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

Mohamed Jleli,¹ Erdal Karapınar,² and Bessem Samet¹

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., [2–9] 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.

References

S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728, pp. 183–197, Annals of the New York Academy of Sciences, 1994.

T. Abdeljawad, E. Karapınar, and K. Taş, “A generalized contraction principle with control functions on partial metric spaces,” Computers & Mathematics with Applications, vol. 63, no. 3, pp. 716–719, 2012.

I. Altun, F. Sola, and H. Simsek, “Generalized contractions on partial metric spaces,” Topology and Its Applications, vol. 157, no. 18, pp. 2778–2785, 2010.

L. Ćirić, B. Samet, H. Aydi, and C. Vetro, “Common fixed points of generalized contractions on partial metric spaces and an application,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2398–2406, 2011.

S. Oltra and O. Valero, “Banach's fixed point theorem for partial metric spaces,” Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 36, no. 1-2, pp. 17–26, 2004.

D. Paesano and P. Vetro, “Suzuki's type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces,” Topology and Its Applications, vol. 159, no. 3, pp. 911–920, 2012.

S. Romaguera, “Fixed point theorems for generalized contractions on partial metric spaces,” Topology and Its Applications, vol. 159, no. 1, pp. 194–199, 2012.

I. A. Rus, “Fixed point theory in partial metric spaces,” Analele Universităţii de Vest din Timişoara. Seria Matematică-Informatică, vol. 46, no. 2, pp. 149–160, 2008.

B. Samet, M. Rajović, R. Lazović, and R. Stojiljković, “Common fixed point results for nonlinear contractions in ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 71, 14 pages, 2011.

R. H. Haghi, Sh. Rezapour, and N. Shahzad, “Be careful on partial metric fixed point results,” Topology and Its Applications, vol. 160, no. 3, pp. 450–454, 2013.

B. Samet, C. Vetro, and F. Vetro, “From metric spaces to partial metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 5, 11 pages, 2013.