Journal of Applied Mathematics

Volume 2013 (2013), Article ID 146794, 6 pages

http://dx.doi.org/10.1155/2013/146794

Research Article

The Point of Coincidence and Common Fixed Point for Three Mappings in Cone Metric Spaces

Anil Kumar,¹ Savita Rathee,¹ and Navin Kumar²

¹Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana 124001, India

²Department of Applied Science, Vaish College of Engineering, Rohtak, Haryana 124001, India

Received 14 December 2012; Accepted 14 February 2013

Academic Editor: Yansheng Liu

Copyright © 2013 Anil Kumar 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.

The aim of this paper is to present the point of coincidence and common fixed point for three mappings in cone metric spaces over normal cone which satisfy a different contractive condition. Our result generalizes the recent related results proved by Stojan Radenović (2009) and Rangamma and Prudhvi (2012).

1. Introduction and Preliminaries

It is well known that the classical contraction mapping principle of Banach is a fundamental result in fixed point theory. Several authors have obtained various extensions and generalizations of Banach’s theorems by considering contractive mappings on different metric spaces. Huang and Zhang [1] have replaced real numbers by ordering Banach space and have defined a cone metric space. They have proved some fixed point theorems of contractive mappings on cone metric spaces. Further generalizations of Huang and Zhang were obtained by Abbas and Jungck [2]. In 2009 Radenović [3] has obtained coincidence point result for two mappings in cone metric spaces which satisfy new contractive conditions. Recently, in this paper we generalized the coincidence point results of Radenović [3] for three maps with different contractive condition.

We recall some definitions and results that will be needed in what follows.

Definition 1. Let be a real Banach space and be a subset of . Then is called a cone if

(1)is closed, nonempty and satisfies ,(2), , and imply ,(3) and imply .

Given a cone , we define a partial ordering with respect to by if and only if . We shall write if and , while if and only if , where is the interior of . A cone is called normal if there is a number such that, for all , implies . The least positive number satisfying the above inequality is called the normal constant of .

In the following we suppose that is a real Banach space and is a cone in with and is a partial ordering with respect to .

Definition 2. Let be a nonempty set. Suppose that the mapping satisfies

(i) for all and if and only if ,(ii) for all ,(iii) for all .

Then is called a cone metric on , and is called a cone metric space.

Example 3. Let , , , and be defined by , where is a constant; then is a cone metric space.

Definition 4. Let be a cone metric space, be a sequence in and . Then converges to if for every lies in with there is an such that for all , . One denotes this by as .

Definition 5. Let be a cone metric space, be a sequence in . If for every lies in with there is an such that for all , , then is called a Cauchy sequence in .

Definition 6. A cone metric space is said to be complete if every Cauchy sequence in is convergent in .

Lemma 7. Let be a cone metric space and be a normal cone. Let be a sequence in . One has the following.

(i) converges to if and only if as .(ii) is a Cauchy sequence if and only if as .(iii) converges to and converges to . Then .

Definition 8. Let and be self-maps on set . If for some in , then is called a coincidence point of and , and is called a point of coincidence of and .

Definition 9. Two self-mappings and of a cone metric space are said to be weakly compatible if whether .

2. Main Result

In this section, we give fixed point theorems for mappings defined on cone metric space with generalized contractive condition.

Theorem 10. Let be a cone metric space and be a normal cone with normal constant K. Suppose that the mappings , and satisfy the condition

for all , where , , , and are nonnegative real numbers satisfying . If the range of contains range of and also range of and is a complete subspace of , then , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, then , , and have a unique common fixed point.

Proof. Let be an arbitrary point. Since and are contained in , there exists such that , and also there exists such that . Continuing this process, a sequence can be chosen such that and , for ; then

This implies that .

Thus

where , as .

Writing = , we obtain

Again

This implies that .

Thus

 where , as .

Therefore

From (4) and (7) we get

Therefore

Now we will show that is a Cauchy sequence. By triangle inequality for , we have

Hence, as is normal cone with normal constant ,

If is even, then from (9) and (12) we have

If is odd, then from (10) and (12) we have

Since , , therefore , so in both cases, as .

From Lemma 7, it follows that is a Cauchy sequence. Since is a complete subspace of , there exists in such that → as ; consequently we can find in such that . We shall show that .

Now using contractive condition (1), we can write

Taking , we have

Hence, , since and .

Again from (1), we can write

Taking , we have

Hence, , since and .

So we get

Therefore is a coincidence point of , , and .

Now we show that , , and have a unique point of coincidence. For this, assume that there exists another point of coincidence in such that .

Consider

Since and , so from (20), .

Therefore, , and hence , , and have unique point of coincidence in .

Now from (1) we have

As is weakly compatible, therefore from (19) and (21) we can write

As and , so from (22), .

Therefore,

Also,

Again from (1) we have

As is weakly compatible, therefore from (19) and (25) we can write

As and , so from (26), .

Hence,

From (23), (24), and (27), it follows that is common fixed point for , , and .

Now we shall prove the uniqueness of common fixed point for , , and . Suppose is another common fixed point for , , and .

Consider

Therefore, , since and . Thus , , and have unique common fixed point in .

Remark 11. (i) If we take , in Theorem 10, then

(ii) If we take , in Theorem 10, then

(iii) If we take , in Theorem 10, then

From Remark 11, it is clear that Theorem 2.1 in [4] is a special case of Theorem 10 with and , where , and Theorem 2.3 in [4] is a special case of Theorem 10 with and , where . Therefore, we can say that Theorem 10 has generalized and unified the main results in [4].

In Theorem 10 if we take , then as immediate consequence of Theorem 10 we obtain the following corollary.

Corollary 12. Let be a cone metric space and be a normal cone with normal constant . Suppose that the mappings satisfy the condition

for all , where , , , and are nonnegative real numbers satisfying . If the range of contains the range of and is a complete subspace of , then and have a unique point of coincidence in . Moreover, if is weakly compatible, then and have a unique common fixed point.

Remark 13. (i) If we take , in Corollary 12, then

(ii) If we take , in Corollary 12, then

(iii) If we take in Corollary 12, then

From Remark 13 it is clear that Theorem 2.3 [3] is a special case of Corollary 12. Therefore we can say that Theorem 10 has generalized and unified the main result of Radenović in [3].

We present now some nontrivial examples that illustrate how general and important is the result given by Theorem 10.

Example 14. Let , with the norm , be a real Banach space and let . If we consider and define by

then is a cone metric space. Let , , and be defined, respectively, as follows:

Then , , and have the properties mentioned in Theorem 10, and also , , and satisfy the inequality (1).

Hence the conditions of Theorem 10 are satisfied. Therefore we conclude that , , and have unique point of coincidence and also unique common fixed point.

Here it is seen that is unique point of coincidence and also the unique common fixed point of , , and .

Remark 15. Example 14 does not satisfy the conditions (29) and (30) at the points , and , , respectively. Therefore, we can say that inequalities of Theorems 2.1 and 2.3 of [4] fail at the points , and , , respectively. Hence, Theorem 2.1 and Theorem 2.3 of [4] cannot apply to Example 14.

Example 16. Let , with the norm , be a real Banach space and let . Let and also define by for all .

Then is a cone metric space. Let be defined, respectively, as follows:

Also

Then and have the properties mentioned in Corollary 12, and also and satisfy the inequality (32).

Hence the conditions of Corollary 12 are satisfied. Therefore we conclude that and have unique point of coincidence and also unique common fixed point.

Here it is seen that 0 is unique point of coincidence and also the unique common fixed point of and .

Remark 17. Example 16 does not satisfy the conditions ((33), (35)), and (34) at the points , and , , respectively. Therefore, we can say that inequalities ((2.4), (2.6)) and (2.5) of [3] fail at the points , and , , respectively. Hence, Theorem 2.3 of [3] cannot apply to Example 16.

Remark 18. Example 14 does not satisfy the inequality 2.8 of [5] at the point , . Therefore, it is clear that Corollary 2.10 of [5] cannot apply to Example 14. Hence Theorem 10 is more general than Corollary 2.10 of [5].

References

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.

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.

S. Radenović, “Common fixed points under contractive conditions in cone metric spaces,” Computers & Mathematics with Applications, vol. 58, no. 6, pp. 1273–1278, 2009.

M. Rangamma and K. Prudhvi, “Common fixed points under contractive conditions for three maps in cone metric spaces,” Bulletin of Mathematical Analysis and Applications, vol. 4, no. 1, pp. 174–180, 2012.

M. Abbas, B. E. Rhoades, and T. Nazir, “Common fixed points for four maps in cone metric spaces,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 80–86, 2010.