 11. Let be a complete metric space, , and . Suppose that the following conditions hold:(1) is lower semicontinuous,(2) there exists such that for all .

Then has a unique fixed point . Moreover, we have .

Proof. It follows from Theorem 7 with .

Corollary 12. Let be a complete metric space, , and . Suppose that the following conditions hold:(1) is lower semicontinuous,(2) there exists such that for all .

Then has a unique fixed point . Moreover, we have .

Proof. It follows from Theorem 7 with .

Corollary 13. Let be a complete metric space, , and . Suppose that the following conditions hold:(1) is lower semicontinuous,(2) there exists such that for all .

Then has a unique fixed point . Moreover, we have .

Proof. It follows from Theorem 7 with .

Corollary 14. Let be a complete metric space, , and . Suppose that the following conditions hold: (1) is lower semicontinuous,(2) there exists such that for all .

Then has a unique fixed point . Moreover, we have .

Proof. It follows from Theorem 7 with .

Many other results can be deduced from Theorem 7 by considering different choices of .

4. Applications to Partial Metric Spaces

We will show that the following fixed-point theorem in partial metric spaces can be deduced from Theorem 7.

Corollary 15. Let be a complete partial metric space, and let . Suppose that there exists such that for all . Then has a unique fixed point . Moreover, one has .

Proof. Let for all and for all . We have where Now, inequality (42) is equivalent to for every . Applying Theorem 7, we obtain that has a unique fixed point with ; that is, .

The following results follow immediately from Corollary 15.

Corollary 16. Let be a complete partial metric space, and let . Suppose that there exists such that for all . Then has a unique fixed point . Moreover, one has .

Corollary 17. Let be a complete partial metric space, and let . Suppose that there exists such that for all . Then has a unique fixed point . Moreover, one has .

Note that Matthews result (see Theorem 4) follows from Corollary 16 by taking , where .

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RGP-VPP-237.

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