#### Abstract

We introduce the concept of a quasi-pseudometric type space and prove some fixed point theorems. Moreover, we connect this concept to the existing notion of quasi-cone metric space.

#### 1. Introduction

Cone metric spaces were introduced in [1] and many fixed point results concerning mappings in such spaces have been established. In [2], Khamsi connected this concept with a generalised form of metric that he named* metric type*. Recently in [3], Shadda and Md Noorani discussed the newly introduced notion of quasi-cone metric spaces and proved some fixed point results of mappings on such spaces. Basically, cone metric spaces are defined by substituting, in the definition of a metric, the real line by a real Banach space that we endowed with a partial order. The fact that the introduced order is not linear does not allow us to always compare any two elements and then gives rise to a kind of duality in the definition of the induced topology, hence the convergence in such space. We introduce a quasi-pseudometric type structure and show that some proofs follow closely the classical proofs in the quasi-pseudometric case but generalize them.

#### 2. Preliminaries

In this section, we recall some elementary definitions from the asymmetric topology which are necessary for a good understanding of the work below.

*Definition 1. *Let be a nonempty set. A function is called a* quasi-pseudometric* on if(i),(ii).Moreover, if , then is said to be a *-quasi-pseudometric.* The latter condition is referred to as the -condition.

*Remark 2. *(i) Let be a quasi-pseudometric on ; then the map defined by whenever is also a quasi-pseudometric on , called the* conjugate* of . In the literature, is also denoted by or .

(ii) It is easy to verify that the function defined by , that is, , defines a* metric* on whenever is a -quasi-pseudometric.

Let be a quasi-pseudometric space. Then for each and , the set denotes the open -ball at with respect to . It should be noted that the collection yields a base for the topology induced by on . In a similar manner, for each and , we define known as the closed -ball at with respect to .

Also the collection yields a base for the topology induced by on . The set is -closed but not -closed in general.

The balls with respect to are often called* forward balls* and the topology is called* forward topology*, while the balls with respect to are often called* backward balls* and the topology is called* backward topology*.

*Definition 3. *Let be a quasi-pseudometric space. The convergence of a sequence to with respect to , called *-convergence* or* left-convergence* and denoted by , is defined in the following way:

Similarly, the convergence of a sequence to with respect to , called *-convergence* or* right-convergence* and denoted by , is defined in the following way:

Finally, in a quasi-pseudometric space , we will say that a sequence *-converges* to if it is both left and right convergent to , and we denote it as or when there is no confusion. Hence

*Definition 4. *A sequence in a quasi-pseudometric is called (a)* left **-Cauchy* with respect to if, for every , there exists such that

(b)* right **-Cauchy* with respect to if, for every , there exists such that

(c)-*Cauchy* if, for every , there exists such that

*Remark 5. *(i) A sequence is left -Cauchy with respect to if and only if it is right -Cauchy with respect to .

(ii) A sequence is -Cauchy if and only if it is both left and right -Cauchy.

*Definition 6. *A quasi-pseudometric space is called* left-complete* provided that any left -Cauchy sequence is -convergent.

*Definition 7. *A quasi-pseudometric space is called* right-complete* provided that any right -Cauchy sequence is -convergent.

*Definition 8. *A -quasi-pseudometric space is called* bicomplete* provided that the metric on is complete.

We now recall some known definitions, notations, and results concerning cones in Banach spaces.

*Definition 9. *Let be a real Banach space with norm and let be a subset of . Then is called a cone if and only if (1) is closed and nonempty and , where is the zero vector in ;(2)for any , and , one has ;(3)for , if , then .

Given a cone in a Banach space , one defines on a partial order with respect to by We also write whenever and , while will stand for (where designates the interior of ).

The cone is called* normal* if there is a number , such that for all , one has
The least positive number satisfying this inequality is called the* normal constant* of . Therefore, one will then say that is a -normal cone to indicate the fact that the normal constant is .

*Definition 10 (compare [3]). *Let be a nonempty set. Suppose the mapping satisfies () for all ;() if and only if ;() for all .Then, is called a* quasi-cone metric* on and is called a* quasi-cone metric space*.

*Definition 11 (compare [3]). *A sequence in a quasi-cone metric space is called(a)*-Cauchy* or* bi-Cauchy* if, for every with , there exists such that
(b)*left (right) Cauchy* if, for every with , there exists such that

*Remark 12. *A sequence is -Cauchy if and only if it is both left and right Cauchy.

We also recall the following lemma, which we take from [4] and we give the proof as it is.

Lemma 13 (compare [4, Lemma 2]). *Let be a cone metric space. Then for each , there exists such that whenever .*

* Proof. *Since , then . Hence, find such that . Now if then and hence .

*Remark 14. *Although the lemma is stated for a cone metric space, it remains valid for a quasi-cone metric space.

#### 3. Some First Results

*Definition 15. *(1) In a quasi-cone metric space , one says that the sequence * left-converges* to if for every with there exists such that, for all , .

(2) Similarly, in a quasi-cone metric space , one says that a sequence * right- converges* to if for every with there exists such that, for all , .

(3) Finally, in a quasi-cone metric space , one says that the sequence * converges* to if for every with there exists such that, for all , and .

*Definition 16. *A quasi-cone metric space is called (1)*left-complete* (resp.,* right-complete*) if every left Cauchy (resp., right Cauchy) sequence in left (resp., right) converges,(2)*bicomplete* if every -Cauchy sequence converges.

*Remark 17. *A quasi-cone metric space is bicomplete if and only if it is left-complete and right-complete.

*Definition 18. *Let be a quasi-cone metric space. A function is said to be* lipschitzian* if there exists some such that
The smallest constant which satisfies the above inequality is called the* lipschitizian constant* of and is denoted by Lip. In particular is said to be* contractive* if Lip and* expansive* if Lip.

Lemma 19. *Let be a quasi-pseudometric space. If a sequence -converges to , then it is -Cauchy.*

* Proof. *Since -converges to , for every , there exist such that for any and such that for any . Hence for any , .

Lemma 20. *Let be a quasi-cone metric space and a -normal cone. Let be a sequence in . Then converges to if and only if and .*

*Proof. *Suppose converges to . For every real , choose with and . Then there exists such that for all and . This implies that when , and . This means that and .

Conversely, suppose that and . For any with , there is such that implies that . For this , there exist and such that for any and for any . Hence, for , and . Therefore converges to .

*Remark 21. *In fact, a sequence left-converges (resp., right-converges) to if and only if (resp., ) .

Lemma 22. *Let be a quasi-cone metric space and let be a sequence in . If converges to , then is a bi-Cauchy sequence.*

*Proof. *For any with , there exists such that, for all , and . Hence
Therefore, is a bi-Cauchy sequence.

Lemma 23. *Let be a quasi-cone metric space, a -normal cone, and a sequence in . Then is a bi-Cauchy sequence if and only if as .*

*Proof. *Suppose that is a bi-Cauchy sequence. For every real , choose with and . Then there exists such that, for all , . Therefore, whenever , . This means that as .

Conversely, suppose that as . For any with , there is such that implies that . For this , there exist such that for any . Hence . Therefore is a bi-Cauchy sequence.

#### 4. First Fixed Points Results

Theorem 24. *Let be a bicomplete quasi-cone metric space and a -normal cone. Suppose that a mapping satisfies the contractive condition
**
where . Then has a unique fixed point. Moreover for any , the orbit converges to the fixed point.*

*Proof. *Take an arbitrary and denote . Then
Similarly,
So for ,
It entails that as .

Similarly for
It entails that as . Hence is a bi-Cauchy sequence. Since is bicomplete, there exists such that converges to .

Moreover since
we have that

Hence . This implies, using property , that . So is a fixed point.

If is another fixed point of , then
Hence, and . Therefore the fixed point is unique.

Corollary 25. *Let be a bicomplete quasi-cone metric space and a -normal cone. For with and , set . Suppose the mapping satisfies the contractive condition
**
where is a constant and . Then has a unique fixed point in .*

*Proof. *We only need to prove that is bicomplete and for all .

Suppose is a bi-Cauchy sequence in . Then is also a bi-Cauchy sequence in . By the bicompleteness of , there is such that converges to . We have

Since converges to , . Hence and . Therefore, is bicomplete.

For every ,
Hence .

*Remark 26. *A weaker version of this corollary is actually sufficient. Indeed, it is enough to consider as a left-complete quasi-cone metric space with the same assumption. In this case, we would just have to prove that is left-complete and for all .

Corollary 27. *Let be a bicomplete quasi-cone metric space and a -normal cone. Suppose a mapping satisfies for some positive integer ,
**
where is a constant. Then has a unique fixed point in .*

*Proof. *From Theorem 24, has a unique fixed point . But , so is also a fixed point of . Hence , is a fixed point of . Since the fixed point of is also a fixed point of , the fixed point of is unique.

Theorem 28. *Let be a quasi-cone metric space over the Banach space with the -normal cone . The mapping defined by satisfies the following properties:**(Q1)** for any ;**(Q2)**, for any points .*

*Proof. *The proof of is immediate by property of the quasi-cone metric. In order to prove , consider as points in . Using property , we get
Since is -normal
which implies that
This completes the proof.

We are therefore led to the following definition.

*Definition 29. *Let be a nonempty set, and let the function satisfy the following properties: for any ; for any points and some constant .Then is called a quasi-pseudometric type space. Moreover, if , then is said to be a *-quasi-pseudometric type space*. The latter condition is referred to as the -condition.

*Remark 30. *(i) Let be a quasi-pseudometric type on ; then the map defined by whenever is also a quasi-pseudometric type on , called the conjugate of . We will also denote by or .(ii)It is easy to verify that the function defined by , that is, , defines a* metric-type* (see [2]) on whenever is a -quasi-pseudometric type.(iii)If we substitute the property by the following property,(D3),

we obtain a -quasi-pseudometric type space directly. For instance, this could be done if the map is obtained from quasi-cone metric.

The concepts of* left **-Cauchy*,* right **-Cauchy*, *-Cauchy,* and* convergence* for a quasi-pseudometric type space are defined in a similar way as defined for a quasi-pseudometric space. Moreover, for , we recover the classical quasi-pseudometric; hence quasi-pseudometric type generalizes quasi-pseudometric.

*Definition 31. *A quasi-pseudometric type space is called* left-complete* provided that any left -Cauchy sequence is -convergent.

*Definition 32. *A -quasi-pseudometric type space is called* bicomplete* provided that the metric type space is complete.

*Definition 33. *Let be a quasi-pseudometric type space. A function is called* lipschitzian* if there exists some such that
The smallest constant will be denoted by .

*Definition 34. *Let be a quasi-pseudometric type space. A function is called *-sequentially continuous* if, for any -convergent sequence with , the sequence -converges to ; that is, .

#### 5. Some Fixed Point Results

In [2], Khamsi proved the following.

Theorem 35. *Let be a complete metric type space. Let be a map such that is lipschitzian for all and . Then has a unique fixed point . Moreover for any , the orbit converges to .*

We state here an analogue of Khamsi's theorem.

Theorem 36. *Let be a bicomplete quasi-pseudometric type. Let be a map such that is lipschitzian for all and . Then has a unique fixed point . Moreover for any , the orbit converges to .*

*Proof. *We just have to prove that is a map such that is lipschitzian for all .

Indeed, since is a map such that is lipschitzian for all , then

Since for any , we have
that is,
we see that is a map such that is lipschitzian for all .

Therefore
for all and for all . Hence
for all and for all , so, is a map such that is lipschitzian for all .

By assumption, is bicomplete; hence is complete. Therefore, by Theorem 35, has a unique fixed point and for any , the orbit converges to .

The connection between a quasi-cone metric space and a quasi-pseudometric type space is given by the following corollary.

Corollary 37. *Let be a bicomplete quasi-cone metric space over the Banach space with the -normal cone . Consider defined by . Let be a contraction with constant . Then
**
for any and . Hence , for any . Therefore is convergent, which implies that has a fixed point and any orbit converges to .*

*Proof. *It is enough to prove that the metric type space is complete. Let be a -Cauchy sequence. Therefore , which implies that the sequence is bi-Cauchy in . Since is bicomplete, there exists such that and . Hence .

Moreover, since is a contraction with constant , we have that
Hence , for any .

#### 6. More Fixed Point Results

We begin with the following lemmas.

Lemma 38. *Let be a sequence in a quasi-pseudometric type space such that
**
for some with . Then is left -Cauchy.*

*Proof. *Let . From the condition in the definition of a quasi-pseudometric type, we can write

From (40) and , the above becomes
It follows that is left -Cauchy.

Similarly, we have the following.

Lemma 39. *Let be a sequence in a quasi-pseudometric type space such that
**
for some with . Then is right -Cauchy.*

Theorem 40. *Let be a Hausdorff left-complete -quasi-pseudometric type space, and let be a -sequentially continuous function such that for some with ,
**
Then has a unique fixed point and for every , the sequence -converges to .*

*Proof. *Take an arbitrary and denote . Then
which implies that

Hence, since , by Lemma 38 we have that is left -Cauchy and since is left-complete and -sequentially continuous, there exists such that and . Since is Hauforff, the limit is unique, hence .

For uniqueness, assume by contradiction that there exists another fixed point . Then
Hence and using the -condition, we conclude that .

Theorem 41. *Let be a Hausdorff left-complete -quasi-pseudometric type space, and let be a -sequentially continuous function such that for some with ,
**
Then has a unique fixed point and for every , the sequence -converges to .*

*Proof . *Take an arbitrary and denote . Then
which implies that

Hence, since , by Lemma 38, we have that is left -Cauchy and since is left-complete and -sequentially continuous, there exists such that and . Since is Hauforff, the limit is unique, hence .

For uniqueness, assume by contradiction that there exists another fixed point . Then
Hence and using the -condition, we conclude that .

Theorem 42. *Let be a Hausdorff left-complete -quasi-pseudometric type space and let be a -sequentially continuous function such that for some with and any ,
**
Then has a unique fixed point and for every , the sequence -converges to .*

Corollary 43. *Let be a Hausdorff left-complete -quasi-pseudometric type space and let be a -sequentially continuous function such that for some with and any **
for all . Then has a unique fixed point and for every , the sequence -converges to .*

#### Conflict of Interests

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