#### Abstract

We introduce a -metric on the cone -metric space and then prove some equivalences between them. As applications, we show that fixed point theorems on cone -metric spaces can be obtained from fixed point theorems on -metric spaces.

#### 1. Introduction and Preliminaries

The fixed point theory in -metric spaces was investigated by Bakhtin , Czerwik , Akkouchi , Olatinwo and Imoru , and Pǎcurar . A -metric space was also called a metric-type space in . The fixed point theory in metric-type spaces was investigated in [6, 7]. Recently, Hussain and Shah introduced the notion of a cone -metric as a generalization of a -metric in . Some fixed point theorems on cone -metric spaces were stated in .

Note that the relation between a cone -metric and a -metric is likely the relation between a cone metric  and a metric. Some authors have proved that fixed point theorems on cone metric spaces are, essentially, fixed point theorems on metric space; see  for example. Very recently, Du used the method in  to introduce a -metric on a cone -metric space and stated some relations between fixed point theorems on cone -metric spaces and on -metric spaces .

In this paper, we use the method in  to introduce another -metric on the cone -metric space and then prove some equivalences between them. As applications, we show that fixed point theorems on cone -metric spaces can be obtained from fixed point theorems on -metric spaces.

Now, we recall some definitions and lemmas.

Definition 1 (see ). Let be a nonempty set and . Then, is called a -metric on if (1) if and only if ;(2) for all ;(3)there exists such that for all .The pair is called a -metric space. A sequence is called convergent to in , written , if . A sequence is called a Cauchy sequence if . The -metric space is called complete if every Cauchy sequence in is a convergent sequence.

Remark 2. On a -metric space , we consider a topology induced by its convergence. For results concerning -metric spaces, readers are invited to consult papers [1, 2].

Remark 3. Let be a -metric space. For each and , we set In , Akkouchi claimed that the topology on associated with is given by setting if and only if, for each , there exists some such that and the convergence of in the -metric space and that in the topological space are equivalent. Unfortunately, this claim is not true in general; see Example 13. Note that; on a -metric space, we always consider the topology induced by its convergence. Most of concepts and results obtained for metric spaces can be extended to the case of -metric spaces. For results concerning -metric spaces, readers are invited to consult papers [1, 2].

In what follows, let be a real Banach space, a subset of , the zero element of , and the interior of . We define a partially ordering with respect to by if and only if . We also write to indicate that and and write to indicate that . Let denote the norm on .

Definition 4 (see ). is called a cone if and only if (1) is closed and nonempty and ;(2) imply that ;(3).
The cone is called normal if there exists such that, for all , we have implies . The least positive number satisfying the above is called the normal constant of .

Definition 5 (see [11, Definition 1]). Let be a nonempty set and satisfy (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 6 (see [8, Definition 2.1]). Let be a nonempty set and satisfy (1) for all and if and only if ;(2) for all ;(3) for some and all . Then is called a cone -metric with coefficient on and is called a cone -metric space with coefficient .

Definition 7 (see [8, Definition 2.4]). Let be a cone -metric space and a sequence in . (1) is called convergent to , written , if for each with , there exists such that for all .(2) is called a Cauchy sequence if for each with there exists such that for all .(3) is called complete if every Cauchy sequence in is a convergent sequence.

Lemma 8 (see [8, Proposition 2.5]). Let be a cone -metric space, a normal cone with normal constant , , and a sequence in . Then one has the following. (1) if and only if  .(2)The limit point of a convergent sequence is unique.(3)Every convergent sequence is a Cauchy sequence.(4) is a Cauchy sequence if  .

Lemma 9 (see [8, Remark 2.6]). Let be a cone -metric space over an ordered real Banach space with a cone . Then one has the following.(1)If and , then .(2)If and , then .(3)If for all , then .(4)If , for all and , then there exists such that for all .(5)If , for all and , then eventually.(6)If for all and , , then .(7)If , , and , then .(8)For each , one has .(9)For each and , there exists such that .(10)For each and , there exists such that and .(11)For each and , there exists such that and .

Remark 10 (see [10, Remark 1.3]). Every cone metric space is a cone -metric space. Moreover, cone -metric spaces generalize cone metric spaces, -metric spaces, and metric spaces.

Example 11 (see [10, Example 2.2]). Let and for all and . Then is a cone -metric space with coefficient , but it is not a cone metric space.

Example 12 (see [10, Example 2.3]). Let be the set of Lebesgue measurable functions on such that , , . Define as for all and . Then is a cone -metric space with coefficient , but it is not a cone metric space.

#### 2. Main Results

The following example shows that the family of all balls does not form a base for any topology on a -metric space .

Example 13. Let and Then we have the following.(1) is a -metric on with coefficient .(2) but for all .

Proof. (1) For all , we have , if and only if and .
If , then If , then If , then If with , then If with and , then If , then If , then If , then
By the previous calculations, we get for all . This proves that is a -metric on with .
(2) We have . Then .
For each , since , we have for being large enough. Note that , so for all . This proves that .

We introduce a -metric on the cone -metric space and then prove some equivalences between them as follows.

Theorem 14. Let be a cone -metric space with coefficient and for all . Then one has the following.(1) is a -metric on .(2) in the cone -metric space if and only if in the -metric space .(3) is a Cauchy sequence in the cone -metric space if and only if is a Cauchy sequence in the -metric space .(4)The cone -metric space is complete if and only if the -metric space is complete.

Proof. (1) For all , it is obvious that and .
If , then .
If , then, for each , there exists such that and . Then , and by Lemma 9(6), we have . It implies that . Therefore, ; that is, .
For each , we have Since and ,, we have Then we have It implies that Now, we have By the previously metioned, is a -metric on .
(2) Necessity. Let in the cone -metric space . For each , by Lemma 9(8), if , then . Then, for each with , there exists such that for all . Using Lemma 9(8) again, we get . It implies that for all . This proves that ; that is, in the -metric space .
Sufficiency. Let in the -metric space . For each , there exists such that . For this , there exists such that Then, there exist and such that . So , and we have . Therefore, for all . By Lemma 9(1), we get for all . This proves that in the cone -metric space .
(3) Necessity. Let be a Cauchy sequence in the cone -metric space . For each , by Lemma 9(6), if , then . Then for each with , there exists such that for all . Using Lemma 9(6) again, we get . It implies that for all . This proves that is a Cauchy sequence in the -metric space .
Sufficiency. Let be a Cauchy sequence in the -metric space . Then . For each , there exists such that . For this , there exists such that for all . Then, there exists , such that . So , and we have . Therefore, for all . By Lemma 9(1), we get for all . This proves that is a Cauchy sequence in the cone -metric space .
(4) It is a direct consequence of (2) and (3).

By choosing in Theorem 14, we get the following results.

Corollary 15 (see [13, Lemma 2.1]). Let be a cone metric space. Then for all is a metric on .

Corollary 16 (see [10, Theorem 2.2]). Let be a cone metric space and for all . Then the metric space is complete if and only if the cone metric space is complete.

The following examples show that Corollaries 15 and 16 are not applicable to cone -metric spaces in general.

Example 17. Let be a cone -metric space as in Example 11. We have It implies that Then is not a metric on . This proves that Corollaries 15 and 16 are not applicable to given cone -metric space .

Example 18. Let be a cone -metric space as in Example 12. We have For , , and for all , we have Then is not a metric on . This proves that Corollaries 15 and 16 are not applicable to given cone -metric space .

Next, by using Theorem 14, we show that some contraction conditions on cone -metric spaces can be obtained from certain contraction conditions on -metric spaces.

Corollary 19. Let be a cone -metric space with coefficient , let be a map, and let be defined as in Theorem 14. Then the following statements hold. (1)If for some and all , then for all .(2)If for some with and all , then for all .

Proof. (1) For each and with , it follows from Lemma 9(8) that Thus, . Then we have It implies that .
(2) Let and satisfy
From Lemma 9(8), we have It implies that Then we have This proves that .

Now, we show that main results in  are consequences of preceding results on -metric spaces.

Corollary 20. Let be a complete cone -metric space with coefficient , and let be a map. Then the following statements hold. (1)(see [9, Theorem 2.1]) If for all , then has a unique fixed point.(2)(see [9, Theorem 2.3]) If for some with and all , then has a unique fixed point.

Proof. Let be defined as in Theorem 14. It follows from Theorem 14(4) that is a complete -metric space.(1)By Corollary 19 (1), we see that satisfies all assumptions of [5, Theorem 2]. Then has a unique fixed point.(2)By Corollary 19 (2), we see that satisfies all assumptions in [6, Theorem 3.7], where , , is the identity, and , and . Note that condition (3.10) in [6, Theorem 3.7] was used to prove (3.16) and at line 3, page 7 in the proof of [6, Theorem 3.7]. These claims also hold if and . Then has a unique fixed point.

Remark 21. By similar arguments as in Corollaries 19 and 20, we may get fixed point theorems on cone -metric spaces in [8, 10] from preceding ones on -metric spaces in [3, 5].

#### Acknowledgments

The authors are thankful for an anonymous referee for his useful comments on this paper. This research was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC Grant no. NRU56000508).