Abstract and Applied Analysis

Volume 2013 (2013), Article ID 986028, 9 pages

http://dx.doi.org/10.1155/2013/986028

Research Article

Fixed Points for Weak --Contractions in Partial Metric Spaces

Poom Kumam,¹ Calogero Vetro,² and Francesca Vetro³

¹Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

²Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy

³Università degli Studi di Palermo, DEIM, Viale delle Scienze, 90128 Palermo, Italy

Received 22 January 2013; Revised 23 May 2013; Accepted 23 May 2013

Academic Editor: Tomas Dominguez

Copyright © 2013 Poom Kumam 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.

Recently, Samet et al. (2012) introduced the notion of --contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notion of weak --contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces. Also, we deduce fixed point results in ordered partial metric spaces. Our results extend and generalize the results of Samet et al.

1. Introduction

The notion of partial metric is one of the most useful and interesting generalizations of the classical concept of metric. The partial metric spaces were introduced in 1994 by Matthews [1] as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle can be generalized to the partial metric context for applications in program verification. Later on, many authors studied the existence of several connections between partial metrics and topological aspects of domain theory (see [2–8] and the references therein). On the other hand, some researchers [9, 10] investigated the characterization of partial metric 0-completeness in terms of fixed point theory, extending the characterization of metric completeness [11–14].

Recently, Samet et al. [15] introduced the notion of --contractive mappings and established some fixed point results in the setting of complete metric spaces. In this paper, we introduce the notion of weak --contractive mappings and give fixed point results for this class of mappings in the setting of partial metric spaces. Also, we deduce fixed point results in ordered partial metric spaces. Our results extend and generalize Theorems 2.1–2.3 of [15] and many others. An application to ordinary differential equations concludes the paper.

2. Preliminaries

In this section, we recall some definitions and some properties of partial metric spaces that can be found in [1, 5, 10, 16, 17]. A partial metric on a nonempty set is a function such that, for all , we have

), (), (), ().

A partial metric space is a pair such that is a non-empty set and is a partial metric on . It is clear that if , then from and it follows that . 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 can 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 .

Let be a partial metric space. A sequence in converges to a point if and only if .

A sequence in is called a Cauchy sequence if there exists (and is finite)

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that

A sequence in is called 0-Cauchy if . We say that is 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that .

On the other hand, the partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an example of a 0-complete partial metric space which is not complete.

It is easy to see that every closed subset of a complete partial metric space is complete.

Notice that if is a partial metric on , then the function given by

is a metric on . Furthermore, if and only if

Lemma 1 (see [1, 16]). 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.

Let be a non-empty set. If is a partial metric space and is a partially ordered set, then is called an ordered partial metric space. Then are called comparable if or holds. Let be a partially ordered set, and let be a mapping. is a non-decreasing mapping if whenever for all .

Definition 2 (see [15]). Let and . One says that is -admissible if

Example 3. Let , and define the function by

Then, every non-decreasing mapping is -admissible. For example the mappings defined by and for all are -admissible.

3. Main Results

Throughout this paper, the standard notations and terminologies in nonlinear analysis are used. We start the main section by presenting the new notion of weak --contractive mappings.

Denote by the family of non-decreasing functions such that and for each , where is the th iterate of .

Remark 4. Notice that the family used in this paper is larger (less restrictive) than the corresponding family of functions defined in [15], see also next Examples 12–13.

Lemma 5. For every function , one has for each .

Definition 6. Let be a partial metric space, and let be a given mapping. We say that is a weak --contractive mapping if there exist two functions and such that

for all . If

for all , then is an --contractive mapping.

Remark 7. If satisfies the contraction mapping principle, then is a weak --contractive mapping, where for all and for all and some .

In the sequel, we consider the following property of regularity. Let be a partial metric space, and let be a function. Then

(r) is -regular if for each sequence , such that for all and , we have that for all , (c) has the property (C) with respect to if for each sequence , such that for all , there exists such that for all .

Remark 8. Let be a non-empty set, and let be a function. Denote

If is a transitive relation on , then has the property (C) with respect to .

In fact, if is a sequence such that for all , then for all . Now, fix and show that

Obviously, (9) holds if . Assume that (9) holds for some . From , since is transitive, we get . This implies that , and so for all ; that is, has the property (C) with respect to .

Remark 9. Let be an ordered partial metric space, and let be a function defined by

Then has the property (C) with respect to . Moreover, if for each sequence , such that for all convergent to some , we have for all , and then is -regular.

By Remark 8, has the property (C) with respect to . Now, let be a sequence such that for all convergent to some , and then for all , and hence for all . This implies that for all , and so is -regular.

Our first result is the following theorem that generalizes Theorem 2.1 of [15].

Theorem 10. Let be a complete partial metric space, and let be a weak --contractive mapping satisfying the following conditions:

(i) is -admissible,(ii)there exists such that , (iii) has the property (C) with respect to ,(iv) is continuous on .

Then, has a fixed point, that is; there exists such that .

Proof. Let such that . Define the sequence in by

If for some , then is a fixed point for . Assume that for all . Since is -admissible, we have

By induction, we get

Applying inequality (6) with and and using (13), we obtain

If , from

we obtain a contradiction; therefore, , and hence

By induction, we get

This implies that

Fix , and let such that

Since has the property (C) with respect to , there exists such that for all . Let with , and we show that

Note that (20) holds for . Assume that (20) holds for some , then

This implies that (20) holds for , and hence

Thus, we proved that is a Cauchy sequence in the partial metric space and hence, by Lemma 1, in the metric space . Since is complete, by Lemma 1, also is complete. This implies that there exists such that as ; that is,

From the continuity of on , it follows that as . By the uniqueness of the limit, we get ; that is, is a fixed point of .

In the next theorem, which is a proper generalization of Theorem 2.2 in [15], we omit the continuity hypothesis of . Moreover, we assume -completeness of the space.

Theorem 11. Let be a -complete partial metric space, and let be a weak --contractive mapping satisfying the following conditions:

(i) is -admissible,(ii)there exists such that ,(iii) has the property (C) with respect to ,(iv) is -regular.

Then, has a fixed point.

Proof. Let such that . Define the sequence in by , for all . Following the proof of Theorem 10, we know that for all and that is a -Cauchy sequence in the -complete partial metric space . Consequently, there exists such that

On the other hand, from for all and the hypothesis (iv), we have

Now, using the triangular inequality, (6) and (25), we get

Since as , for great enough, we have

and hence

This is a contradiction, and so we obtain ; that is, .

The following example illustrates the usefulness of Theorem 10.

Example 12. Let and be defined by for all . Clearly, is a complete partial metric space. Define the mapping by

At first, we observe that we cannot find such that

for all , since we have

for all . Now, we define the function by

Clearly is a weak --contractive mapping with for all . In fact, for all , we have

Moreover, there exists such that . In fact, for , we have

Obviously, is continuous on since , and so we have to show that is -admissible. In doing so, let such that . This implies that , and by the definitions of and , we have

Then, is -admissible. Moreover, if is a sequence such that for all , then for all , and hence for all . Thus, has the property (C) with respect to .

Now, all the hypotheses of Theorem 10 are satisfied, and so has a fixed point. Notice that Theorem 10 (also Theorem 11) guarantees only the existence of a fixed point but not the uniqueness. In this example, and are two fixed points of .

Moreover, , and so is not an --contractive mapping in the sense of [15] with respect to the complete metric space ; that is, Theorem 2.1 of [15] cannot be applied in this case.

Now, we give an example involving a mapping that is not continuous. Also, this example shows that our Theorem 11 is a proper generalization of Theorem 2.2 in [15].

Example 13. Let and as in Example 12. Clearly, is a -complete partial metric space which is not complete. Then, Theorem 10 is not applicable in this case. Define the mapping by

It is clear that is not continuous at with respect to the metric . Define the function by

Clearly is a weak --contractive mapping with for all . In fact, for all , we have

Proceeding as in Example 12, the reader can show that all the required hypotheses of Theorem 11 are satisfied, and so has a fixed point. Here, and are two fixed points of .

Moreover, since is not complete, where for all , we conclude that neither Theorem 2.1 nor Theorem 2.2 of [15] can be applied to cover this case, also because .

To ensure the uniqueness of the fixed point, we will consider the following hypothesis:

(H)for all with , there exists such that , , and .

Theorem 14. Adding condition to the hypotheses of Theorem 10 (resp., Theorem 11), one obtain the uniqueness of the fixed point of .

Proof. Suppose that and are two fixed points of with . If , using (6), we get

which is a contradiction, and so . If by (H), there exists such that

Since is -admissible, from (40), we get

Let for all . Using (41) and (6), we have

Now, let . If is an infinite subset of , then

Then, letting with in the previous inequality, we get

If is a finite subset of , then there exists such that

This implies that

Then, letting , we get

Similarly, using (41) and (6), we get

Since , using (47) and (48), we deduce that

Now, the uniqueness of the limit gives us . This finishes the proof.

From Theorems 10 and 11, we obtain the following corollaries.

Corollary 15. Let be a complete partial metric space, and let be an --contractive mapping satisfying the following conditions:

(i) is -admissible, (ii)there exists such that , (iii) has the property with respect to , (iv) is continuous on .

Then, has a fixed point.

Corollary 16. Let be a -complete partial metric space, and let be an --contractive mapping satisfying the following conditions:

(i) is -admissible,(ii)there exists such that ,(iii) has the property with respect to ,(iv) is -regular.

Then, has a fixed point.

From the proof of Theorem 14, we deduce the following corollaries.

Corollary 17. One adds to the hypotheses of Corollary 15 (resp., Corollary 16) the following condition:

(HC) for all with , there exists such that and ,

and one obtains the uniqueness of the fixed point of .

4. Consequences

Now, we show that many existing results in the literature can be deduced easily from our theorems.

4.1. Contraction Mapping Principle

Theorem 18 (Matthews [1]). Let be a -complete partial metric space, and let be a given mapping satisfying

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

Proof. Let be defined by , for all , and let be defined by . Then is an --contractive mapping. It is easy to show that all the hypotheses of Corollaries 16 and 17 are satisfied. Consequently, has a unique fixed point.

Remark 19. In Example 12, Theorem 18 cannot be applied since . However, using our Corollary 15, we obtain the existence of a fixed point of .

4.2. Fixed Point Results in Ordered Metric Spaces

The existence of fixed points in partially ordered sets has been considered in [18]. Later on, some generalizations of [18] are given in [19–24]. Several applications of these results to matrix equations are presented in [18]; some applications to periodic boundary value problems and particular problems are given in [22, 23], respectively.

In this section, we will show that many fixed point results in ordered metric spaces can be deduced easily from our presented theorems.

4.2.1. Ran and Reurings Type Fixed Point Theorem

In 2004, Ran and Reurings proved the following theorem.

Theorem 20 (Ran and Reurings [18]). Let be a partially ordered set, and suppose that there exists a metric in such that the metric space is complete. Let be a continuous and non-decreasing mapping with respect to . Suppose that the following two assertions hold:

(i)there exists such that for all with ,(ii)there exists such that ,(iii) is continuous.

Then, has a fixed point.

From Theorem 10, we deduce the following generalization and extension of the Ran and Reurings theorem in the framework of ordered complete partial metric spaces.

Theorem 21. Let be an ordered complete partial metric space, and let be a non-decreasing mapping with respect to . Suppose that the following assertions hold:

(i)there exists such that for all with ,(ii)there exists such that ,(iii) is continuous on .

Then, has a fixed point.

Proof. Define the function by

From (i), we have

Then, is a weak --contractive mapping. Now, let such that . By the definition of , this implies that . Since is a non-decreasing mapping with respect to , we have , which gives us that . Then is -admissible. From (ii), there exists such that , and so . Moreover, by Remark 9, has the property (C) with respect to .

Therefore, all the hypotheses of Theorem 10 are satisfied, and so has a fixed point.

Example 22. Let and be defined by for all . Clearly, is a complete partial metric space. Define the mapping by

Clearly is a continuous mapping with respect to the metric . We endow with the usual order of real numbers. Now, condition of Theorem 21 is not satisfied for . In fact, if we assume the contrary, then

which is a contradiction. Then, we cannot apply Theorem 21 to prove the existence of a fixed point of .

Define the function by

It is clear that

Then, is a weak --contractive mapping with for all . Now, let such that . By the definition of , this implies that . Then we have , and so is -admissible. Also, for , we have . Consequently, all the hypotheses of Theorem 10 are satisfied, then we deduce the existence of a fixed point of . Here is a fixed point of .

4.2.2. Nieto and Rodríguez-López Type Fixed Point Theorem

In 2005, Nieto and Rodríguez-López proved the following theorem.

Theorem 23 (Nieto and Rodríguez-López [22]). Let be a partially ordered set, and suppose that there exists a metric in such that the metric space is complete. Let be a non-decreasing mapping with respect to . Suppose that the following assertions hold:

(i)there exists such that for all with ,(ii)there exists such that ,(iii)if is a non-decreasing sequence in such that as , then for all .

Then, has a fixed point.

From Theorem 11, we deduce the following generalization and extension of the Nieto and Rodríguez-López theorem in the framework of ordered -complete partial metric spaces.

Theorem 24. Let be an ordered -complete partial metric space, and let be a non-decreasing mapping with respect to . Suppose that the following assertions hold:

(i)there exists such that for all with , (ii)there exists such that , (iii)if is a non-decreasing sequence in such that as , then for all .

Then,  has a fixed point.

Proof. Define the function by

The reader can show easily that is a weak --contractive and -admissible mapping. Now, by Remark 9, has the property (C) with respect to and is -regular. Thus all the hypotheses of Theorem 11 are satisfied, and has a fixed point.

Remark 25. In, Example 22, also Theorem 24 cannot be applied since condition is not satisfied.

Remark 26. To establish the uniqueness of the fixed point, Ran and Reurings, Nieto and Rodríguez-López [18, 22] considered the following hypothesis:

(u) for all , there exists such that and .

Notice that in establishing the uniqueness it is enough to assume that (u) holds for all that are not comparable. This result is also a particular case of Corollary 17. Precisely, if are not comparable, then there exists such that and . This implies that and , and here, we consider the same function used in the previous proof. Then, hypothesis of Corollary 17 is satisfied, and so we deduce the uniqueness of the fixed point. For establishing the uniqueness of the fixed point in Theorems 21 and 24, we consider the following hypothesis:

(U) for all that are not comparable, there exists such that , , and .

5. Application to Ordinary Differential Equations

In this section, we present a typical application of fixed point results to ordinary differential equations. In fact, in the literature there are many papers focusing on the solution of differential problems approached via fixed point theory (see, e.g., [15, 25, 26] and the references therein). For such a case, even without any additional problem structure, the optimal strategy can be obtained by finding the fixed point of an operator which satisfies a contractive condition in certain spaces.

Here, we consider the following two-point boundary value problem for second order differential equation:

where is a continuous function. Recall that the Green's function associated to (58) is given by

Let () be the space of all continuous functions defined on . It is well known that such a space with the metric given by

is a complete metric space.

Now, we consider the following conditions:

(i)for all , for all with , we have

where , (ii)there exists such that , (iii)for all ,

Theorem 27. Suppose that conditions – hold. Then (58) has at least one solution .

Proof. Consider endowed with the partial metric given by

where . It is easy to show that is -complete but is not complete. In fact,

and consequently is not complete.

On the other hand, it is well known that , and is a solution of (58), is equivalent to is a solution of the integral equation

Define the operator by

Then solving problem (58) is equivalent to finding that is a fixed point of . Now, let such that . From (i), we have

Note that for all , , which implies that

Then, for all such that , we have

Define the function by

For all , we have

Then, is an --contractive mapping. From condition (iii), for all , we get

Then, is -admissible. From conditions (ii) and (iii), there exists such that . Thus, all the conditions of Corollary 16 are satisfied, and hence we deduce the existence of such that ; that is, is a solution to (58).

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under Grant no. NRU56000508).

References

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

M. A. Bukatin and J. S. Scott, “Towards computing distances between programs via Scott domains,” in Logical Foundations of Computer Science, S. Adian and A. Nerode, Eds., vol. 1234 of Lecture Notes in Computer Science, pp. 33–43, Springer, Berlin, Germany, 1997.

M. A. Bukatin and S. Yu. Shorina, “Partial metrics and co-continuous valuations,” in Foundations of Software Science and Computation Structures, M. Nivat, Ed., vol. 1378 of Lecture Notes in Computer Science, pp. 33–43, Springer, Berlin, Germany, 1998.

P. Chaipunya, W. Sintunavarat, and P. Kumam, “On -contractions in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 219, 2012.

S. J. O'Neill, “Partial metrics, valuations and domain theory,” in Proceedings of the 11th Summer Conference on General Topology and Applications, vol. 806 of Annals of the New York Academy of Sciences, pp. 304–315, 1996.

S. Romaguera and M. Schellekens, “Partial metric monoids and semivaluation spaces,” Topology and Its Applications, vol. 153, no. 5-6, pp. 948–962, 2005.

W. Sintunavarat and P. Kumam, “Weak condition for generalized multi-valued -weak contraction mappings,” Applied Mathematics Letters, vol. 24, no. 4, pp. 460–465, 2011.

W. Sintunavarat, J. K. Kim, and P. Kumam, “Fixed point theorems for a generalized almost -contraction with respect to in ordered metric spaces,” Journal of Inequalities and Applications, vol. 2012, article 263, 11 pages, 2012.

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, “A Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory and Applications, Article ID 493298, 6 pages, 2010.

W. A. Kirk, “Caristi's fixed point theorem and metric convexity,” Colloquium Mathematicum, vol. 36, no. 1, pp. 81–86, 1976.

S. Park, “Characterizations of metric completeness,” Colloquium Mathematicum, vol. 49, no. 1, pp. 21–26, 1984.

P. V. Subrahmanyam, “Completeness and fixed-points,” vol. 80, no. 4, pp. 325–330, 1975.

T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.

B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for --contractive type mappings,” Nonlinear Analysis, vol. 75, no. 4, pp. 2154–2165, 2012.

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.

O. Valero, “On Banach fixed point theorems for partial metric spaces,” Applied General Topology, vol. 6, no. 2, pp. 229–240, 2005.

A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2003.

R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008.

I. Altun and G. Durmaz, “Some fixed point theorems on ordered cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, vol. 58, no. 2, pp. 319–325, 2009.

H. K. Nashine and B. Samet, “Fixed point results for mappings satisfying -weakly contractive condition in partially ordered metric spaces,” Nonlinear Analysis, vol. 74, no. 6, pp. 2201–2209, 2011.

J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.

J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007.

J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2517, 2007.

A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2238–2242, 2010.

J. Chu and P. J. Torres, “Applications of Schauder's fixed point theorem to singular differential equations,” Bulletin of the London Mathematical Society, vol. 39, no. 4, pp. 653–660, 2007.