#### Abstract

In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space , and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties of , and we discuss their extension to . We introduce the notion of -metric spaces and characterize cone metric space. Afterwards, we get generalizations of notions of convergence and Cauchy theory. In particular, we get a fixed point theorem of a contractive mapping in -metric spaces. Finally, by extending the notion of a contractive sequence in a real-valued metric space, we show that in -metric spaces, a contractive sequence is Cauchy.

#### 1. Introduction

Recall that the space of real numbers is a normed space which having the usual ordering ≤, such that it is translation invariant, that is, for all , and in implies that . Also, for any space , a metric (real-valued metric) defines a metric space .

Recently, the concept of a cone metric space has been studied in [115], and others. Indeed, they proved some fixed-point theorems of generalized contractive mappings. In particular, Huang and Zhang in [7] introduced the following.

Definition 1.1 (see [7]). Let be a subset of a normed space with . Then is called a cone if(1) is closed and .(2).(3) and .

A cone induces a partial ordering ≤ with respect to by if and only if . The notion means that and , while stands for . A cone is called normal if there exists a number such that for all , we have The least positive number , satisfying (1.1), is called the normal constant of . Vandergraft in [16] (or see Example 2.1 of [10]) presented an example of a nonnormal cone, that is, a cone of a normed space which does not satisfy (1.1).

Definition 1.2 (See [7]). Let be a nonempty set, and let be a normed space having a normal cone with . Suppose, the mapping satisfies the following:(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.

Remark 1.3. In [7] and in order to define a cone metric space, the authors started by a Banach space (see [7, page 1469]). But in fact, and up to the results of the current paper, they just need to be a normed space only.

More generally, Karapınar in [17] and Shatanawi in [18, 19] studied the couple fixed-point theorems in cone metric spaces. Also, Shatanawi in [20] studied common coincidence point in cone metric spaces.

In this paper, we start with different approach, and without starting by cones, we introduce the notion of normed ordered spaces in general (see Definition 2.1) which generalizes most of the properties of (also the notion of ordered field is already defined, see [21, page 7]). We show that many results can be extended to a normed ordered space ; however, some properties cannot, as depending on crucial properties of . In the second part, and considering a nonempty set , we replace the real-valued metric with an -valued metric (denoted by ), and is called -metric space, we discuss some examples. We generalize results of real-valued metric spaces (see, e.g., [22]) concerning sequences to the case of -metric spaces. Afterwards, we introduce the notions of convergence and Cauchy sequences in -metric space. Considering to be a normed ordered space and that has an -valued matric, we give a characterization of cone metric spaces in the sense of Huang and Zhang in [7]. Then we get results concerning convergence and Cauchy theory in the case of -metric space. In particular, we get a fixed-point theorem of a contractive mapping. Finally, and in a similar way of a real-valued metric space, we introduce the notion of a contractive sequence in an -metric space, and we prove that every contractive sequence is Cauchy.

Notice that ordered spaces need not be totally ordered. In fact, many properties of are deduced as being totally ordered. We say that has the completeness property (CP) if every subset of an upper bound has a supremum in and every subset of a lower bound has an infemum in . If is an ordered space, then all types of intervals can be defined. Indeed the unbounded intervals (left and right rays) are

#### 2. Normed Ordered Spaces

In this section, we define normed ordered spaces, and we give different examples. Then we discuss generalizations of main results in the real space to normed ordered spaces. In particular, the Bolzano-Weierstrass theorem and intermediate value theorem are not true in general normed ordered spaces. The properties (CP) and totally ordered of , are in fact seems to be crucial.

Definition 2.1. An ordered space is a vector space over the real numbers, with a partial order relation ≤ such that(O1)for all , and in , implies (translation invariant),(O2)for all and with , .Moreover, if is equipped with a norm such that(O3)there exists a real number and for all , implies , then is called a normed ordered space.

By the translation invariant (O1), means . The strict inequality stands for and .

Proposition 2.2. Let be an ordered space. Then for all and in , we have(i) implies that ,(ii)If and , then ,(iii)If and , then ,(iv)for all and , ,(v) and implies that ,(vi) and implies that .

Proof. (i) As , by translation invariant, we have , hence . (ii) Assuming that implies by (i) that so we get , and as the relation is an antisymmetric, we have . (iii) As , we get , hence the result holds by transitivity. (iv) As , we have and by (O2) this implies that . Using (i) , hence . (v) As , then , and by (O2) we get , hence . (vi) is similar.

Now let us introduce the following examples of normed ordered spaces.

Example 2.3. (a) The set of real numbers with usual ordering and the absolute value.(b)The set with the ordering defined by This ordering is called the simplicial ordering of . Then together with simplicial ordering and the Euclidean norm is a normed ordered space.(c)The set of rational numbers as a vector space over itself together with absolute value and usual ordering.(d)The complex space together with the modulus, and the order defined by is a normed ordered space.(e)The space of all continuous real-valued functions on together with the supremum norm, and the pointwise ordering, is a normed ordered space.(f)Let be any measure space. Then for all , the space with the norm, and the pointwise ordering is a normed order space.

Definition 2.4. Let be an ordered space. Then an element is called positive if , and it is called strict positive if . The set of all positive elements in is denoted by .

Remark 2.5. The following result is valid for any ordered space having a norm, and not necessarily satisfying the property (O3). So the space needs not to be a normed ordered space. Then according to Definition 2.1 with the characterization in Theorem 3.8, this is equivalent to say that a cone is nonnormal (indeed this exists, see Example 2.1 of [10]).

Proposition 2.6. Let be an ordered space having a norm. Then is a closed set.

Proof. We will prove that is an open subset of . Given that . Define that , then is an open ball containing . Now we claim that : if not, then there exists , which means that , and , , hence we get a contradiction. Then the claim is proved.

The interior of is denoted by . In the following example, we show that consists of functions that never touch the -axis.

Example 2.7. If , then .

Proof. Suppose that and , for all . Choose such that . Now given any in the open ball , and assume that for some . Then Therefore, is an open subset, so we get that . Conversely, if , then there exists such that . If for some , , then define that . Therefore, and as which is a contradiction, so , for all .

The following example shows that can be empty.

Example 2.8. In , consider the subspace , with the induced norm and simplicial ordering as in Example 2.3(b). Then , hence .

For all and in , by we mean , so means . Then we have the following properties.

Proposition 2.9. Let be an ordered space having a norm. Then(i)The zero element is not in ,(ii)If , then ,(iii)If and , then ,(iv)If and , then .

Proof. (i) Assume that , then there is an -neighborhood of 0 such that so this open neighborhood contains a nonzero (if not then ; hence, is an open subset which is a contradiction, as metric spaces are Hausdorff). Also, we have , hence by Proposition 2.2(ii), we get a contradiction. If , then ; hence, this proves (ii). To prove (iii) assume that and let . There exists an -neighborhood of such that Consider that ball of , we claim that it is contained in . If , then which means that and therefore so by (O2), we get , which proves the claim. (iv) follows by (ii) and (iii), hence the proposition has been checked.

Proposition 2.10. Let be an ordered space having a norm with . If , then there exists , such that .

Proof. Given that and let , by Proposition 2.9(i), we have and by Proposition 2.9(iii), we have . Choose a real number and put . Then and , hence the lemma is checked.

Now let us consider a sequence in a normed ordered space . We recall some properties that hold for and try to extend it to . As is totally ordered, we know that have a monotonic subsequence. This is not true in general even though is assumed to be bounded sequence, the following example explains this.

Example 2.11. Consider that , with the simplicial ordering. Let be a bijection from onto and define the sequence by . Then is a bounded sequence in ; for example, is an upper bound, with no monotonic subsequence, as every two terms are not comparable. Also, it is a divergent sequence.

Remark 2.12. Recall that by the completeness property (CP), we know that has the bounded monotone convergence theorem. Also, the spaces and (Example 2.3(b), (e), resp.) have the bounded monotone convergence theorem.

The following example shows that the bounded monotone convergence theorem is not true in a normed ordered space , in general. The (CP) does not hold in the normed ordered space . Moreover, the Bolzano-Weierstrass theorem (B-W) is not true in general.

Example 2.13. Consider the normed ordered space in Example 2.3(c) and the sequence . It is bounded and increasing sequence but has no limit in . As every subsequence of is convergent to , so no subsequence is convergent in , that is, (B-W) fails.

Now, let us consider a continuous function defined on a normed ordered space into a normed ordered space . The following examples show that the maximum-minimum theorem (Max-Min) and the intermediate value theorem (IVT) in the case of real-valued functions do not hold for .

Example 2.14. Consider that the continuous function , which is defined by , and consider that . Then is a compact subset of and is the line segment from to . Indeed, and and both do not belong to .

More precisely, using known theorems about normed spaces and compactness (see, e.g., [22]), one may easily deduce the following.

Proposition 2.15. Let be a normed ordered space having (CP), let be a metric space and be any continuous function. If is any compact subset of , then attains its sup and inf in .

Example 2.16 (IVT is invalid). Consider the function , which is defined by Let be the rotation mapping by the angle , that is, for any . Set , so is a continuous function defined on real numbers with values in . Now but no with .

#### 3. -Metric Spaces

Recall that in order to define a metric on a set , it is necessary to have an ordered space only. In this section, we define -metric spaces, and we give main examples. We characterize cone metric spaces as in [7], by using the notion of -metric space of a normed ordered space . Then we generalize main theorems in real-valued metric spaces such as Cauchy, convergence theories, and contractive sequences (see, e.g., [21, 22]) to the case of -metric spaces. Moreover, we get a fixed point theorem of a contractive mapping.

Definition 3.1. Let be any nonempty set and let be any ordered space, over the real scalars. An ordered -metric on is an -valued function such that for all , and in , we have(i) and if and only if ,(ii), (iii).Then the pair is called -metric space.

Now, consider an ordered space with a norm and consider an -metric space , let be a point in and . Then the open ball in centered at of radius is

Example 3.2. (a) Fixing , with the usual distance, is reduced to the usual metric space of real numbers, having the open intervals as open balls.
(b) Consider that and with the simplicial ordering. Define the function from into by Then is a metric, and hence is an -metric space. Moreover, the open balls are realized as the open rectangles in space.
(c) Recall Example 2.3(e), let , and consider the normed space . Define the function by . Then is an -metric space. Moreover, fix and let be a function with all its values are strictly positive; that is, never touch the -axis, so by Example 2.7 we have . Then the open ball is a fiber that consists of all functions with values

Definition 3.3. Let be a normed ordered space, and let be an -metric space. Then a sequence in is called convergent to a point , if for all , there exists a positive integer such that , for all .

If a sequence converges to a point , then we write

Definition 3.4. Let be a normed ordered space, and let be an -metric space. Then a sequence in is called Cauchy, if for all , there exists a positive integer such that , for all .

Proposition 3.5. Let be a sequence in a -metric space , where is a normed ordered space. If is convergent, then it is a Cauchy sequence.

Proof. Assume that for some , . Let . Then there exists a positive integer such that , for all . Therefore for all , .

Proposition 3.6. Let be a normed ordered space with and let be an -metric space. Then any convergent sequence in has a unique limit.

Proof. Assume that and and let . Then there exist positive integers and such that Choose that , then we get the following: As is a normed ordered space, there exists such that . As an arbitrary, then , which gives , then , hence .

Definition 3.7. Let be a normed ordered space, and let be an -metric space. Then is called complete -metric space if every Cauchy sequence in is convergent.

Now, using the concepts of -metric spaces, let us have the following characterization of cone metric spaces (Definition 1.2) in the sense of Huang and Zhang as in [7].

Theorem 3.8. Let be any nonempty set and be a space over the real scalars. Then the following are equivalent.(a)The pair is an -metric space, where is a normed ordered space, with ,(b)The pair is a cone metric space.

Proof. First (a) implies (b). Consider that , then by Proposition 2.6, is closed, and as , we have . If real numbers and , then, by (O2) of Definition 2.1, we get , then, by Proposition 2.2(iii), we have . Now if and , then by Proposition 2.2(ii), we get , so is a cone in the sense of Definition 1.1. As is a normed space, then by definition, is a normal cone. Then considering that , we get that is a cone metric space.

Conversely, assume that is a cone metric space. Then we have a cone in , and the order with respect to () defines a partial ordering on . To check the translation invariant (O1): given any and , with . Then by the definition of the order, , therefore, , hence . To prove (O2): by (ii) of Definition 1.1, we know that , and taking , we get , which means . (O3) is direct, hence is a normed ordered space. By the definition of cone metric space, . Finally, the cone metric on is chosen to be the ordered -metric as in Definition 3.1; hence, the theorem has been checked.

Now, using the above characterization, we have the following results, including a fixed-point theorem of a contractive mapping. Indeed, Proposition 2.10 is used in the proof of Lemmas 1 and of Haung and Zhang in [7].

Corollary 3.9. Let be a normed ordered space with , and let be an -metric space. Then a sequence in converges to if and only if in .

Proof. Direct by using Theorem 3.8 together with Lemma  1 in [7].

Corollary 3.10. Let be a normed ordered space with , and let be an -metric space. Then a sequence is Cauchy if and only if in as and .

Proof. The proof is directed by using Theorem 3.8 together with Lemma  4 in [7].

Finally, we have the following fixed-point theorem in normed ordered spaces.

Theorem 3.11. Let be a normed ordered space with , and let be a complete -metric space. If a function satisfies the following contractive condition: then has a unique fixed point in .

Proof. It follows directly by using Theorem 3.8 and Theorem  1 in [7].

Finally, and extending the notion in real-valued metric spaces, let us introduce the notion of a contractive sequence in -metric space, then proving that it is indeed a sufficient to be a Cauchy sequence.

Definition 3.12. Let be an ordered space, and let be an -metric space. Then a sequence in is called contractive if there exists a real number , such that .

Theorem 3.13. Let be a normed ordered space with , and let be an -metric space. Then every contractive sequence in is Cauchy.

Proof. Suppose that is a contractive sequence in . Then for some real number , we have Therefore assuming that , we get the following: Therefore, for some , which implies that and then in , hence by Corollary 3.10, is a Cauchy sequence.

#### Acknowledgment

The authors would like to thank the editor and the referee for their valuable comments and suggestions.