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Sila, Eriola; Duraj, Sidite; Hoxha, Elida

doi:https://doi.org/10.1155/2021/5542787

Journal of Mathematics

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Abstract Introduction and Preliminaries Conclusions Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5542787 | https://doi.org/10.1155/2021/5542787

Equivalent Cauchy Sequences in -Quasi Metric-Like Space and Applications to Fixed-Point Theory

Eriola Sila,¹Sidite Duraj,²and Elida Hoxha¹

Academic Editor: Ji Gao

Received09 Jan 2021

Revised26 Mar 2021

Accepted20 Apr 2021

Published06 May 2021

Abstract

This paper introduces the concept of -quasi metric-like space and delves into some topological properties of it. A necessary and sufficient condition related to equivalent Cauchy sequences and Cauchy sequences in have been proved. Furthermore, the results of the fixed point are shown in the setting of -quasi metric-like space as applications of conditions of equivalent Cauchy sequences. Besides, some instances are inclined to epitomize the examined consequences.

1. Introduction and Preliminaries

The theory of fixed point has a crucial role not only in scientific and engineering applications [1–5] but also in theory.

The development of this theory is accomplished in two directions, either by generalizing the Banach contraction or by giving new metric type spaces. There are a lot of generalizations of the metric space based on reducing or modifying the metric axioms.

Some generalized metric spaces which are obtained by the above changes are, -metric space [6–8], quasi-metric spaces, partial-metric space [9], partial-metric spaces [10, 11], dislocated metric spaces [12, 13], rectangular partial metric spaces [14], and so on.

In 2012, Harandi [15] introduced the notion of metric-like space which was given as dislocated metric space in [12, 13].

Later, in 2016, Arutyunov and Greshnov [16] introduced the notion of quasi-metric space and examined their properties. Moreover, they proved some results on fixed points in this space. In addition, they extended their results in [17, 18].

Henceforth, many mathematicians have researched several masterwork involving fixed points for self-mappings in generalized metric spaces.

Necessarily, a technical tool in the proof of fixed-point theorems is the Cauchy and the equivalent Cauchy sequences.

In 1983, Leader [19] obtained a necessary and sufficient condition as a characterization of equivalent Cauchy sequences in the metric space. Nevertheless, the generalizations of the consequences are found in spaces, [20, 21], obtaining fixed-point results for the contractive function of Mein-Keeler type.

As a consequence, this invigorated us to introduce the notion of the -quasi metric-like space. Further, there are shown some features of topology induced by a -quasi metric-like space.

The highlight of this paper is Theorem 3, which gives a necessary and sufficient condition that two sequences are equivalent Cauchy in the -quasi metric-like space. An application of this result is Theorem 1, which extends the existing results for the Mein-Keeler contractions in the -quasi metric-like space. One example is given to authenticate the effectiveness and applicability of our results.

Amini [15] defined the metric-like space as the forthcoming definition.

Definition 1 (see [15]). Let be a nonempty set. A metric like in is a mapping such that for all , it satisfies the following:(i) implies (ii)(iii) The pair is called a metric-like space.
Alghamdi et al. [22] extended the concept of the metric-like space to metric-like space as follows.

Definition 2 (see [22]). Let be a nonempty set and is a given real number. A function is -metric like if for all , the following conditions are accomplished:(i) implies (ii)(iii) The pair is called metric-like space. The number is called the coefficient of .
This definition was generalized by Arutyunov and Greshov [16], who gave the concepts of -metric space and quasi-metric space as in Definitions 3 and 4.

Definition 3 (see [16]). Let be a nonempty set, be two positive numbers, and be a given function satisfying the following conditions for all :(i) if and only if (ii)(iii) The function is called -metric. The pair is called -metric space.

Definition 4 (see [16]). Let be a nonempty set, two positive numbers, and be a given function that satisfies the following conditions:(i) if and only if , for each (ii) , for every The function is called -quasi metric, and the pair is called quasi-metric space.
These definitions motivated us to define a new space and give some topological aspects of it.

2. (q₁, q₂)-Quasi Metric-Like Space and Its Topology

In this section, there is definition of -quasi metric-like space. Furthermore, the topology induced by -quasi metric-like space is given and its properties are proved.

Definition 5. Let be a set and be two positive numbers. The function is called -quasi metric-like if for all , and the following conditions are satisfied:(i)(p₁) implies (ii)(p₂) The pair is called the -quasi metric-like space.

Remark 1. If as a result, the concept of -quasi metric-like space coincides with the concept of quasi metric-like space [22].
If , consequently, the concept of -quasi metric-like space coincides with the concept of -quasi metric-like space [16].
If is a -quasi metric-like space and , then is a b-quasi metric-like space [22].
If as a consequence is a quasi metric-like space [22].
The above facts prove that the -quasi metric-like space is a generalization of space mentioned in [16, 22].
If is a -quasi metric-like space, then for and implies . However, the converse may not be concordant, and may be positive for .

Example 1. Let , and be defined byThis prompts the couple to be a -quasi metric-like space, and , for every .
The following definitions are given to introduce the topology induced by a -quasi metric-like space.

Definition 6. The set is called a right open ball with center and radius The set is called a left open ball with center and radius It is clear-cut from Definitions 5 and 6 that the center of a ball may not be an element of it. For example, in the -quasi metric-like space of Example 2, the element is not in because .
Furthermore, is different from a left open ball . For example, in the -quasi metric-like space given in Example 2, there isSince for each , there is implied .
Additionally, , and since for every , . This leads to .

Definition 7. The set is called a right closed ball with center and radius The set is called a left closed ball with center and radius

Proposition 1. Let be a -quasi metric-like space. The family , is a topology in . It is called the right topology in obtained from -quasi metric-like
The family , is a topology in . It is called left topology obtained from -quasi metric-like
All following propositions related to right topology can be proved using analog methods for left topology induced by -quasi metric-like .

Definition 8. A set is said to be right (left) open if for every point , there exists a right (left) open ball (this means ).

Definition 9. A set is said to be right (left) closed if .

Proposition 2. Every -quasi metric-like space is .

Proof. Let be a -quasi metric-like space. To prove that space is , it needs to show that, for every , the set is right closed or the set is a right open.
Indeed, for every , . Taking , there exists . This is true because for every , it yields It means that ; consequently, .

Proposition 3. The topology satisfies the first axiom of countability.

Proof. The set forms a countable base of .

Proposition 4. Let be a -quasi metric-like space. The family is a topology in .

Proof. Constructing the family we assure that it is topology.
The topology is called the topology induced by -quasi metric-like

Proposition 5. If is a -quasi metric-like space, then the function where is a metric like.

Proof. This is clear since satisfies also the symmetry condition.
The topology induced by metric like is the topology defined above.
In the following paragraphs, there are defined Cauchy sequences and equivalent Cauchy sequences in -quasi metric-like space.

Definition 10. Let be a -quasi metric-like space and be a sequence in (1)The sequence is called right Cauchy in if for every , there exists , such that, for every , . It is denoted that .(2)The sequence is called left Cauchy in if for every , there exists , such that, for every , ().The concepts of the right Cauchy sequence and the left Cauchy sequence are independent of each other. This means that a sequence may be right Cauchy but not left Cauchy or vice versa.
The following example illustrates it.

Example 2. Let and .
The function is a -quasi metric-like with and .
Nevertheless, it is not a quasi metric-like because for , and , and we attain , .
As a result, .
Taking the sequence in , for it yields and .
Consequently, and which proves that the sequence is left Cauchy.
In addition, . Hereof, it yields that is not a right Cauchy sequence.

Definition 11. Let be a -quasi metric-like space and be a sequence in The sequence in is called Cauchy if it is left Cauchy and right Cauchy.

Definition 12. Let be a -quasi metric-like space and be a sequence in (1)The sequence is called right convergent to if, for every , there exists , such that, for every , . It is denoted .(2)The sequence is called left convergent to if for every , there exists , such that, for every , ().(3)The sequence is called convergent to if it is left convergent and right convergent to . Consequently, for every , there exists , such that, for every , and or .Since and could not be space, there is conceded that these spaces do not satisfy the uniqueness of the limit. As a result, a right (left) convergent sequence might converge to more than one limit.
However, space , where is the metric like obtained from -quasi metric-like , is , and the convergent sequences in converge only to a unique limit.
Note that every convergent sequence in is a Cauchy sequence.
Indeed, if is a convergent sequence to a point in , then the following inequalities are veritable: and .
Consequently, .

Definition 13. Let be a metric-like space and and be two sequences in .(1)The sequences and are left equivalents if for every , there exists , such that, for every , . It is denoted .(2)The sequences and are right equivalents if for every , there exists , such that, for every , ().(3)The sequences and are equivalents if they are left equivalent and right equivalent or .

Definition 14. Let be a metric-like space and and be two sequences in .(1)The sequences and are left equivalent Cauchy if they are left equivalent and left Cauchy(2)The sequences and are right equivalent Cauchy if they are right equivalent and right Cauchy(3)The sequences and are equivalent Cauchy if they are left equivalent Cauchy and right equivalent Cauchy

3. Main Results

3.1. A Necessary and Sufficient Condition for Equivalent Cauchy Sequences in the Metric-Like Space

Leader [19] has given the following necessary and sufficient condition for the characterization of equivalent Cauchy sequences in the metric space.

Theorem 1. (see [19]). Two sequences and in metric space are equivalent Cauchy if and only if for every , there exists and such that , implying , for all .

Taking for every , Theorem 3.2.7 in [19], it gives a necessary and sufficient condition that a sequence is Cauchy in the metric space.

Besides, Pasicki [20] and Hoxha and Duraj [21] have given necessary and sufficient conditions in order for a sequence to be Cauchy in dislocated spaces and quasi-dislocated spaces.

In this section, we derive a result for equivalent Cauchy sequences and Cauchy sequences in the metric-like space.

Theorem 2. Let be a metric-like space and and be two sequences in . If the sequences and are equivalent and convergent to and , respectively, then .

Proof. Taking limits in the following inequalities:where is yielded. Consequently, .

Theorem 3. Let be a metric-like space where and and be two sequences in .
The sequences and are equivalent Cauchy if and only if for every , there exists and such that (1) , implying , for all .

Proof. Denote The set , where is fixed, is lower bound from 0. As a result, .
If , then for in (1), there exist and such that implying .
From the characteristic property of inferior for there exists such that . Besides,
Accordingly,
and from (1), is yielded for every meaning that which is absurd.
As a result,For every ,Taking , there exist and which satisfies condition (1). Using (4), for these and , there exists , such thatSuppose that there exists ,where is the smallest index that satisfies this inequality.
Consequently,From (6), there is obtained , and .
If then . As a result, Using this fact and (8), there is resulted and :Using (6), it results inAdditionally, and . Using (1), is obtained, which contradicts (7).
Hence, for every . Considering (5) for every , we obtainMoreover, , resulting inSubsequently, the sequences and are equivalent.
The next step is to show that each of them is Cauchy.
Indeed, and .
Taking limits in the above inequalities results in
Using the same method, it can be proved that
Thus, the sequences and are Cauchy.
Conversely, if and are two equivalent Cauchy sequences, using Definitions 10, 11, and 12, then for every , there exist and such that , for .
This assures that condition (1) is accomplished for .

Corollary 1. Let be a -quasi metric-like space, where . The sequence in is Cauchy if and only if for every , there exist and such that implying .

Proof. Replacing in Theorem 3, the corollary is true.

Remark 2. Corollary 1 generalizes Lemma 2.3 and Corollary 6.9 in [20] for the -quasi metric-like space.

3.2. Fixed-Point Results in the -Quasi Metric-Like Space

Many authors have given fixed-point theorems for the contractive function of the Mein-Keeler type in different spaces [23–25]. In this section, we acquire a fixed-point result related to the Mein-Keeler type contraction in the -quasi metric-like space.

Theorem 4. Let be a -quasi metric-like space where . The following propositions are equivalent.(1)The mapping has a fixed point, and for every , the iterative sequences converge to the fixed point of (2)For every and for each , there exist and such that implying.

Proof. The first step is to prove that condition (1) implies condition (2).
Let be the unique fixed point of mapping , , and for every , the sequences and converge to .
Using the definition of convergence in , it yielded the following:From the definition of -quasi metric-like, it is obtained thatTaking limit when and using (5), it is taken thatSignificantly, the sequences and are equivalent.
Since the sequences and are convergent, therefore they are Cauchy. Consequently, the sequences and are equivalent Cauchy. From Theorem 3, condition (2) is yielded.
The next step is to prove that condition (2) implies condition (1).
Let be a point in and be an iterative sequence in . If there exists such that then is a fixed point of .
Suppose that, for all , . Since satisfies condition (2) and replacing in (2), we obtain that, for every , there exist and such thatimplying ,implying .
Consequently, the sequence satisfies the condition of Corollary 1. So, is the Cauchy sequence in . Since is complete, then there exists such that .
As satisfies condition (2), then for the sequences and are equivalent Cauchy.
Due to from Theorem 2, is yielded.
Using condition (2) for , the following results are obtained.
For every , there exist and such that implying (6).
Since for , there exists such that, for every and are obtained.
Consequently, .
Furthermore, due to the fact that , it is implied thatConsidering (6), it is yielded for , there exists and such that, for every it results inThis means that .
In addition, and .
Consequently, .
Since and , then is the fixed point of , and for every , the iterative sequences converge to the fixed point of .
An example to illustrate the validity of Theorem 1 is introduced below.

Example 3. Let and , where .
The couple is a -quasi metric-like where and .
Define the function such that and .
For , the function accomplishes condition 2 of Theorem 1.

Case 1. and such thatSince and , and are obtained for each .
For every , there exist , such thatand .

Case 2. and such thatwhere for each . Since , is obtained as a result, for each
For every , there exist , such thatand .

Case 3. and such thatand , for each , and for each
For every , there exist , such thatand .
It is discernible that the function accomplishes the conditions of Theorem 1.
Thus, it has a fixed point . Since and for every then for each , the sequence is constant and converges to 0.

Remark 3. Condition (2) in Theorem 1 is the contractive condition of the Meir-Keeler type [26]. As a result, Theorem 1 is an application of Mein-Keeler contraction in the metric spaces mentioned in [16, 20, 22], which are generalized by the -quasi metric-like space.

4. Conclusions

This scrutiny elucidates the -quasi metric-like space and enunciates some topological properties of it. Some compelling results are acquired correlated with equivalent Cauchy sequence and Cauchy sequences (Theorem 3, Corollary 1), by generalizing Lemma 2.3 and Corollary 6.9 in [20] for -quasi metric-like spaces.

In addition, the existence of fixed-point theorem for the Meir-Keeler type contractions in -quasi metric-like spaces is attained. Moreover, some of the provided examples ascertain the application of the fixed-point theorem and the topological attributions. Accordingly, all obtained results are veridical in quasi-metric spaces, -quasi metric-like spaces, -metric spaces since the -quasi metric-like space generalizes the cited spaces.

Data Availability

All data required for this study are included within this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Eriola Sila et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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