Abstract and Applied Analysis

Volume 2013 (2013), Article ID 815289, 7 pages

http://dx.doi.org/10.1155/2013/815289

## Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia

Received 28 February 2013; Accepted 18 April 2013

Academic Editor: Abdelghani Bellouquid

Copyright © 2013 Sahar Mohammad Abusalim and Mohd Salmi Md Noorani. 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 concept of a cone b-metric space has been introduced recently as a generalization of a b-metric space and a cone metric space in 2011. The aim of this paper is to establish some fixed point and common fixed point theorems on ordered cone b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Some supporting examples are given.

#### 1. Introduction

Fixed point theory has attracted many researchers since 1922 with the admired Banach fixed point theorem. This theorem supplies a method for solving a variety of applied dilemma in mathematical sciences and engineering. A large literature on this subject exists, and this is a very active area of research at present. Banach contraction principle has been generalized in dissimilar directions in different spaces by mathematicians over the years; for more details on this and related topics, we refer to [1–6] and references therein.

In contemporary time, fixed point theory has evolved speedily in partially ordered cone metric spaces; that is, cone metric spaces equipped with a partial ordering, for some new results in ordered metric spaces see [7]. A coming early result in this bearing was constituted by Altun and Durmaz [8] under the condition of normality for cones. Then, Altun et al. [9] generalized the results of Altun and Durmaz [8] by omitting the assumption of normality condition for cones. Afterward, several authors have studied fixed point and common fixed point problems in ordered cone metric spaces; for more details see [10–17].

In 2011, Hussain and Shah [18] presented cone b-metric spaces as a generalization of b-metric spaces and cone metric spaces; for some new results in b-metric spaces see [19]. They not only constructed some topological properties in such spaces but also ameliorated some current results about KKM mappings in the setting of a cone b-metric space. After some time, many authors have been motivated to demonstrate fixed point theorems as well as common fixed point theorems for two or more mappings on cone b-metric spaces by the incipient work of Hussain and Shah [18] (see [20–23] and the references therein).

In [8], Altun and Durmaz proved the following results under the condition of normality for cones.

Theorem 1 (see [8]). *Let be a partially ordered set, suppose that there exists a cone metric in such that the cone metric space is complete, and let be a normal cone with normal constant . Let be a continuous and nondecreasing mapping with respect to . Suppose that the following three assertions hold: ** there exists such that for all with ;** there exists such that .**Then has a fixed point in . *

In [9], Altun et al. generalized the above theorem and proved it without normality condition for cones.

Theorem 2 (see [9]). *Let be a partially ordered set and suppose that there exists a cone metric in such that the cone metric space is complete over a solid cone . Let be a continuous and nondecreasing mapping with respect to . Suppose that the following two assertions hold: ** there exist with such that
for all with ;** there exists such that .**Then has a fixed point in . *

Theorem 3 (see [9]). *Let be a partially ordered set and suppose that there exists a cone metric in such that the cone metric space is complete over a solid cone . Let be a nondecreasing mapping with respect to . Suppose that the following three assertions hold: ** there exist with such that
for all with ;** there exists such that ;** if an increasing sequence converges to in , then for all . **Then has a fixed point in . *

In the same paper, they also presented the following two common fixed point results in ordered cone metric spaces.

Theorem 4 (see [9]). *Let be a partially ordered set and suppose that there exists a cone metric in such that the cone metric space is complete over a solid cone . Let be two weakly increasing mappings with respect to . Suppose that the following three assertions hold:** there exist with such that
for all comparative ;** or is continuous. **Then and have a common fixed point . *

Theorem 5 (see [9]). *Let be a partially ordered set and suppose that there exists a cone metric in such that the cone metric space is complete over a solid cone . Let be two weakly increasing mappings with respect to. Suppose that the following three assertions hold: ** there exist with such that
for all comparative ;** if an increasing sequence converges to in , then for all . **Then and have a common fixed point . *

In this paper, we prove some fixed point and common fixed point theorems on ordered cone b-metric spaces. Our results extend and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Throughout this paper, we do not impose the normality condition for the cones, but the only assumption is that the cone is solid, that is, int .

The following definitions and results shall be needed in the sequel.

Let be a real Banach space and denotes the zero element in . A cone is a subset of such that is nonempty closed set and ; if , are nonnegative real numbers and , then ; and imply .For any cone , the partial ordering with respect to is defined by if and only if . The notation of stands for but . Also, we use to indicate that int , where int denotes the interior of . A cone is called normal if there exists the number such that for all . The least positive number satisfying the above condition is called the normal constant of .

*Definition 6 (see [18]). *Let be a nonempty set and a real Banach space equipped with the partial ordering with respect to the cone . A vector-valued function is said to be a cone b-metric function on with the constant if the following conditions are satisfied: for all and if and only if ; for all ; for all . Then pair is called a cone b-metric space (or a cone metric type space); we shall use the first mentioned term.

Observe that if , then the ordinary triangle inequality in a cone metric space is satisfied; however, it does not hold true when . Thus the class of cone b-metric spaces is effectively larger than that of the ordinary cone metric spaces. That is, every cone metric space is a cone b-metric space, but the converse need not be true. The following examples show the above remarks.

*Example 7. *Let ,. Define by for all ,,, and ,. Then is a complete cone b-metric space but the triangle inequality is not satisfied. Indeed, we have that . It is not hard to verify that .

*Example 8. *Let , , and. Define by . Then, it is easy to see that is a cone b-metric space with the coefficient . But it is not a cone metric spaces since the triangle inequality is not satisfied.

*Definition 9 (see [18]). *Let be a cone b-metric space, a sequence in and . For all with , if there exists a positive integer such that for all , then is said to be convergent and is the limit of . One denotes this by . For all with , if there exists a positive integer such that for all , then is called a Cauchy sequence in . A cone metric space is called complete if every Cauchy sequence in is convergent.

The following lemma is useful in our work.

Lemma 10 (see [24]). * If is a real Banach space with a cone and where and , then . ** If , , and , then there exists a positive integer such that for all . ** If and , then . ** If for each , then . *

#### 2. Fixed Point Results

In this section, we prove some fixed point theorems on ordered cone b-metric space. We begin with a simple but a useful lemma.

Lemma 11. *Let be a sequence in a cone b-metric space with the coefficient relative to a solid cone such that
**
where and . Then is a Cauchy sequence in . *

*Proof. *Let . It follows that
Now, (6) and imply that
According to Lemma 10, and for any with , there exists such that for any , . Furthermore, from (8) and for any , Lemma 10 shows that
Hence, by Definition 9 is a Cauchy sequence in .

Theorem 12. *Let be a partially ordered set and suppose that there exists a cone b-metric in such that the cone b-metric space is complete with the coefficient relative to a solid cone . Let be a continuous and nondecreasing mapping with respect to . Suppose that the following three assertions hold: ** there exist , such that with ,
for all with ; ** there exists such that . **Then has a fixed point .*

*Proof. *If , then the proof is finished. Suppose that . Since and is nondecreasing with respect to, we obtain by induction that . Then we have,
Then, one can assert that
On the other hand, we have
Then, one can assert that
Adding (12) and (14), we get
where . According to Lemma 11, we have is a Cauchy sequence in . Since is complete, there exists such that . Since is continuous, then . Therefore, is a fixed point of .

If we use condition (iii) instead of the continuity of in Theorem 12, we have the following result.

Theorem 13. *Let be a partially ordered set and suppose that there exists a cone b-metric in such that the cone b-metric space is complete with the coefficient relative to a solid cone . Let be a nondecreasing mapping with respect to. Suppose that the following three assertions hold: ** there exist , such that with ,
for all with ;** there exists such that ; ** if an increasing sequence converges to in , then for all .**Then has a fixed point .*

*Proof. *As in the Theorem 12, we can construct an increasing sequence and prove that there exists such that . Now, condition (iii) implies for all . Therefore, we can use condition (i) and so
Taking , we have . Since , Lemma 10(1) shows that ; that is, . Therefore is a fixed point of .

#### 3. Common Fixed Point Results

Now, we give two common fixed point theorems on ordered cone b-metric spaces. We need the following definition.

*Definition 14 (see [9]). *Let be a partially ordered set. Two mappings are said to be weakly increasing if and hold for all .

Theorem 15. *Let be a partially ordered set and suppose that there exists a cone b-metric in such that the cone b-metric space is complete with the coefficient relative to a solid cone . Let be two weakly increasing mappings with respect to. Suppose that the following three assertions hold:** there exist , such that with ,
for all comparative ;** or is continuous. **Then and have a common fixed point . *

*Proof. *Let be an arbitrary point of and define a sequence in as follows: and for all . Note that, since and are weakly increasing, we have and , and continuing this process we have . That is, the sequence is nondecreasing. Now, since and are comparative, we can use the inequality (18), and then we have
Hence,

On the other hand and by symmetry we have
Hence,
Adding inequalities (20) and (22), we get
where . Similarly, it can be shown that
Therefore,
According to Lemma 11, we have is a Cauchy sequence in . Since is complete, there exists such that . Suppose that is continuous. Then, . Therefore, is a fixed point of . Now, we need to show that is a fixed point of . Since , we can use the inequality (18) for . Then we have
Hence,
Since , Lemma 10 shows that ; that is, . Therefore is a fixed point of . Therefore, and have a common fixed point. The proof is similar when is a continuous mapping.

Theorem 16. *Let be a partially ordered set and suppose that there exists a cone b-metric in such that the cone b-metric space is complete with the coefficient relative to a solid cone . Let be two weakly increasing mappings with respect to. Suppose that the following three assertions hold: ** there exist such that with ,
for all comparative ; ** if an increasing sequence converges to in , then for all . **Then and have a common fixed point . *

*Proof. *As in Theorem 15, we can construct an increasing sequence and prove that there exists such that , also; by the construction of , . Now, condition (iii) implies for all . Putting and in (28), we get
Hence,
Since and , then by Definition 9 and for there exists such that for all , , and . Then we have
Now again, according to Definition 9 it follows that . It follows that . In a similar way and using that , we can prove that . Therefore, and have a common fixed point.

Now, we present two examples to illustrate our results. In the first example (the case of a normal cone), the conditions of Theorem 12 are fulfilled, but Theorem 2 of Altun et al. [9, Theorem 12] cannot be applied. In the second example (the case of a nonnormal cone), the conditions of Theorem 12 are fulfilled, but Theorem 3 of Altun et al. [9, Theorem 13] cannot be applied.

*Example 17. *Let endowed with the standard order and and let . Define as in Example 8. Define by . Then is a continuous and nondecreasing mapping with respect to . Then we have
where , . It is clear that the conditions of Theorem 12 are satisfied. Therefore, has a fixed point .

*Example 18. *Let endowed with the standard order and with and let . It is well known that this cone is solid, but it is not normal. Define a cone metric by . Then is a complete cone b-metric space with the coefficient . Let us define by . Then is a continuous and nondecreasing mapping with respect to . Then we have is an increasing mapping; also we have
where , , . It is clear that the conditions of Theorem 12 are satisfied. Therefore, has a fixed point .

#### Acknowledgments

The authors would like to acknowledge the financial support received from Ministry of Higher Education, Malaysia, under the Research Grant no. ERGS/1/2011/STG/UKM/01/13. The authors thank the referee for his/her careful reading of the paper and useful suggestions.

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