International Journal of Mathematics and Mathematical Sciences

Volume 2012, Article ID 686801, 10 pages

http://dx.doi.org/10.1155/2012/686801

## Generalization of Some Coupled Fixed Point Results on Partial Metric Spaces

^{1}Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan^{2}Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Naradha, Mandir Hasaud, Chhattisgarh Raipur 492101, India

Received 21 March 2012; Accepted 3 May 2012

Academic Editor: Heinz Gumm

Copyright © 2012 Wasfi Shatanawi 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 partial metric spaces, we prove some coupled fixed point results. Our results generalize several well-known comparable results of H. Aydi (2011). Also, we introduce an example to support our results.

#### 1. Introduction and Preliminaries

The notion of coupled fixed point of a mapping was introduced by Gnana Bhaskar and Lakshmikantham in [1]. Later on, many authors investigated many coupled fixed point results in different spaces such as usual metric spaces, fuzzy metric spaces, generalized metric spaces, partial metric spaces, and partially ordered metric spaces (see [1–20]).

*Definition 1.1 (see [1]). *An element is called a coupled fixed point of a mapping if

Matthews [21] in 1994 introduced the notion of partial metric spaces in such a way that each object does not necessarily have to have a zero distance from itself. Consistent with Matthews [21], the following definitions and results will be needed in the sequel.

*Definition 1.2 (see [21]). *A partial metric on a nonempty set is a function such that for all :,
,
,
.

A partial metric space is a pair such that is a nonempty set and is a partial metric on .

Each partial metric on generates a topology on . The set , where for all and , forms the base of .

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

*Definition 1.3 (see [21]). *Let be a partial metric space. Then:(1)a sequence in a partial metric space converges, with respect to , to a point if and only if ,(2)a sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) ,(3)A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

Lemma 1.4 (see [21]). *Let be a partial metric space.*(1)* is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .*(2)*A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if
*

Abdeljawad et al. [22–24], Altun et al. [25], Karapinar and Erhan [26–28], Oltra and Valero [29] and Romaguera [30] studied fixed point theorems in partial metric spaces. For more works in partial metric spaces, we refer the reader to [31–40].

Aydi [2] proved the following coupled fixed point theorems in partial metric spaces.

Theorem 1.5. *Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all :
**
where are nonnegative constants with . Then has a unique coupled fixed point.*

Theorem 1.6. *Let be a complete partial metric space. Suppose that the mapping satisfies the following contractive condition for all :
**
where are nonnegative constants with . Then has a unique coupled fixed point.*

In this paper, we prove some coupled fixed point results. Our results generalize Theorems 1.5 and 1.6. Also, we introduce an example to support our results.

#### 2. The Main Result

Theorem 2.1. *Let be a complete partial metric space. Suppose that the mapping satisfies
**
for all . If , then has a unique coupled fixed point.*

*Proof. *Choose . Let and . Again let and . By continuing in the same way, we construct two sequences and in such that
Then by (2.1), we have
Thus from (2.3), we have
By repeating (2.4) -times, we get that
Letting in (2.5), we get that
Therefore, we have
For with , we have
By (2.5) and (2.8), we have
Letting in (2.9), we get that
Thus exists and is finite. Hence is a Cauchy sequence in . Similarly, we may show that
and hence is a Cauchy sequence in . By Lemma 1.4 there exist such that (resp., ) if and only if
Now, we prove that . By (2.1), we have
Letting in the above inequality and using (2.12), we get that

Since , we conclude that . By and , we have . Similarly, we may show that . Thus is a coupled fixed point of . To prove the uniqueness of the fixed point, we let be a coupled fixed point of . We will show that and . By (2.1), we have
Since and , we have
Also, from (2.1), we have
Since and , we have
From (2.16) and (2.18), we have
Since , we have . Hence and . By and , we have and .

Corollary 2.2. *Let be a complete partial metric space. Suppose that there are with such that the mapping satisfies
**
for all . Then has a unique coupled fixed point.*

*Proof. *The proof follows from Theorem 2.1 by noting that:

*Remarks.*(1)Theorem 1.5 [2, Theorem 2.1] is a special case of Corollary 2.2.(2)[2, Corollary 2.2] is a special case of Corollary 2.2.(3)Theorem 1.6 [2, Theorem 2.4] is a special case of Corollary 2.2.(4)[2, Corollary 2.6] is a special case of Corollary 2.2.Now, we introduce an example satisfying the hypotheses of Theorem 2.1 but not the hypotheses of Theorems 2.1 and 2.4 of [2].

*Example 2.3. *Define by . Then is a complete partial metric space. Let be the mapping defined by
Then,(a) for all .(b) There are no with such that for all .(c) There are no with such that for all .

*Proof. *To prove part (a), given . Then:
To prove part (b), suppose that there are with such that for all .

Since
we have , which is a contradiction.

To prove part (c), suppose that there are with such that for all .

Since
we have , which is a contradiction.

Thus by Theorem 2.1, has a unique coupled fixed point. Here, is the unique fixed point of .

#### Acknowledgments

The authors thank the editor and the referees for their valuable comments and suggestions.

#### References

- T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 65, no. 7, pp. 1379–1393, 2006. View at Publisher · 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 - H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered $G$-metric spaces,”
*Mathematical and Computer Modelling*, vol. 54, no. 9-10, pp. 2443–2450, 2011. View at Publisher · View at Google Scholar - H. Aydi, E. Karapınar, and W. Shatanawi, “Coupled fixed point results for $(\psi ,\phi )$-weakly contractive condition in ordered partial metric spaces,”
*Computers & Mathematics with Applications*, vol. 62, no. 12, pp. 4449–4460, 2011. View at Publisher · View at Google Scholar - Y. J. Cho, B. E. Rhoades, R. Saadati, B. Samet, and W. Shatanawi, “Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type,”
*Fixed Point Theory and Applications*, vol. 2012, article 8, 2012. View at Publisher · View at Google Scholar - B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized metric spaces,”
*Mathematical and Computer Modelling*, vol. 54, no. 1-2, pp. 73–79, 2011. View at Publisher · View at Google Scholar - E. Karapınar, “Couple fixed point theorems for nonlinear contractions in cone metric spaces,”
*Computers & Mathematics with Applications*, vol. 59, no. 12, pp. 3656–3668, 2010. View at Publisher · View at Google Scholar - V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 12, pp. 4341–4349, 2009. View at Publisher · View at Google Scholar - N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 3, pp. 983–992, 2011. View at Publisher · View at Google Scholar - H. K. Nashine and W. Shatanawi, “Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces,”
*Computers & Mathematics with Applications*, vol. 62, no. 4, pp. 1984–1993, 2011. View at Publisher · View at Google Scholar - B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 12, pp. 4508–4517, 2010. View at Publisher · View at Google Scholar - S. Sedghi, I. Altun, and N. Shobe, “Coupled fixed point theorems for contractions in fuzzy metric spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1298–1304, 2010. View at Publisher · View at Google Scholar - W. Shatanawi and Z. Mustafa, “On coupled random fixed point results in partially ordered metric spaces,”
*Matematicki Vesnik*, vol. 64, no. 2, pp. 139–146, 2012. View at Google Scholar - W. Shatanawi, “Coupled fixed point theorems in generalized metric spaces,”
*Hacettepe Journal of Mathematics and Statistics*, vol. 40, no. 3, pp. 441–447, 2011. View at Google Scholar - W. Shatanawi, B. Samet, and M. Abbas, “Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces,”
*Mathematical and Computer Modelling*, vol. 55, pp. 680–687, 2012. View at Google Scholar - W. Shatanawi, “Some common coupled fixed point results in cone metric spaces,”
*International Journal of Mathematical Analysis*, vol. 4, no. 45–48, pp. 2381–2388, 2010. View at Google Scholar - W. Shatanawi, “Partially ordered cone metric spaces and coupled fixed point results,”
*Computers & Mathematics with Applications*, vol. 60, no. 8, pp. 2508–2515, 2010. View at Publisher · View at Google Scholar - W. Shatanawi, “On
*w*-compatible mappings and common coupled coincidence point in cone metric spaces,”*Applied Mathematics Letters*, vol. 25, no. 6, pp. 925–931, 2012. View at Publisher · View at Google Scholar - W. Shatanawi, “Fixed point theorems for nonlinear weakly $C$-contractive mappings in metric spaces,”
*Mathematical and Computer Modelling*, vol. 54, no. 11-12, pp. 2816–2826, 2011. View at Publisher · View at Google Scholar - W. Shatanawia and H. K. Nashine, “A generalization of Banach's contraction principle for nonlinear contraction in a partial metric space,”
*Journal of Nonlinear Science and Applications*, vol. 2012, no. 5, pp. 37–43, 2012. View at Publisher · View at Google Scholar - S. G. Matthews, “Partial metric topology,” in
*Papers on General Topology and Applications*, vol. 728, pp. 183–197, The New York Academy of Sciences, New York, NY, USA, 1994. 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. 1900–1904, 2011. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar - T. Abdeljawad, “Fixed points for generalized weakly contractive mappings in partial metric spaces,”
*Mathematical and Computer Modelling*, vol. 54, no. 11-12, pp. 2923–2927, 2011. View at Publisher · View at Google Scholar - 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 - E. Karapinar, “Weak
*φ*-contraction on partial contraction,”*Journal of Computational Analysis and Applications*. In press. - E. Karapinar, “Generalizations of Caristi Kirk's theorem on partial metric spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 7, 2011. View at Google Scholar - E. Karapınar and M. Erhan, “Fixed point theorems for operators on partial metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 11, pp. 1894–1899, 2011. View at Publisher · View at Google Scholar - 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 - 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 - I. Altun and A. Erduran, “Fixed point theorems for monotone mappings on partial metric spaces,”
*Fixed Point Theory and Applications*, Article ID 508730, 10 pages, 2011. View at Publisher · View at Google Scholar - I. Altun and H. Simsek, “Some fixed point theorems on dualistic partial metric spaces,”
*Journal of Advanced Mathematical Studies*, vol. 1, no. 1-2, pp. 1–8, 2008. View at Google Scholar - H. Aydi, “Some fixed point results in ordered partial metric spaces,”
*The Journal of Nonlinear Science and Applications*, vol. 4, no. 2, pp. 1–12, 2011. View at Google Scholar - H. Aydi, “Fixed point results for weakly contractive mappings in ordered partial metric spaces,”
*Journal of Advanced Mathematical Studies*, vol. 4, no. 2, pp. 1–12, 2011. View at Google Scholar - 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. \'Cirić, 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 - R. Heckmann, “Approximation of metric spaces by partial metric spaces,”
*Applied Categorical Structures*, vol. 7, no. 1-2, pp. 71–83, 1999. View at Publisher · 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 - 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 - 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