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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 171624, 8 pages

http://dx.doi.org/10.1155/2014/171624
Research Article

Remarks on Some Recent Fixed Point Results on Quaternion-Valued Metric Spaces

1Department of Mathematics, Texas A&M University-Kingsville, MSC 172, Rhode Hall 217B, Kingsville, TX 78363-8202, USA

2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

3Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia

4Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

5Department of Mathematics, Islamic Azad University, Arak Branch, Arak, Iran

Received 17 March 2014; Accepted 18 April 2014; Published 21 May 2014

Academic Editor: Wei-Shih Du

Copyright © 2014 Ravi P. Agarwal 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

Very recently, Ahmed et al. introduced the notion of quaternion-valued metric as a generalization of metric and proved a common fixed point theorem in the context of quaternion-valued metric space. In this paper, we will show that the quaternion-valued metric spaces are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be derived as a consequence of the corresponding existing fixed point result in the setting cone metric spaces.

1. Introduction

Recently, Azam et al. [1] introduced the notion of complex-valued metric space, as a generalization of Banach-valued metric space which is also known as a cone metric space. The authors [1] proved several fixed point theorems in the context of complex-valued metric space. Inspired from these results, Ahmed et al. [2] defined the concept of quaternion-valued metric space, as a generalization of complex-valued metric space, and proved a common fixed point theorem in the context of such spaces.

In this paper, we announce that the quaternion-valued metric spaces, introduced by Ahmed et al. [2], are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be concluded from the classical versions in cone metric spaces. Consequently, the fixed point results in such spaces can be concluded from the classical versions in cone metric spaces. On the other hand, several results have been reported on the equivalence of cone metric space and metric space; see, for example, [39]. In particular, by the help of scalarization function, Du [3] proved that several fixed point results in the context of cone metric spacecan be concluded from the existing associated results in the setting of metric space. Furthermore, if the cone is normal, then there is a metric induced by Banach-valued metric. Hence, most of the announced fixed point results in the setting cone metric space can be deduced from related existing results in the literature in the context of the metric space.

1.1. Complex-Valued Metric Spaces

First we recall the concept of complex-valued metric space which is given by Azam et al. in [1].

Let be the set of complex numbers and . Define a partial order on as follows: It follows that if one of the following conditions is satisfied: In particular, we will write if and one of , , and is satisfied and we will write if only is satisfied. Note that where represents modulus or magnitude of , and

Definition 1 (see [1]). Let be a nonempty set. A function is called a complex-valued metric on , if it satisfies the following conditions: : for all and , if and only if , : , for all , : , for all .Here, the pair is called a complex-valued metric space.

Let be a sequence in and . If for every , with , there is such that, for all , , then is said to be convergent, converges to , and is the limit point of . We denote this by , or , as . If for every with there is such that for all , , then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete complex-valued metric space.

Lemma 2 (see [1, Lemma 2, Azam et al.]). Let be a complex-valued metric space and let be a sequence in. Then converges to if and only if as .

Lemma 3 (see [1, Lemma 3, Azam et al.]). Let be a complex-valued metric space and let be a sequence in. Then is a Cauchy sequence if and only if as .

1.2. Quaternion Metric Space

Now, we recollect the basic definitions and concept on quaternion-valued metric spaces.

The skew field of quaternion denoted by means to write each element in the form ; , where , , , and are the basis elements of and . For these elements we have the multiplication rules , , , and . The conjugate element is given by .

The quaternion modulus has the form of . A quaternion q may be viewed as a four-dimensional vector .

Define a partial order on as follows.

if and only if , , , where , , and . It follows that if one of the following conditions holds:(I) ; where ; ,(II) ; where ; ,(III) ; where ; ,(IV) ; ; ,(V) ; ; ,(VI) ; ; ,(VII) ; ,(VIII) ; ,(IX) ; ; ,(X) ; ; ,(XI) ; ; ,(XII) ; ; ,(XIII) ; ; ,(XIV) ; ; ,(XV) ; ,(XVI) ; .

Remark 4. In particular, we write if and one from to is satisfied. Also, we will write if only is satisfied. It should be remarked that

Ahmed et al. [2] introduced the definition of the quaternion-valued metric space as follows.

Definition 5. Let be a nonempty set. A function is called a quaternion-valued metric on , if it satisfies the following conditions: : for all and , if and only if, , : , for all , : , for all .Then, is called a quaternion-valued metric space.

Let be a sequence in and . If for every , with , there is such that, for all , , then is said to be convergent, converges to , and is the limit point of . We denote this by , or , as . If for every with there is such that, for all , , then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete quaternion-valued metric space.

Lemma 6 (see [2, Lemma 2.1, Ahmed et al.]). Let be a quaternion-valued metric space and let be a sequence in. Then converges to if and only if as .

Lemma 7 (see [2, Lemma 2.2, Ahmed et al.]). Let be a quaternion-valued metric space and let be a sequence in. Then is a Cauchy sequence if and only if as .

1.3. Cone Metric Space

Let be a real Banach space. A subset of is called a cone, if and only if the following holds: : is closed, nonempty, and , : , , and imply that , : and imply that .Given a cone , we define a partial ordering with respect to by , if and only if . We write to indicate that but , while stands for , where denotes the interior of .

The cone is called normal, if there exist a number such that implies , for all . The least positive number satisfying this is called the normal constant [10]. It is proved that the normal constant can not be less then 1 (see [11]). For more details on cone metric space, we refer, for example, to [1016].

In this paper, denotes a real Banach space, denotes a cone in with , and denotes partial ordering with respect to .

Definition 8 (see [10]). Let be a nonempty set. A function is called a cone metric on , if it satisfies the following conditions: : for all and , if and only if, , : , for all , : , for all .Then, is called a cone metric space.

The following definitions and lemmas have been chosen from [10, 16].

Definition 9. Let be a cone metric space and let be a sequence in and . If, for all with , there is such that for all , , then is said to be convergent, converges to , and is the limit of .

Definition 10. Let be a cone metric space and let be a sequence in . If for all with , there is such that, for all , , then is called a Cauchy sequence in .

Definition 11. Let be a cone metric space. If every Cauchy sequence is convergent in , then is called a complete cone metric space.

Definition 12. Let be a cone metric space. A self-map on is said to be continuous, if implies for all sequence in .

Lemma 13. Let be a normal cone metric space and let be a normal cone. Let be a sequence in . Then, converges to , if and only if

Lemma 14. Let be a cone metric space and let be a sequence in . If is convergent, then it is a Cauchy sequence.

Lemma 15. Let be a cone metric space and let be a normal cone in . Let be a sequence in . Then, is a Cauchy sequence, if and only if .

2. Main Result

Let be a quaternion-valued metric space where is the skew field of quaternion number ; that is, Define It is apparent that . Assume is the zero of from now on. Note that is a real Banach space.

Lemma 16. is a normal cone in real Banach space .

Proof. Precisely, is nonempty, closed and . Also for all , and we have and . Notice that the normality of the cone follows from Remark 4.

Lemma 17. Any quaternion-valued metric space is a cone metric space.

Proof. For all define defines a partial ordered on and one can easily verify that is a cone metric space with respect to .

Lemma 18. The partial ordered defined in Lemma 17 is equivalent to .

Proof. Assume and . , if and only if , if and only if , , , and . In other words, , , where , and and , and , if and only if .

Lemma 19. A sequence in is convergent in the context of quaternion-valued metric space if and only if is convergent in the setting of of cone metric space.

Proof. Let be sequence in . converges to as the concept of quaternion-valued metric space if and only if as (see Lemma 6) if and only if converges to as the concept of cone metric space by considering as the Banach space endowed with the cone (see Lemma 13).

Let be a complex-valued metric space where is the skew field of complex number ; that is, Define It is apparent that . Assume is the zero of from now on. Note that is a real Banach space.

Lemma 20. is a normal cone in real Banach space .

Proof. Precisely, is nonempty, closed and . Also for all , and we have and .

Lemma 21. Any complex-valued metric space is a cone metric space.

Proof. For all define defines a partial ordered on and one can easily verify that is a cone metric space with respect to .

Lemma 22. The partial ordered defined in Lemma 21 is equivalent to .

We omitted the proof of Lemma 22 since it is the mimic of the proof Lemma 18.

Lemma 23. A sequence in is convergent as the concept of complex-valued metric space if and only if is convergent as the concept of cone metric space.

We omit the proof of Lemma 23 above due to Lemma 19.

Definition 24. Let be a complete cone metric space. For all . A cone metric space is said to be metrically convex if has the property that, for each with , there exists , such that

The following lemma finds immediate applications which is straightforward from [17].

Lemma 25. Let be a metrically convex quaternion-valued metric space, and let be a nonempty closed subset of . If and , then there exists a point (where stands for the boundary of ) such that

Definition 26. Let be a nonempty subset of a cone metric space and . The pair is said to be weakly commuting if, for each such that and , we have (see also [12, Hadžić and Gajić]).

Denote by the collection of all continuous and increasing mappings such that such that .

Lemma 27. Let be an increasing function. Then,

Proof. Suppose that and . Then there exists and such that , for all . Since is increasing, we have and this is a contradiction since .

Definition 28. Let be a nonempty subset of a cone metric space and let be two mappings. We say that is generalized -contractive if For all , with , , , and let .

Proposition 29. Let be a complete Banach-valued metric space, which is metrically onvex. Let be a nonempty closed subset of and , and let be such that is generalized -contractive. Suppose also we have(i) and ,(ii) ,(iii) and are weakly commuting,(iv) is continuous on .Then, there exists a unique common fixed point in such that .

Proof. We construct the sequences and in the following way.

Let . Then there exists a point such that as . From and the implication , we conclude that . Now, let be such that Let and assume that , and then which implies that there exists a point such that . Suppose , and then there exists a point (using Lemma 25), such that Since , there exists a point such that and so Let . Thus, repeating the forgoing arguments, we obtain two sequences and such that (i) ,(ii) , (iii) , and

Denote Obviously, the two consecutive terms of cannot lie in . Let us denote . We have the following three cases.

Case 1. If .

Case 2. If and .

Case 3. If    and so .

Proving the above cases are similar to [2, Theorem 3.1]. Also we see that for all we get Letting , we have . Since , we have . So that is a Cauchy sequence and hence it converges to a point . Now there exists a subsequence of which is contained in . Without loss of generality, we may denote . Since is continuous, converges to . We are going now to show that and have common fixed point . Using the weak commutativity of and , we obtain that and then This implies that On letting , we obtain It means that .

Now, consider Taking limit on both sides of (31) yields which is a contradiction, thus giving which implies , so that and hence .

To show that , consider Taking limit on both sides of (33) yields which is a contradiction, thereby giving which implies , so that and hence .

Thus, we have shown that , so is a common fixed point of and . To show that is unique, let be another fixed point of and , and then which is a contradiction, therefore giving which implies that , so that ; thus, .

Theorem 30 (see [2]). Let be a complete quaternion-valued metric space which is metrically convex and a nonempty closed subset of and . Let be such that is generalized -contractive satisfying the following conditions: (i) and ,(ii) ,(iii) and are weakly commuting,(iv) is continuous on , and then there exists a unique common fixed point in such that .

Theorem 31. Theorem 30 is a consequence of Proposition 29.

Proof. Let be a complete quaternion-valued metric space and a nonempty closed subset of. Then is a complete cone-valued metric space with cone . Further, we have that is generalized -contractive in cone metric space if and only if is generalized -contractive in quaternion metric space. The rest follows from Proposition 29.

3. Further Comment on Cone Metric Spaces

3.1. By Using a Scalarization Function

In 2010, Du [3] introduced the notion of TVS-valued metric space, also known as TVS-cone metric space (TVS-CMS), as a real generalization of Banach-valued metric space. Let be a locally convex Hausdorff t.v.s. with its zero vector , a proper, closed, convex pointed cone in with , , and a partial ordering with respect to .

Definition 32 (see [3]). Let be a nonempty set. Suppose that a vector-valued function satisfies the following: : for all and if and only if , : for all , : , forall .Then, the function is called TVS-cone metric on . Furthermore, the pair is called a TVS-cone metric space (in short, TVS-CMS).

On his paper, Du [3] concluded that, for a TVS-valued metric space , one can define a function by forming a metric, where , defined by is a nonlinear scalarization function (see e.g., [1820]). In this part, whenever we write , we mean .

Proposition 33 (see [3, 4]). Let be a TVS-CMS, , and a sequence in . Set . Then the following statements hold:(i) converges to in TVS-CMS if and only if as ,(ii) is a Cauchy sequence in TVS-CMS if and only if is a Cauchy sequence in ,(iii) is a complete TVS-CMS if and only if is a complete metric space.

From Proposition 33, the following result was derived easily.

Proposition 34 (see [3]). Let is complete TVS-CMS and satisfies the contractive condition: for all and . Then, has a unique fixed point in . Moreover, for each , the iterative sequence converges to fixed point.

Proposition 35 (see [3]). The Banach contraction principle and Proposition 34 are equivalent.

Definition 36. Let be a nonempty subset of a metric space and let be two mappings. We say that is generalized -contractive of type A if For all , with , , , and let .

Proposition 37. Let be a complete metric space, let be a nonempty closed subset of and , and let be such that is generalized -contractive of type A. If the following are satisfied:(i) and ,(ii) ,(iii) and are weakly commuting,(iv) is continuous on ,then there exists a unique common fixed point in such that .

We skip the proof of Proposition 37 since it can be derived by the mimic of Proposition 29. On the other hand, regarding Proposition 35, we can conclude that Proposition 37 implies Proposition 29.

3.2. By Using a Metric-Type Space

Definition 38 (see, e.g., [5]). Let be a set. Let be a function which satisfies(1) if and only if ,(2) , for any ,(3) for any ,for some constant . The pair is called a metric-type space.

Proposition 39. Let be a metric cone over the Banach space with the cone which is normal with the normal constant . The mapping defined by is a function which satisfies(1) if and only if ,(2) , for any ,(3) for any .

Remark 40. In Definition 38, (3), the term needs not to be distinct. Hence, metric type space turns into -metric space when we deal with cone metric space (see, e.g., [21, 22]).

Remark 41. Furthermore, by Lemma 20, is a normal cone. Hence, some resuls of [2] and Theorem 30, are equivalent to the corresponding results in the context of metric-type space (see also [5]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

This research was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.

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