International Journal of Mathematics and Mathematical Sciences

Volume 2012, Article ID 597074, 11 pages

http://dx.doi.org/10.1155/2012/597074

## A Fixed Point Result for Boyd-Wong Cyclic Contractions in Partial Metric Spaces

^{1}Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, Hammam Sousse, 4002 Sousse, Tunisia^{2}Department of Mathematics, Atilim University, 06836 İncek, Turkey

Received 31 March 2012; Accepted 31 May 2012

Academic Editor: Billy Rhoades

Copyright © 2012 Hassen Aydi and Erdal Karapinar. 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

A fixed point theorem involving Boyd-Wong-type cyclic contractions in partial metric spaces is proved. We also provide examples to support the concepts and results presented herein.

#### 1. Introduction and Preliminaries

Partial metric spaces were introduced by Matthews [1] to the study of denotational semantics of data networks. In particular, he proved a partial metric version of the Banach contraction principle [2]. Subsequently, many fixed points results in partial metric spaces appeared (see, e.g., [1, 3–19] for more details).

Throughout this paper, the letters and will denote the sets of all real numbers and positive integers, respectively. We recall some basic definitions and fixed point results of partial metric spaces.

*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 .

If is a partial metric on , then the function given by is a metric on .

*Example 1.2 (see, e.g., [1, 3, 11, 12]). *Consider with . Then, is a partial metric space. It is clear that is not a (usual) metric. Note that in this case .

*Example 1.3 (see, e.g., [1]). * Let , and define . Then, is a partial metric space.

*Example 1.4 (see, e.g., [1, 20]). *Let , and define by
Then, is a complete partial metric space.

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

*Definition 1.5. *Let be a partial metric space and a sequence in . Then, (i) converges to a point if and only if ,(ii) is called a Cauchy sequence if exists and is finite.

*Definition 1.6. *A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point , such that .

Lemma 1.7 (see, e.g., [3, 11, 12]). * 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)* is complete if and only if the metric space is complete. Furthermore, if and only if
*

Lemma 1.8 (see, e.g., [3, 11, 12]). * Let be a partial metric space. Then, *(a)*if , then , *(b)*if , then . *

*Remark 1.9. *If , may not be .

Lemma 1.10 (see, e.g., [3, 11, 12]). *Let as in a partial metric space where . Then, for every . *

Let be the set of functions such that(i) is upper semicontinuous (i.e., for any sequence in such that as , we have ),(ii) for each .

Recently, Romaguera [21] obtained the following fixed point theorem of Boyd-Wong type [22].

Theorem 1.11. *Let be a complete partial metric space, and let be a map such that for all **
where and
**
Then, has a unique fixed point. *

In 2003, Kirk et al. [23] introduced the following definition.

*Definition 1.12 (see [23]). * Let be a nonempty set, a positive integer, and a mapping. is said to be a cyclic representation of with respect to if (i) are nonempty closed sets, (ii).

Recently, fixed point theorems involving a cyclic representation of with respect to a self-mapping have appeared in many papers (see, e.g., [24–28]).

Very recently, Abbas et al. [24] extended Theorem 1.11 to a class of cyclic mappings and proved the following result, but with being a continuous map.

Theorem 1.13. *Let be a complete partial metric space. Let be a positive integer, nonempty closed subsets of , and . Let be a mapping such that*(i)* is a cyclic representation of with respect to , *(ii)*there exists such that is continuous and for each , satisfying for any , , , where and is defined by (1.5). **Then, has a unique fixed point . *

In the following example, , but it is not continuous.

*Example 1.14. *Define by for all and for , . Then, is upper semicontinuous on with for all . However, it is not continuous at for all .

Following Example 1.14, the main aim of this paper is to present the analog of Theorem 1.13 for a weaker hypothesis on , that is, with . Our proof is simpler than that in [24]. Also, some examples are given.

#### 2. Main Results

Our main result is the following.

Theorem 2.1. *Let be a complete partial metric space. Let be a positive integer, nonempty closed subsets of , and . Let be a mapping such that*(1)* is a cyclic representation of with respect to ,*(2)*there exists such that
for any , , , where and is defined by (1.5).**Then, has a unique fixed point . *

*Proof. *Let . Consider the Picard iteration given by for . If there exists such that , then and the existence of the fixed point is proved.

Assume that , for each . Having in mind that , so for each , there exists such that and . Then, by (2.1)
where
Therefore,
If for some , we have , so by (2.2)
which is a contradiction. It follows that
Thus, from (2.2), we get that
Hence, is a decreasing sequence of positive real numbers. Consequently, there exists such that . Assume that . Letting in the above inequality, we get using the upper semicontinuity of
which is a contradiction, so that , that is,
By (1.1), we have for all , and then from (2.9)
Also, by ,
In the sequel, we will prove that is a Cauchy sequence in the partial metric space . By Lemma 1.7, it suffices to prove that is Cauchy sequence in the metric space . We argue by contradiction. Assume that is not a Cauchy sequence in . Then, there exists for which we can find subsequences and of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (2.12). Then,
We use (2.13) and the triangular inequality
Letting in (2.14) and using (2.10), we find
On the other hand
Letting in the two above inequalities and using (2.10) and (2.15),
Similarly, we have
Also, by (1.1), (2.11), and (2.15)–(2.18), we may find
On the other hand, for all , there exists , , such that . Then, (for large enough, ) and lie in different adjacently labeled sets and for certain . Using the contractive condition (2.1), we get
where
As (2.19), using (2.9), we may get
By (2.22) and (2.23), we get that
Letting in (2.20), we get using (2.22), (2.24), and the upper semicontinuity of
which is a contradiction.

This shows that is a Cauchy sequence in the complete subspace equipped with the metric . Thus, there exists . Notice that the sequence has an infinite number of terms in each , , so since is complete, from each , one can extract a subsequence of that converges to . Because , are closed in , it follows that
Thus, .

For simplicity, set . Clearly, is also closed in , so it is a complete subspace of and then is a complete partial metric space. Consider the restriction of on , that is, . Then, satisfies the assumptions of Theorem 1.11, and thus has a unique fixed point in .

#### 3. Examples

We give some examples illustrating our results.

*Example 3.1. *Let and . It is obvious that is a complete partial metric space.

Set , , and . Define by

Notice that and , and hence . Analogously, and , and hence .

Take Clearly, satisfies condition (2.1). Indeed, we have the following cases.

*Case 1. * and . Inequality (2.1) turns into
which is necessarily true.

*Case 2. * and . Inequality (2.1) becomes
It is clear that for all . Hence, (3.4) holds.

*Case 3. * and . Inequality (2.1) turns into
which is true again by the fact that for all .

*Case* *4. * and . Inequality (2.1) becomes
Let use examine all possibilities:
Thus, (2.1) holds for .

The rest of the assumptions of Theorem 2.1 are also satisfied. The function has as a unique fixed point.

However, since is not a continuous function, we could not apply Theorem 1.13.

*Example 3.2. *Let and for all . Then, is a complete partial metric space. Take . Define by . Consider given by Example 1.14.

For all , we have Then all the assumptions of Theorem 2.1 are satisfied. The function has as a unique fixed point.

Similarly, Theorem 1.13 is not applicable.

#### 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. - S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux éequations intégrales,”
*Fundamenta Mathematicae*, vol. 3, pp. 133–181, 1922. View at Google Scholar - T. Abdeljawad, E. Karapınar, and K. Taş, “Existence and uniqueness of a common fixed point on partial metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 11, pp. 1894–1899, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Altun and A. Erduran, “Fixed point theorems for monotone mappings on partial metric spaces,”
*Fixed Point Theory and Applications*, vol. 2011, Article ID 508730, 10 pages, 2011. View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Aydi, “Some fixed point results in ordered partial metric spaces,”
*The Journal of Nonlinear Science and Applications*, vol. 4, no. 3, pp. 210–217, 2011. View at Google Scholar - H. Aydi, “Some coupled fixed point results on partial metric spaces,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2011, Article ID 647091, 11 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Aydi, “Fixed point results for weakly contractive mappings in ordered partial metric spaces,”
*Journal of Advanced Mathematical Studies*, vol. 4, no. 2, pp. 1–15, 2011. View at Google Scholar · View at Zentralblatt MATH - H. Aydi, “Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces,”
*Journal of Nonlinear Analysis and Optimization*, vol. 2, no. 2, pp. 33–48, 2011. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Karapınar and I. M. Erhan, “Existence and uniqueness of a common fixed point on partial metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 11, pp. 1900–1904, 2011. View at Google Scholar - E. Karapınar, “Weak
*ϕ*-contraction on partial contraction,”*Journal of Computational Analysis and Applications*, vol. 14, no. 2, pp. 206–210, 2012. View at Google Scholar - E. Karapınar, “Weak $\varphi $-contraction on partial metric spaces and existence of fixed points in partially ordered sets,”
*Mathematica Aeterna*, vol. 1, no. 3-4, pp. 237–244, 2011. View at Google Scholar · View at Zentralblatt MATH - E. Karapınar, “Generalizations of Caristi Kirk's theorem on partial metric spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 4, 2011. View at Google Scholar - S. J. O'Neill, “Two topologies are better than one,” Tech. Rep., University of Warwick, Coventry, UK, 1995.
- 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. View at Google Scholar · View at Zentralblatt MATH - S. Romaguera, “A Kirk type characterization of completeness for partial metric spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 493298, 6 pages, 2010. View at Google Scholar - B. Samet, M. Rajović, R. Lazović, and R. Stoiljković, “Common fixed point results for nonlinear contractions in ordered partial metric spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 71, 2011. View at Google Scholar - O. Valero, “On Banach fixed point theorems for partial metric spaces,”
*Applied General Topology*, vol. 6, no. 2, pp. 229–240, 2005. View at Google Scholar · View at Zentralblatt MATH - D. Ilić, V. Pavlović, and V. Rakočević, “Some new extensions of Banach's contraction principle to partial metric space,”
*Applied Mathematics Letters*, vol. 24, no. 8, pp. 1326–1330, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Romaguera, “Fixed point theorems for generalized contractions on partial metric spaces,”
*Topology and its Applications*, vol. 159, no. 1, pp. 194–199, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,”
*Proceedings of the American Mathematical Society*, vol. 20, pp. 458–464, 1969. View at Publisher · View at Google Scholar - W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,”
*Fixed Point Theory*, vol. 4, no. 1, pp. 79–89, 2003. View at Google Scholar - M. Abbas, T. Nazir, and S. Romaguera, “Fixed point results for generalized cyclic contraction mappings in partial metric spaces,”
*Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales A*. In press. View at Publisher · View at Google Scholar - E. Karapınar, “Fixed point theory for cyclic weak
*ϕ*-contraction,”*Applied Mathematics Letters*, vol. 24, no. 6, pp. 822–825, 2011. View at Publisher · View at Google Scholar - M. Păcurar and I. A. Rus, “Fixed point theory for cyclic $\varphi $-contractions,”
*Nonlinear Analysis A*, vol. 72, no. 3-4, pp. 1181–1187, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. A. Rus, “Cyclic representations and fixed points,”
*Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity*, vol. 3, pp. 171–178, 2005. View at Google Scholar - M. De la Sen, “Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 572057, 23 pages, 2010. View at Google Scholar · View at Zentralblatt MATH