`Abstract and Applied AnalysisVolume 2012, Article ID 639713, 12 pageshttp://dx.doi.org/10.1155/2012/639713`
Research Article

New Fixed Point Results of Single-Valued Mapping for c-Distance in Cone Metric Spaces

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia
2Teacher Education School in Prizren-Leposavić, Nemanjina b.b, 38218 Leposavić, Serbia

Received 13 July 2012; Accepted 20 August 2012

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 results in literature for c-distance in cone metric spaces by replacing the constants in contractive conditions with functions. Some supporting examples are given.

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 [18]. 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 [9] 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 [1015].

In [16], Cho et al. introduced a new concept of the c-distance in cone metric spaces (also see [17]) and proved some fixed point theorems in ordered cone metric spaces. This is more general than the classical Banach contraction mapping principle. Then, Sintunavarat et al. [18] extended and developed the Banach contraction theorem on c-distance of Cho et al. [16]. They gave some illustrative examples of main results. Their results improve, generalize, and unify the results of Cho et al. [16] and some results of the fundamental metrical fixed point theorems in the literature, for some new results for c-distance see [1924].

In [21], Fadail et al. proved the following theorems for c-distance in cone metric spaces.

Theorem 1.1. Let be a complete cone metric space and is a c-distance on . Suppose the mapping satisfies the following 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.

Theorem 1.2. Let be a complete cone metric space and is a c-distance on . Suppose the mapping is continuous and satisfies the following contractive condition: for all , where are none negative 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.

Theorem 1.3. Let be a complete cone metric space and is a c-distance on . Suppose the mapping satisfies the folowing contractive condition: for all , where are none negative 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.

The aim of this paper is to continue the study of common coupled fixed points of mappings but now for c-distance in cone metric space. Our results extend and develop some theorems on c-distance of Fadail et al. [21]. In this paper, we do not impose the normality condition for the cones; the only assumption is that the cone is solid, that is, .

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 implies . For any cone , the partial ordering with respect to is defined by if and only if . The notation of stand 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 . It is clear that .

Definition 2.1 (see [9]). Let be a nonempty set and 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 [9]). Let be a cone metric space, 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 [25]). (1) If be 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 c-distance on a cone metric space which is a generalization of -distance of Kada et al. [26] with some properties.

Definition 2.4 (see [16]). Let be a cone metric space. A function is called a c-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 [16]). 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 [16]). 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 [16]). (1) does not necessarily for all .
(2) is not necessarily equivalent to for all .

3. Main Results

In this section, we generalize some fixed point results from [21] by replacing the constants in contractive conditions with functions.

Theorem 3.1. Let be a complete cone metric space and is a c-distance on . Let be a mapping and suppose that there exists mapping such that the following hold:(a) for all ,(b) for all . 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 . Then we have
Let . Then it follows that Thus, Lemma 2.6(3) shows that is a Cauchy sequence in . Since is complete, there exists such that as . By q3, we have
On the other hand, we have By Lemma 2.6(3), (3.3), and (3.4), we have . Thus, is a fixed point of .
Suppose that , then we have . Since , Lemma 2.3(1) shows that .
Finally, suppose there is another fixed point of , then we have: . Since , Lemma 2.3(1) shows that , and also we have , hence by Lemma 2.6(1), . Therefore, the fixed point is unique.

In the above theorem, if is constant, then we have the following corollary.

Corollary 3.2 ([21, theorem 3.1]). Let be a complete cone metric space and is a c-distance on . Suppose the mapping satisfies the following 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.

Theorem 3.3. Let be a complete cone metric space and is a c-distance on . Let be a continuous mapping and suppose that there exists mapping such that the following hold:(a), , for all , (b) for all , (c) for all . 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 . Then we have So where .
Let . Then it follows that Thus, Lemma 2.6(3) 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 . Since , Lemma 2.3(1) shows that .
Finally, suppose there is another fixed point of , then we have Since , Lemma 2.3(1) shows that , and also we have , hence by Lemma 2.6(1), . Therefore, the fixed point is unique.

In Theorem 3.3, if , and are constant, then we have the following corollary.

Corollary 3.4 ([21, theorem 3.3]). Let be a complete cone metric space and is a c-distance on . Suppose the mapping is continuous and satisfies the following contractive condition: for all , where are none negative 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.

Theorem 3.5. Let be a complete cone metric space and is a c-distance on . Let be a mapping and suppose that there exists mapping such that the following hold:(a), , for all , (b) for all , (c) for all . 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(3) shows that is a Cauchy sequence in . Since is complete, there exists such that as .
By q3, we have
On the other hand, we have So By Lemma 2.6(1), (3.16), and (3.18), we have . Thus, is a fixed point of .
Suppose that , then we have Since , Lemma 2.3(1) shows that .
Finally, suppose there is another fixed point of , then we have Since , Lemma 2.3(1) shows that and also we have , hence by Lemma 2.6(1), . Therefore, the fixed point is unique.

In Theorem 3.5, if , and are constants, then we have the following corollary.

Corollary 3.6 ([21, Theorem 3.5]). Let be a complete cone metric space and is a c-distance on . Suppose the mapping satisfies the following contractive condition: for all , where are none negative 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.

Example 3.7. Let and . Let and define a mapping by for all . Then is a complete cone metric space. Define a mapping by for all . Then is a c-distance on . Define the mapping by for all . Take , . Observe that(a) for all . (b)For all , we have Therefore, the conditions of Theorem 3.1 are satisfied. Hence has a unique fixed point with .

Acknowledgments

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant OUP-UKM-FST-2012. Third author 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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