`Discrete Dynamics in Nature and SocietyVolume 2012, Article ID 184534, 9 pageshttp://dx.doi.org/10.1155/2012/184534`
Research Article

## Coupled Fixed Point Theorems under Weak Contractions

1Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
3Department of Mathematics, Iran University of Science and Technology, Behshahr, Iran
4Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

Received 6 October 2011; Accepted 28 December 2011

Copyright © 2012 Y. J. Cho 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

Cho et al. [Comput. Math. Appl. 61(2011), 1254–1260] studied common fixed point theorems on cone metric spaces by using the concept of c-distance. In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of c-distance in cone metric spaces.

#### 1. Introduction

Many fixed point theorems have been proved for mappings on cone metric spaces in the sense of Huang and Zhang [1]. For some more results on fixed point theory and applications in cone metric spaces, we refer the readers to [215]. Recently, Bhaskar and Lakshmikantham [16] introduced the concept of a coupled coincidence point of a mapping from into and a mapping from into and studied fixed point theorems in partially ordered metric spaces. For some more results on couple fixed point theorems, refer to [1723].

Recently, Cho et al. [7] introduced a new concept of -distance in cone metric spaces, which is a cone version of -distance of Kada et al. [24] (see also [25]) and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the -distance.

In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of -distance.

#### 2. Preliminaries

In this paper, assume that is a real Banach space. Let be a subset of with . Then is called a cone if the following conditions are satisfied:(1) is closed and ;(2), implies ;(3) implies .

For a cone , define the partial ordering with respect to by if and only if . We write to indicate that but , while stand for .

It can be easily shown that for all positive scalars .

Definition 2.1 (see [1]). Let be a nonempty set. Suppose that the mapping satisfies the following conditions:(1) for all and if and only if ;(2) for all ;(3) for all .Then is called a cone metric on , and is called a cone metric space.

Definition 2.2 (see [1]). Let be a cone metric space. Let be a sequence in and .(1)If, for any with , there exists such that for all , then is said to be convergent to a point and is the limit of . We denote this by or as .(2)If, for any with , there exists such that for all , then is called a Cauchy sequence in .(3)The space is called a complete cone metric space if every Cauchy sequence is convergent.

Definition 2.3 (see [7]). Let be a partially ordered set, and let be a function. Then the mapping is said to have the mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in ; that is, for all and for all .

Definition 2.4 (see [7]). An element is called a coupled fixed point of a mapping if and .

Recently, Cho et al. [7] introduced the concept of -distance on cone metric space which is a generalization of -distance of Kada et al. [24].

Definition 2.5 (see [7]). Let be a cone metric space. Then a function is called a -distance on if the following are satisfied:(q1) for all ;(q2) for all ;(q3) for any , if there exists such that for each , then whenever is a sequence in converging to a point ;(q4) for any with , there exists with such that and imply .

Cho et al. [7] noticed the following important remark in the concept of -distance on cone metric spaces.

Remark 2.6 (see [7]). Let be a -distance on a cone metric space . Then(1) does not necessarily hold for all ,(2) is not necessarily equivalent to for all .

The following lemma is crucial in proving our results.

Lemma 2.7 (see [7]). Let be a cone metric space, and let be a -distance on . Let and be sequences in and . Suppose that is a sequence in converging to . Then the following hold:(1)if and , then ;(2)if and , then converges to a point ;(3)if for each , then is a Cauchy sequence in ;(4)If , then is a Cauchy sequence in .

#### 3. Main Results

In this section, we prove some coupled fixed point theorems by using -distance in partially ordered cone metric spaces.

Theorem 3.1. Let be a partially ordered set, and suppose that is a complete cone metric space. Let be a -distance on , and let be a continuous function having the mixed monotone property such that for some and all with or . If there exist such that and , then has a coupled fixed point . Moreover, one has and .

Proof. Let be such that and . Let and . Since has the mixed monotone property, we have and . Continuing this process, we can construct two sequences and in such that Let . Now, by (3.1), we have From (3.3), it follows that
Similarly, we have Thus it follows from (3.4) and (3.5) that Repeating (3.6) -times, we get Thus we have Let with . Since and , we have From Lemma 2.7 (3), it follows that and are Cauchy sequences in . Since is complete, there exist such that and . Since is continuous, we have By the uniqueness of the limits, we get and . Thus is a coupled fixed point of .
Moreover, by (3.1), we have Therefore, we get Since , we conclude that , and hence and . This completes the proof.

Theorem 3.2. In addition to the hypotheses of Theorem 3.1, suppose that any two elements and in are comparable. Then the coupled fixed point has the form , where .

Proof. As in the proof of Theorem 3.1, there exists a coupled fixed point . Here and . By the additional assumption and (3.1), we have Thus we have Since , we get . Hence and . Let and . Then From Lemma 2.7 (1), we have . Hence the coupled fixed point of has the form . This completes the proof.

Theorem 3.3. Let be a partially ordered set, and suppose that is a complete cone metric space. Let be a -distance on , and let be a function having the mixed monotone property such that for some and all with or . Also, suppose that has the following properties:(a)if is a nondecreasing sequence in with , then for all ;(b)if is a nonincreasing sequence in with , then for all .Assume there exist such that and . If , then has a coupled fixed point.

Proof. As in the proof of Theorem 3.1, we can construct two Cauchy sequences and in such that
Moreover, we have that converges to a point and converges to , for each . By (q3), we have and so By the properties (a) and (b), we have By (3.17), we have Thus we have By (3.21), we get Therefore, we have By using (3.20) and (3.26), Lemma 2.7 (1) shows that and . Therefore, is a coupled fixed point of . This completes the proof.

Example 3.4. Let with and . Let (with usual order), and let be defined by . Then is an ordered cone metric space (see [7, Example 2.9]). Further, let be defined by . It is easy to check that is a -distance. Consider now the function defined by Then it is easy to see that for all with or . Note that and . Thus, by Theorem 3.1, it follows that has a coupled fixed point in . Here is a coupled fixed point of .

#### Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.: 2011–0021821).

#### References

1. L. G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.
2. M. Abbas, Y. J. Cho, and T. Nazir, “Common fixed point theorems for four mappings in TVS-valued cone metric spaces,” Journal of Mathematical Inequalities, vol. 5, no. 2, pp. 287–299, 2011.
3. M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416–420, 2008.
4. M. Abbas and B. E. Rhoades, “Fixed and periodic point results in cone metric spaces,” Applied Mathematics Letters, vol. 22, no. 4, pp. 511–515, 2009.
5. I. Altun and V. Rakočević, “Ordered cone metric spaces and fixed point results,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1145–1151, 2010.
6. A. Amini-Harandi and M. Fakhar, “Fixed point theory in cone metric spaces obtained via the scalarization method,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3529–3534, 2010.
7. Y. J. Cho, R. Saadati, and S. Wang, “Common fixed point theorems on generalized distance in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1254–1260, 2011.
8. E. Graily, S. M. Vaezpour, R. Saadati, and Y. J. Cho, “Generalization of fixed point theorems in ordered metric spaces concerning generalized distance,” Fixed Point Theory and Applications, vol. 2011, p. 30, 2011.
9. D. Ilić and V. Rakočević, “Common fixed points for maps on cone metric space,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 876–882, 2008.
10. S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 7, pp. 2591–2601, 2011.
11. Z. Kadelburg, M. Pavlović, and S. Radenović, “Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3148–3159, 2010.
12. S. Radenović and B. E. Rhoades, “Fixed point theorem for two non-self mappings in cone metric spaces,” Computers & Mathematics with Applications, vol. 57, no. 10, pp. 1701–1707, 2009.
13. W. Sintunavarat, Y. J. Cho, and P. Kumam, “Common fixed point theorems for c-distance in ordered cone metric spaces,” Computers and Mathematics with Applications, vol. 62, no. 4, pp. 1969–1978, 2011.
14. D. Turkoglu and M. Abuloha, “Cone metric spaces and fixed point theorems in diametrically contractive mappings,” Acta Mathematica Sinica, vol. 26, no. 3, pp. 489–496, 2010.
15. D. Turkoglu, M. Abuloha, and T. Abdeljawad, “KKM mappings in cone metric spaces and some fixed point theorems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 1, pp. 348–353, 2010.
16. T. G. 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.
17. Y. J. Cho, G. He, and N.-j. Huang, “The existence results of coupled quasi-solutions for a class of operator equations,” Bulletin of the Korean Mathematical Society, vol. 47, no. 3, pp. 455–465, 2010.
18. Y. J. Cho, M. H. Shah, and N. Hussain, “Coupled fixed points of weakly F-contractive mappings in topological spaces,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1185–1190, 2011.
19. M. E. Gordji, Y. J. Cho, and H. Baghani, “Coupled fixed point theorems for contractions in intuitionistic fuzzy normed spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 1897–1906, 2011.
20. W. Sintunavarat, Y. J. Cho, and P. Kumam, “Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces,” Fixed Point Theory and Applications, vol. 2011, p. 81, 2011.
21. V. Lakshmikantham and L. Cirić, “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.
22. 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.
23. W. Shatanawi, “Partially ordered cone metric spaces and coupled fixed point results,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2508–2515, 2010.
24. O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.
25. D. Ilić and V. Rakočević, “Common fixed points for maps on metric space with w-distance,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 599–610, 2008.