#### Abstract

We point out some equivalence between the results in (Sedghi et al., 2012) and (Khamsi, 2010). Then, we introduce the notion of a general distance between three arbitrary points and study some of its properties. In the final section, some fixed point results are proposed.

#### 1. Introduction

The literature of a distance for any triple of points in a space was first considered during the sixties by Gähler [1, 2]. It is known as a 2-metric, the concept of which was later extended by Dhage [3] into a -metric. Both notions are in no easy ways related to the classical concept of a metric. This led to the -metric due to Mustafa and Sims [4].

On the other hands, Sedghi et al. [5] had introduced the notion of a -metric, which has later been generalized by Sedghi et al*.* [6] into an -metric.

These developments confirm that this kind of measurement is recently of many mathematicians’ interests. One of the area that exploited these establishments largely is the fixed point theory, especially the ones involving some generalized contractions; see, for example, Sedghi et al. [5], Sedghi et al. [6], Mustafa et al. [7], Mustafa and Sims [8], Aydi et al. [9], Alghamdi and Karapinar [10], Chandok et al. [11], and Abbas et al. [12].

In this present paper, we divide our interests into three parts. Firstly, we give a remark on the existing fixed point result endowed in a -metric space. Secondly, we propose and study a very general principle in measuring the distance between three arbitrary points called a . Thirdly, we construct some fixed point theorems by utilizing the and its properties.

#### 2. Preliminaries

This section is devoted to the recollection of important definitions and lemmas. We start with a sequence of definitions of -, -, and -metric spaces.

*Definition 1 (see [4]). *Let be a nonempty set. A function is said to be a -metric if the following conditions are satisfied:(1)for , if ; (2)for with , ; (3)for with , ; (4)for , , where is any permutation of ; (5)for , .

The pair is called a -metric space. Moreover, if for all , then is said to be *symmetric*.

*Definition 2 (see [5]). *Let be a nonempty set. A function is said to be a -metric if the following conditions are satisfied:(1)for , if and only if ;(2)for , , where is any permutation of ;(3)for , .

The pair is called a -metric space.

*Definition 3 (see [6]). *Let be a nonempty set. A function is said to be a -metric if the following conditions are satisfied:(1)for , if and only if ;(2)for , .

The pair is called a -metric space.

Lemma 4 (see [6]). *Let be a -metric space, then for all .*

It can be seen that each symmetric -metric is a -metric and that each -metric is a -metric. In case of nonsymmetric -metric, the concepts of -metric and -metric are independent.

*Definition 5 (see [6]). *Let be a -metric space. For and we define the open ball as follows:

As in [6], one may consider the topology for which is generated from the base containing all open balls in . Some concepts are also introduced.

*Definition 6 (see [6]). *Let be a -metric space. A sequence in is called(i)*Cauchy* if for any , we may find such that
(ii)*convergent* if there is a point in which

Moreover, if every Cauchy sequence in is also convergent, is said to be *complete*.

#### 3. A -Metric Space as a Metric Type Space

In this section, we shall be giving a small remark on a fixed point theorem due to [6]. According to [6], a self-operator on a -metric space is called a contraction if it satisfies the following inequality: for all , where . The following result was introduced subsequently.

Theorem 7 (see [6]). *Let be a complete -metric space and let be a contraction on . Then, has a unique fixed point. *

To expedite our remark, we shall first recall the notion of a metric type space, which was introduced by Khamsi in [13].

*Definition 8 (see [13]). *Let be a nonempty set and let be a function satisfying the following conditions:(1)for , if and only if ;(2)for , ;(3)there exists a constant such that for ,
The triple is called a metric type space.

In particular, if , then is a metric space. A self-mapping operator on a metric type space is called *Lipschizian* if there exists such that
for all . The smallest satisfying such condition is denoted by Lip. Moreover, the following fixed point theorem was proposed.

Theorem 9 (see [13]). *Let be a complete metric type space and let be an operator such that the composition is Lipschizian for each and Lip. Then, has a unique fixed point.*

It follows that if is a Lipschizian operator with Lip, then has a unique fixed point.

Now, let be a -metric space. Suppose that a function is given by for , it is obvious that if and only if and for all . Now, observe for each that

Thus, is a metric type space. Moreover, the balls and coincide.

Notice that we may rewrite the inequality (4) as follows: for all with the same . Also notice that the definition of Cauchyness, convergence and completeness in a -metric space may be rewriten in terms of metric type spaces. So, these notions are transferred to the corresponding metric type space .

Now, if is an operator satisfying (9), then each , where , is Lipschizian with Lip. Therefore, Theorem 7 is obtained via Theorem 9.

Even though we set a new condition for an operator , where , to be for each , the unique fixed point can still be obtained via Theorem 9, anyway. Note that not only the mentioned theorem, but also many theorems in the literature may be proved via this concept in metric type spaces. We shall give some results which seem more general than the setting of Theorem 7 but however equivalent.

Beforehand, we give the following useful lemma without proof since it is straight forward.

Lemma 10. *Let be a -metric space and let . Then, the following inequalities hold:*(i)*;*(ii)*.*

Theorem 11. *Let be a complete -metric space and let be an operator such that there exists a sequence of nonnegative reals satisfying the condition:
**
for all , where and for each , is a fixed permutation in . Then, has a unique fixed point.*

*Proof. *We shall show that (11) implies that is Lipschizian with Lip in metric type space . For each , it is easy to verify that no matter which permutations are defined. Thus, we obtain that

Apply Theorem 9 (or Theorem 7) to obtain the desired result.

#### 4. A General Distance between Three Arbitrary Points

In this section, we shall be dealing with a new concept of a general distance between three arbitrary points (or ).

To be able to define the , we first consider a nonempty set together with a function for which if and only if . Given and , we define an open ball in the usual sense:
To be natural, we say that a subset is *bounded* if . Certainly, the assertion “* if and only if *” is not enough to guarantee that open balls in are bounded. We shall illustrate in the following.

*Example 12. *Let and let be a function given by
It is clear that if and only if .

Let and let . Note that if , then
Let be a sequence in such that for each . Since , we have for all . The same conclusion can be deduced also when . Therefore, is not bounded at each and .

This is not quite natural and does not meet the requirements we would like to have. So, we may add one more assumption at this stage and define the space as follows.

*Definition 13. *Let be a nonempty set. A function is said to be a if the following conditions are satisfied:(g1)for , if and only if ;(g2)there exists some such that the balls are bounded for all .The pair is called a * space*.

Next, we shall give a characterization of a space.

Lemma 14. *Let be a nonempty set and let be a function satisfying (g1). Then, the following are equivalent:*(i)* satisfies (g2);*(ii)*there exist some constants such that for any , one has
*

*Proof. *[(i) (ii)] Assume that (i) holds. Set and let arbitrarily. If , then . Thus, setting and it follows that .

[(ii) (i)] Assume that (ii) holds. Let and suppose that . Therefore, we have
From (ii), we obtain that . Thus, . Since is arbitrary, the balls are bounded for every .

*Remark 15. *Suppose that is a -metric space and . Then, we have
Thus, we can choose any and let to fulfill the assumptions in Lemma 14. Hence, every -metric space is in turn a space.

*Remark 16. *Suppose that is a -metric space and . Then, we have
The same argument is to be considered as in the previous remark. So, a -metric space is a space. This immediately implies that a -metric space is also a space.

Denoted by the family of all open balls in . Throughout this paper, we shall assume that represents the topology having as its subbase. Also, we write to denote the base generated by .

*Remark 17. *The topology here is defined using different idea from those given in symmetric spaces or in semimetric spaces (see e.g., [14–17]).

Proposition 18. *The topology for a space is -separable. *

*Proof. *Let with . So, we have and , for some . Observe that and . The desired result is then followed.

We shall now explicate an example of a space. In particular, this next example will even show that a space is no need to be -separable.

*Example 19. *Let and define a function in the following:
It is clear that any subset in this space is always bounded. Hence, is a space. Observe that for and , we have
Therefore, any two balls intersect one another, implying that is not -separable.

We next introduce a new concept of convergence and compare it with the classical topological ones.

*Definition 20. *Let be a space. A sequence in is said to be(1)*Cauchy* if for any , there exists such that
(2)*-convergent* if we can find a point in which for any , there exists satisfying

In this case, we say that -converges to and write .

*Remark 21. *Given a space , a sequence in -converges to if and only if for any set with , there exists such that

Lemma 22. *Let be a space and let be a sequence in . Then, the following are equivalent:*(i)* converges to in the topology ;*(ii)*for any neighborhood of , on can find such that
*(iii)*for any set containing , we can find such that
*

*Proof. *(i) (ii) is by definition. So, we only need to show that (ii) (iii).

[(ii) (iii)] Since , we again apply (ii) to obtain our desired result.

[(iii) (ii)] By the definition of , for every , we can find in which . Suppose that is a neighborhood of , then we can find some such that . Applying (ii), we obtain that for every for some fixed .

We conclude the following lemma immediately from Remark 21 and Lemma 22.

Lemma 23. *Let be a space and let be a sequence in . If converges to some point in the topology , then it also -converges to . *

This lemma shows that the concept of -convergence is weaker than convergence in topology. However, in case when is a -metric space or when is a -metric space, the two concepts coincide. Along this paper, we shall deal with this new kind of convergence, rather than those topological ones.

*Definition 24. *A space is called(i)*-Hausdorff* if every -convergent sequence -converges to at most one point; (ii)*-complete* if every sequence satisfying -converges; (iii)*Cauchy-complete* if every Cauchy sequence -converges.

*Remark 25. *A space is -Hausdorff if and only if for any two distinct points , there exist two disjoint sets such that and .

#### 5. Some Fixed Point Theorems

Under this section, we propose some fixed point theorems in the framework of a space.

We first introduce a lemma which will be used in our main theorems.

Lemma 26. *Let be a space. Suppose that be an operator such that
**
for all , where and is a fixed permutation on . Then, the following hold for every :*(i)*;*(ii)*.*

*Proof. *Let be arbitrary. We shall consider the permutation case-by-case.

(i) *Case I*: or .

Observe that

(ii) *Case II*: .

Observe that
where .

(iii) *Case III*: .

Observe that
where .

(iv) *Case IV*: or .

Observe that

In each case, we may conclude that

Similarly, we may prove that
where .

Now, we consider our fixed point results, which exploited the above lemma.

Theorem 27. *Let be a -Hausdorff -complete space. Suppose that is a -sequentially continuous operator (i.e., , for every sequence in ) satisfying
**
for all , where and is a fixed permutation on . Then, has exactly one fixed point. *

*Proof. *Let be arbitrary. By (i) in Lemma 26, we have
Since is -complete, -converges to some point . Now, since is -sequentially continuous, . Since is -Hausdorff, we have .

Assume that is also a fixed point of . Observe that
The only possibility of the value of allows us to conclude that . So, the theorem is proved.

Theorem 28. *Let be a -Hausdorff Cauchy-complete space. Suppose that is a -sequentially continuous operator satisfying
**
for all , where and are given as in Lemma 14. Then, has exactly one fixed point.*

*Proof. *Let . From Lemma 26, we also have and . So, we may choose such that
Consequently, we obtain that
We will show that for all via the mathematical induction. Now, we assume that
for some . Then, observe that
Hence, from Lemma 14, we have . Therefore, we have for all .

Let be given and let with . Thus, we may write
for some . Note that . It follows that
Since , if is chosen large enough so that , then we ended up with . Therefore, is Cauchy. By mean of the Cauchy-completeness of , it converges to some point . Since is -sequentially continuous, we have . Moreover, since is -Hausdorff, .

Assume that is also a fixed point of . Observe that
This implies that . Therefore, the uniqueness is proved.

*Example 29. *Let and define a function by
It is easy to verify that is a space which is -Hausdorff and -complete. Note also that is neither a -metric nor a -metric.

Now, let us consider the map given by
Obviously, is -sequentially continuous.

Our results (Theorems 27 and 28) then guarantee the existence and uniqueness of the fixed point .

#### 6. Conclusions

In this work, we pointed out that the results in [6] are obtainable through a metric type space. In addition, a -metric generalizes a -metric only in the case when is symmetric. We then fill this gap by introducing a new space, namely, a space, which covers a -metric space and also a -metric space in which the symmetric is absent. We also study the underlying topology for this new space in the new direction, totally different from those studied in symmetric and semimetric spaces. We lastly give the sufficient conditions for a fixed point to exist and to be unique.

#### Acknowledgments

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC no. 56000508).