Journal of Applied Mathematics

Volume 2012 (2012), Article ID 312078, 15 pages

doi:10.1155/2012/312078

Research Article

Coupled Coincidence Points in Partially Ordered Cone Metric Spaces with a c-Distance

Wasfi Shatanawi,¹ Erdal Karapınar,² and Hassen Aydi³

¹Department of Mathematics, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

²Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey

³Institut Supérieur d'Informatique et des Technologies de Communication De Hammam Sousse, Université de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia

Received 22 May 2012; Revised 25 June 2012; Accepted 25 June 2012

Academic Editor: Alexander Timokha

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.

Cho et al. (2012) proved some coupled fixed point theorems in partially ordered cone metric spaces by using the concept of a c-distance in cone metric spaces. In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of a c-distance. Our results generalize several well-known comparable results in the literature. Also, we introduce an example to support the usability of our results.

1. Introduction and Preliminaries

Fixed point theory is an essential tool in functional nonlinear analysis. Consequently, fixed point theory has wide applications areas not only in the various branches of mathematics (see, e.g., [1, 2]) but also in many fields, such as, chemistry, biology, statistics, economics, computer science, and engineering (see, e.g., [3–11]). For example, fixed point results are incredibly useful when it comes to proving the existence of various types of Nash equilibria (see, e.g., [7]) in economics. On the other hand, fixed point theorems are vital for the existence and uniqueness of differential equations, matrix equations, integral equations (see, e.g., [1, 2]). Banach contraction mapping principle (Banach fixed point theorem) is one of the most powerful theorems of mathematics and hence fixed point theory. Huang and Zhang [12] generalized the Banach contraction principle by replacing the notion of usual metric spaces by the notion of cone metric spaces. Then many authors obtained many fixed and common fixed point theorems in cone metric spaces. For some works in cone metric spaces, we may refer the reader (as examples) to [13–24]. The concept of a coupled fixed point of a mapping was initiated by Bhaskar and Lakshmikantham [25], while Lakshmikantham and Ćirić [26] initiated the notion of coupled coincidence point of mappings and and studied some coupled coincidence point theorems in partially ordered metric spaces. For some coupled fixed point and coupled coincidence point theorems, we refer the reader to [27–34].

In the present paper, is the set of positive integers and stands for a real Banach space. Let be a subset of . We will always assume that the cone has a nonempty interior (such cones are called solid). Then is called a cone if the following conditions are satisfied:

(1)is closed and ,(2), implies , (3) implies .

For a cone , define a partial ordering with respect to by if and only if . We will write to indicate that but , while will stand for . It can be easily shown that for all positive scalar .

Definition 1.1 (see [12]). Let be a nonempty set. Suppose the mapping satisfies

(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.

Bhaskar and Lakshmikantham [25] introduced the notion of mixed monotone property of the mapping .

Definition 1.2 (see [25]). Let be a partially ordered set and be a mapping. Then the mapping is said to have mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in ; that is, for any ,

Inspired by Definition 1.2, Lakshmikantham and Ćirić in [26] introduced the concept of a -mixed monotone mapping.

Definition 1.3 (see [26]). Let be a partially ordered set and . Then the mapping is said to have mixed -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in ; that is:

Definition 1.4 (see [25]). An element is called a coupled fixed point of a mapping if

Definition 1.5 (see [26]). An element is called a coupled coincidence point of the mappings and if

Recently, Cho et al. [35] introduced the concept of -distance on cone metric space which is a generalization of -distance of Kada et al. [36] (see also [37, 38]).

Definition 1.6 (see [35]). Let be a cone metric space. Then a function is called a -distance on if the following is satisfied:

(q1) for all , (q2) for all , (q3) for each and , if for some , then whenever is a sequence in converging to a point , (q4) for all with , there exists with such that , and implies .

Cho et al. [35] noticed the following important remark in the concept of -distance on cone metric spaces.

Remark 1.7 (see [35]). 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 .

Very recently, Cho et al. [39] proved the following existence theorems.

Theorem 1.8 (see [39]). 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 .

Theorem 1.9 (see [39]). 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:

(1)if is a nondecreasing sequence in with , then for all , (2)if is a nonincreasing sequence in with , then for all .

If there exist such that and , then has a coupled fixed point . Moreover, one has .

For other fixed point results using a -distance, see [40].

In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of -distance. Our results generalize Theorems 1.8 and 1.9. We consider an application to illustrate our result is useful (see Section 3).

2. Main Results

The following lemma is essential in proving our results.

Lemma 2.1 (see [35]). Let be a cone metric space, and let be a cone distance on . Let and be sequences in and . Suppose that is a sequence in converging to . Then the following holds.

(1)If and , then . (2)If and , then converges to . (3)If for , then is a Cauchy sequence in . (4)If , then is a Cauchy sequence in .

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

Theorem 2.2. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let and be two mappings such that

for some and for all with or . Assume that and satisfy the following conditions:

(1) is continuous, (2) is continuous and commutes with , (3), (4) is complete, (5) has the mixed -monotone property.

If there exist such that and , then and have a coupled coincidence point . Moreover, one has and .

Proof. Let be such that and . Since , we can choose such that and . Again since , we can choose such that and . Since has the mixed -monotone property, we have and . Continuing this process, we can construct two sequences and in such that

Let . Then by (2.1), we have

Repeating (2.3) -times, we get

Thus, we have

Let with . Then by and (2.5), we have

Similarly, we have

From part (3) of Lemma 2.1, we conclude that and are Cauchy sequences in in . Since is complete, there are such that and . Using the continuity of , we get and . Also, by continuity of and commutativity of and , we have

Hence, is a coupled coincidence point of and . Moreover, by (2.1) we have

Since , we conclude that , and hence and .

The continuity of in Theorem 2.2 can be dropped. For this, we present the following useful lemma which is a variant of Lemma 2.1, (1).

Lemma 2.3. Let be a cone metric space, and let be a -distance on . Let be a sequence in . Suppose that and are sequences in converging to . If and , then .

Proof. Let be arbitrary. Since , so there exists such that for all . Similarly, there exists such that for all . Thus, for all , we have

Take , so by , we get that for each ; hence .

Theorem 2.4. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let and let be two mappings such that

for some and for all with or . Assume that and satisfy the following conditions:

(1), (2) is a complete subspace of , (3) has the mixed -monotone property.

Suppose that has the following properties:

(i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all .

Assume there exist such that and . Then and have a coupled coincidence point, say . Also, .

Proof. As in the proof of Theorem 2.2, we can construct two Cauchy sequences and in the complete cone metric space . Then, there exist such that and . Similarly we have for all

By (q3), we get that

By summation, we get that

Since is nondecreasing and is nonincreasing, using the properties (i), (ii) of , we have

From this and (2.14), we have

Therefore

By (2.16), we have

This implies that

By (2.14), (2.21) and Lemma 2.3, we obtain . Similarly, by (2.15), (2.22), and Lemma 2.3, we obtain . Also, adjusting as the proof of Theorem 2.2, we get that

Corollary 2.5. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let , and let be two mappings such that

for some with and for all with or . Assume that and satisfy the following conditions:

(1) is continuous, (2) is continuous and commutes with , (3), (4) is complete, (5) has the mixed -monotone property.

If there exist such that and , then and have a coupled coincidence point . Moreover, one has and .

Proof. Given such that . By (2.24), we have

Thus

Since , the result follows from Theorem 2.2.

Corollary 2.6. Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on . Let be a continuous mapping having the mixed monotone property such that

for some with and for all with or . If there exist such that and , then has a coupled fixed point . Moreover, one has and .

Proof. It follows from Corollary 2.5 by taking (the identity map).

Corollary 2.7. Let be a partially ordered set and suppose that is a cone metric space. Let be a -distance on . Let and be two mappings such that

for some with and for all with or . Assume that and satisfy the following conditions:

(1), (2) is a complete subspace of , (3) has the mixed -monotone property.

Suppose that has the following properties:

(i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all .

Assume there exist such that and . Then and have a coupled coincidence point.

Proof. It follows from Theorem 2.4 by similar arguments to those given in proof of Corollary 2.5.

Corollary 2.8. Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on . Let be a mapping having the mixed monotone property such that

for some with and for all with or . Suppose that has the following properties:

(i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all .

Assume there exist such that and . Then has a coupled fixed point.

Proof. It follows from Corollary 2.7 by taking (the identity map).

Corollary 2.9. 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 mapping having the mixed monotone property such that

for some and for all with or .

If there exist such that and , then has a coupled fixed point . Moreover, we have and .

Proof. It follows from Theorem 2.2 by taking .

Corollary 2.10. Let be a partially ordered set and suppose that is a complete cone metric space. Let be a -distance on . Let be a mapping having the mixed monotone property such that

for some and for all with or . Suppose that has the following properties:

(i)if a nondecreasing sequence , then for all , (ii)if a nonincreasing sequence , then for all .

Assume there exist such that and . Then has a coupled fixed point.

Proof. It follows from Theorem 2.4 by taking .

Example 2.11. Let with and . Let with usual order . Define by for all . Then is a partially ordered cone metric space. Define by for all . Then is a -distance. Define by

Then,

(1), for all and , (2)there is no such that for all and , (3)there is no such that for all and .

Note that and . Thus by Corollary 2.10, we have which has a coupled fixed point. Here is a coupled fixed point of .

Proof. The proof of (2.1) is easy. To prove (2.3), suppose the contrary; that is, there is such that for all and . Take and . Then

Thus

Hence is a contradiction. The proof of (2.5) is similar to proof of (2.3).

Remark 2.12. Note that Theorems 3.1 and 3.2 of [39] are not applicable to Example 2.11.

Remark 2.13. Theorem 3.1 of [39] is a special case of Corollary 2.6 and Corollary 2.9.

Remark 2.14. Theorem 3.3 of [39] is a special case of Corollary 2.8 and Corollary 2.10.

3. Application

Consider the integral equations

where and . Let denote the space of -valued continuous functions on . The purpose of this section is to give an existence theorem for a solution to (3.1) that belongs to , by using the obtained result given by Corollary 2.10. Let , and let be the cone defined by

We endow with the cone metric defined by

It is clear that is a complete cone metric space. Let for all . Then, is a -distance.

Now, we endow with the partial order given by

Also, the product space can be equipped with the partial order (still denoted ) given as follows:

It is easy that and given in Corollary 2.10 are satisfied.

Now, we consider the following assumptions:

(a) is continuous, (b)for all , the function has the mixed monotone property, (c)for all , for all with and , we have

where is continuous nondecreasing an satisfies the following condition: There exists such that

(d)there exists such that

We have the following result.

Theorem 3.1. Suppose that hold. Then, (3.1) has at least one solution .

Proof. Define the mapping by

We have to prove that has at least one coupled fixed point .

From (b), it is clear that has the mixed monotone property.

Now, let such that ( and ) or ( and ). Using (c), for all , we have

which implies that

Similarly, one can get

We deduce

Thus

Thus, we proved that condition (2.31) of Corollary 2.10 is satisfied. Moreover, from (d), we have and . Finally, applying our Corollary 2.10, we get the desired result.

References

A. Yantir and S. G. Topal, “Positive solutions of nonlinear -point BVP on time scales,” International Journal of Difference Equations, vol. 3, no. 1, pp. 179–194, 2008.

Y. Guo, “A generalization of Banachs contraction principle for some non-obviously contractive operators in a cone metric space,” Turkish Journal of Mathematics, vol. 36, pp. 297–304, 2012.

A. Cataldo, E. A. Lee, X. Liu, E. D. Matsikoudis, and H. Zheng, “A constructive fixed point theorem and the feedback semantics of timed systems,” Tech. Rep. UCB/EECS-2006-4, EECS Department, University of California, Berkeley, Calif, USA, 2006.

A. Noumsi, S. Derrien, and P. Quinton, “Acceleration of a content based image retrieval application on the RDISK cluster,” in Proceedings of the IEEE International Parallel and Distributed Processing Symposium, April 2006.

A. Hyväarinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Transactions on Neural Networks, vol. 10, no. 3, pp. 626–634, 1999.

D. P. Mandic, J. A. Chambers, and M. M. Bozic, “On global asymptotic stability of fully connected recurrent neural networks,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (ICASSP '00), vol. 6, pp. 3406–3409, 2000.

K. C. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, New York, NY, USA, 1985.

M. H. Escardó, “PCF extended with real numbers,” Theoretical Computer Science, vol. 162, no. 1, pp. 79–115, 1996.

J. B. Gillespie and C. J. Houghton, “A metric space approach to the information channel capacity of spike trains,” Journal of Computational Neuroscience, vol. 30, no. 1, pp. 201–209, 2011.

K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942.

D. H. Fenwick and R. P. Batycky, “Using metric space methods to analyse reservoir uncertainty,” in Proceedings of the Gussow Conference, Alberta, Canada, 2011.

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.

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.

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.

H. Aydi, H. K. Nashine, B. Samet, and H. Yazidi, “Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations,” Nonlinear Analysis, vol. 74, no. 17, pp. 6814–6825, 2011.

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, article 30, 2011.

S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey,” Nonlinear Analysis, vol. 74, no. 7, pp. 2591–2601, 2011.

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.

Z. Kadelburg, P. P. Murthy, and S. Radenović, “Common fixed points for expansive mappings in cone metric spaces,” International Journal of Mathematical Analysis, vol. 5, no. 25–28, pp. 1309–1319, 2011.

W. Sintunavarat, Y. J. Cho, and P. Kumam, “Common fixed point theorems for -distance in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1969–1978, 2011.

W. Shatanawi and F. Awawdeh, “Some fixed and coincidence point theorems for expansive maps in cone metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 19, 2012.

W. Shatanawi, “On w-compatible mappings and common coupled coincidence point in cone metric spaces,” Applied Mathematics Letters, vol. 25, pp. 925–931, 2012.

W. Shatanawi, “Some coincidence point results in cone metric spaces,” Mathematical and Computer Modelling, vol. 55, pp. 2023–2028, 2012.

D. Turkoglu, M. Abuloha, and T. Abdeljawad, “KKM mappings in cone metric spaces and some fixed point theorems,” Nonlinear Analysis, vol. 72, no. 1, pp. 348–353, 2010.

T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, vol. 65, no. 7, pp. 1379–1393, 2006.

V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis, vol. 70, no. 12, pp. 4341–4349, 2009.

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.

H. Aydi, M. Abbas, and M. Postolache, “Coupled coincidence points for hybrid pair of mappings via mixed monotone property,” Journal of Advanced Mathematical Studies, vol. 5, no. 1, pp. 118–126, 2012.

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.

Y. J. Cho, M. H. Shah, and N. Hussain, “Coupled fixed points of weakly -contractive mappings in topological spaces,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1185–1190, 2011.

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.

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.

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, article 81, 2011.

W. Shatanawi, “Partially ordered cone metric spaces and coupled fixed point results,” Computers & Mathematics with Applications, vol. 60, no. 8, pp. 2508–2515, 2010.

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.

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.

L. Cirić, H. Lakzian, and V. Rakočević, “Fixed point theorems for w-cone distance contraction mappings in tvs-cone metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 3, 2012.

Z. Kadelburg and S. Radenovic, “Coupled fxed point results under tvs-cone metric and w-cone distance,” Advances in Fixed Point Theory, vol. 2, no. 1, pp. 29–46, 2012.

Y. J. Cho, Z. Kadelburg, R. Saadati, and W. Shatanawi, “Coupled fixed point theorems under weak contractions,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 184534, 9 pages, 2012.

M. Dornevic, D. Doric, Z. Kadelburg, S. Radenovic, and D. Spaic, “Fixed point results under c-distance in tvs-cone metric spaces,” Fixed Point Theory and Its Applications, vol. 2011, article 29, 2011.