On Quasi-Pseudometric Type Spaces

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.