Research Article | Open Access

# Fixed Point Theorems of Single-Valued Mapping for *c*-Distance in Cone Metric Spaces

**Academic Editor:**Gerd Teschke

#### Abstract

A new concept of the *c*-distance in cone metric space has been introduced recently in 2011. The aim of this paper is to extend and generalize some fixed point theorems on *c*-distance in cone metric space.

#### 1. Introduction

The concept of cone metric spaces is a generalization of metric spaces, where each pair of points is assigned to a member of a real Banach space with a cone, for new results on cone metric spaces see [1–6]. This cone naturally induces a partial order in the Banach spaces. The concept of cone metric space was introduced in the work of Huang and Zhang [7], where they also established the Banach contraction mapping principle in this space. Then, several authors have studied fixed point problems in cone metric spaces. Some of these works are noted in [8–13].

In [14], Cho et al. introduced a new concept of the *c*-distance in cone metric spaces and proved some fixed point theorems in ordered cone metric spaces. This is more general than the classical Banach contraction mapping principle.

In [15], Sintunavarat et al. extended and developed the Banach contraction theorem on *c*-distance of Cho et al. [14]. They gave some illustrative examples of the main results. Their results improve, generalize, and unify the results of Cho et al. [14] and some results of the fundamental metrical fixed point theorems in the literature.

In this paper we proved some fixed point theorems for *c*-distance in cone metric space. These theorems extend and develop some theorems in literature on *c*-distance of Cho et al. [14] in cone metric space.

The following theorems are the main results given in [7, 14, 16].

Theorem 1.1 (see [16]). *Let be a complete cone metric space. Suppose that the mapping satisfies the contractive condition:
**
for all , where is a constant. Then has a unique fixed point in and for any , iterative sequence converges to the fixed point.*

Theorem 1.2 (see [7]). *Let be a complete cone metric space and be a normal cone with normal constant . Suppose that the mapping satisfies the contractive condition:
**
for all , where is a constant. Then has a unique fixed point in and for any , iterative sequence converges to the fixed point.*

Theorem 1.3 (see [7]). *Let be a complete cone metric space and be a normal cone with normal constant . Suppose that the mapping satisfies the contractive condition:
**
for all , where is a constant. Then has a unique fixed point in and for any , iterative sequence converges to the fixed point.*

Theorem 1.4 (see [14]). *Let be a partially ordered set and suppose that is a complete cone metric space. Let is a c-distance on and be a continous and nondecreasing mapping with respect to . Suppose that the following two assertions hold: *(1)*there exist with such that
for all with ,*(2)* there exists such that .**Then has a fixed point . If then .*

#### 2. Preliminaries

Let be a real Banach space and denote to the zero element in . A cone is a subset of such that(1) is nonempty set closed and ,(2) if are nonnegative real numbers and then ,(3) and imply that .

For any cone , the partial ordering with respect to is defined by if and only if . The notation of stands for but . Also, we used to indicate that , where denotes the interior of . A cone is called normal if there exists a number such that for all . The least positive number satisfying the above condition is called the normal constant of .

*Definition 2.1 (see [7]). *Let be a nonempty set and be a real Banach space equipped with the partial ordering with respect to the cone . 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 [7]). *Let be a cone metric space, be a sequence in , and *. *(1)For all with , if there exists a positive integer such that for all , then is said to be convergent and is the limit of . We denote this by .(2) For all with , if there exists a positive integer such that for all , then is called a Cauchy sequence in .(3) A cone metric space is called complete if every Cauchy sequence in is convergent.

Lemma 2.3 (see [17]). (1)*If is a real Banach space with a cone and , where and , then .*(2)* If , and , then there exists a positive integer such that for all .*

Next we give the notation of -distance on a cone metric space which is a generalization of -distance of Kada et al. [18] with some properties.

*Definition 2.4 (see [14]). *Let is a cone metric space. A function is called a -distance on if the following conditions hold:(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 imply .

*Example 2.5 (see [14]). *Let and . let and define a mapping by for all . then is a cone metric space. define a mapping by for all . Then is a *c*-distance on .

Lemma 2.6 (see [14]). *Let be a cone metric space and is a c-distance on . Let and be sequences in and . Suppose that is a sequences in converging to . Then the following hold: *(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 .*

*Remark 2.7 (see [14]). *(1) does not necessarily for all .(2) is not necessarily equivalent to for all .

#### 3. Main Results

In this section we prove some fixed point theorems using *c*-distance in cone metric space. In whole paper cone metric space is over nonnormal cone with nonempty interior.

Theorem 3.1. *Let be a complete cone metric space and is a -distance on . Suppose that the mapping satisfies the contractive condition:
**
for all , where is a constant. Then has a fixed point and for any , iterative sequence converges to the fixed point. If then . The fixed point is unique.*

*Proof. *Choose . Set . We have:
Let . Then it follows that

Thus, Lemma 2.6 shows that is a Cauchy sequence in . Since is complete, there exists such that as . By (q3) we have:

On the other hand,

By Lemma 2.6 part 1, (3.4) and (3.5), we have . Thus, is a fixed point of .

Suppose that , then we have the following: . Since , Lemma 2.3 show that .

Finally suppose there is another fixed point of , then we have the following: . Since , Lemma 2.3 show that and also we have . Hence by Lemma 2.6 part 1, . Therefore the fixed point is unique.

Corollary 3.2. *Let be a complete cone metric space and is a c-distance on . Suppose that the mapping satisfies the contractive condition:
**
for all , where is a constant. Then has a unique fixed point . If then .*

*Proof. *From Theorem 3.1 has a unique fixed point . But , so is also a fixed point of . Hence . Thus, is a fixed point of . Since the fixed point of is also fixed point of , the fixed point of is unique.

Suppose that . From above the fixed point of is also fixed point of , then we have the following: . Since , Lemma 2.3 show that .

The following result is generalized from Theorem 1.4. We prove a fixed point theorem and we do not require that is a partially ordered set.

Theorem 3.3. *Let be a complete cone metric space and is a -distance on . Suppose that the mapping is continuous and satisfies the contractive condition:
**
for all , where , , are nonnegative real numbers such that . Then has a fixed point and for any , iterative sequence converges to the fixed point. If then . The fixed point is unique.*

*Proof. *Choose . Set . We have the following:
So
where .

Let . Then it follows that

Thus, Lemma 2.6 shows that is a Cauchy sequence in . Since is complete, there exists such that as . Since is continuous, then . Therefore is a fixed point of .

Suppose that , then we have
since , Lemma 2.3 show that .

Finally, suppose that, there is another fixed point of , then we have the following:
Since , Lemma 2.3 shows that and also we have . Hence by Lemma 2.6 part 1, . Therefore the fixed point is unique.

If and , then we have the following result.

Corollary 3.4. *Let be a complete cone metric space and is a c-distance on . Suppose that the mapping is continuous and satisfies the contractive condition:
**
for all ,where is a constant. Then has a fixed point and for any , iterative sequence converges to the fixed point. If then . The fixed point is unique.*

Finally, we provide another result and we do not require that is continuous.

Theorem 3.5. *Let be a complete cone metric space and is a -distance on . Suppose that the mapping satisfies the contractive condition:
**
for all , where are nonnegative real numbers such that . Then has a fixed point and for any , iterative sequence converges to the fixed point. If then . The fixed point is unique.*

*Proof. *Choose . Set . Observe that
equivalently
Then we have:
So,
where .

Let . Then it follows that

Thus, Lemma 2.6 shows that is a Cauchy sequence in . Since is complete, there exists such that as .

By (q3) we have:

On the other hand,
So,

By Lemma 2.6 part 1, (3.20) and (3.22), we have . Thus, is a fixed point of .

Suppose that , then we have
Since , Lemma 2.3 shows that .

Finally, suppose that, there is another fixed point of , then we have

Since , Lemma 2.3 shows that and also we have . Hence by Lemma 2.6 part 1, . Therefore the fixed point is unique.

*Example 3.6. *Consider Example 2.5. Define the mapping by and for all with . Since , there is not such that . Since Theorem 2.3 of Rezapour and Hamlbarani [16] cannot be applied to this example on cone metric space. To check this example on *c*-distance we have:(1)If , then we have the following.(2) If , then we have
(3) If , then we have
(4) If , then we have
Hence for all . Therefore, the condition of Theorem 3.1 are satisfied and then has a unique fixed point , with .

#### Acknowledgments

The authors Zaid Mohammed Fadail and Abd Ghafur Bin Ahmad would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant ERGS/1/2011/STG/UKM/01/13. Zoran Golubović is thankful to the Ministry of Science and Technological Development of Serbia. The authors thank the referee for his/her careful reading of the paper and useful suggestions.

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#### Copyright

Copyright © 2012 Zaid Mohammed Fadail 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.