Research Article | Open Access

Abdullah Shoaib, Sabeena Kazi, Asifa Tassaddiq, Shaif S. Alshoraify, Tahair Rasham, "Double Controlled Quasi-Metric Type Spaces and Some Results", *Complexity*, vol. 2020, Article ID 3460938, 8 pages, 2020. https://doi.org/10.1155/2020/3460938

# Double Controlled Quasi-Metric Type Spaces and Some Results

**Academic Editor:**Hassan Zargarzadeh

#### Abstract

Abdeljawad et al. (2018) introduced a new concept, named double controlled metric type spaces, as a generalization of the notion of extended *b*-metric spaces. In this paper, we extend their concept and introduce the concept of double controlled quasi-metric type spaces with two incomparable functions and prove some unique fixed point results involving new types of contraction conditions. Also, we introduce the concept of double controlled contraction and prove some related fixed point results. We give several examples to show that our results are the proper generalization of the existing works.

#### 1. Introduction and Preliminaries

The theory of fixed points takes an important place in the transition from classical analysis to modern analysis. One of the most remarkable work on fixed point theory was done by Banach [1]. Various generalizations of Banach fixed point theorem were made by numerous mathematicians, see [2–4]. One of the abstraction of the metric spaces is the quasi-metric space that was introduced by Wilson [5]. The commutativity condition does not hold in general in quasi-metric spaces. Several authors used this concept to prove some fixed point results, see [6–18]. On the other hand, Bakhtin [19] and Czerwik [20] established the idea of -metric spaces. Lateral, many authors got several fixed point results, for instance, see [21–25]. Kamran et al. [26] introduced a new idea to generalized -metric spaces, named as extended -metric spaces, see also [27–30]. They replaced the parameter in the triangle inequality by the control function . Nurwahyu [31] introduced dislocated quasi-extended -metric space and obtained several fixed point results. Mlaiki et al. [32] generalized the triangle inequality in -metric spaces by using control function in a different style and introduced controlled metric type spaces. Recently, Abdeljawad et al. [33] generalized the idea of extended -metric spaces as well as controlled metric type spaces and introduced double controlled metric type spaces. They replaced the control function in triangle inequality by two control functions and . Now, we recall some basic definitions and examples that will be used in this paper.

*Definition 1 (see [33]). *Given noncomparable functions . If satisfies(q1) if and only if (q2)(q3), for all Then, is called a double controlled metric type with the functions , and the pair is called double controlled metric type space with the functions .

Theorem 1 (see [33]). *Let be a complete double controlled metric type space with the functions and let be a given mapping. Suppose that the following conditions are satisfied.*

There exists such that

For , choose . Assume that

In addition, for every , we have

Then, has a unique fixed point .

*Definition 2. *Given noncomparable functions . If satisfies (Q1) if and only if (Q2) , for all Then, is called a double controlled quasi-metric type with the functions and and is called a double controlled quasi-metric type space. If , then is called a controlled quasi-metric type space.

*Remark 1. *Any quasi-metric space, double controlled metric type space, and controlled quasi-metric type space are also double controlled quasi-metric type space, but the converse is always not true (see Examples 1–3).

*Example 1. *Let . Define by , , , , , and .

Define as , , , , , , , , and .

To show that the usual triangle inequality in quasi-metric is not satisfied. Let , , and , then we havethis shows that is a double controlled quasi-metric type for all , but it is not a controlled quasi-metric type. Indeed,Also, it is not a double controlled metric type space because we have

*Definition 3. *Let be a double controlled quasi-metric type space with two functions. A sequence is convergent to some in if and only if .

*Definition 4. *Let be a double controlled quasi-metric type space with two functions:(i)The sequence is a left Cauchy if and only if for every , we obtain a positive integer such that , for all or (ii)The sequence is a right Cauchy if and only if for every , we obtain a positive integer such that , for all or (iii)The sequence is a dual Cauchy if and only if it is left Cauchy as well as right Cauchy

*Definition 5. *Let be a double controlled quasi-metric type space. Then, is(i)Left complete if and only if each left Cauchy sequence in is convergent(ii)Right complete if and only if each right Cauchy sequence in is convergent(iii)Dual complete if and only if every left Cauchy as well as right Cauchy sequence in is convergentNote that each dual complete double controlled quasi-metric type space is left complete as well as right complete, but converse is not true in general.

#### 2. Main Results

In this section, we generalize the definition of the fixed point for double controlled quasi-metric type spaces with two incomparable functions and which are given as follows.

Theorem 2. *Let be a left complete double controlled quasi-metric type space with the functions and let be a given mapping. Suppose that the following conditions are satisfied.*

There exists such that

For , choose . Assume that

In addition, for every , we have

Then, has a unique fixed point .

*Proof. *Let be an arbitrary element and be the sequence defined as above. If , then be a fixed point of . By (7), we haveFor all natural numbers , we haveLetHence, we haveLet By using (8), we have . By the ratio test, the infinite series is convergent, and let tend to infinity in (13), which implies thatSince is a left complete double controlled quasi-metric type space, there exists some such thatBy using (Q2) and (7), we haveBy taking limit tends to infinity together with (9) and (15), we get , that is, . Now, we have to show that the fixed point of is unique for this; let be such that and , so we haveSo, . Hence, is a unique fixed point of .

*Example 2. *Let . Define byGiven asIt is easy to see that is a double controlled quasi-metric type and the given function is not a controlled metric type for the function . Indeed,Take , , and . We observe the following cases:(i)If and , we have So, inequality (7) holds. Also, it holds if and.(ii)Inequality (7) holds trivially in the cases when and and if and .(iii)If and , we getSimilarly, in the case when and , we have . So, (7) holds for all cases. Now, let , so we have and , That is, (8) holds. In addition, for each , we haveThat is, (9) holds. Hence, all conditions of Theorem 2 are satisfied, and is a unique fixed point.

#### 3. Further Results

In this section, we introduce the concept of double controlled contraction and prove related fixed point results with some examples.

*Definition 6. *Let be a nonempty set, be a left complete double controlled quasi-metric type space with the functions , and be a given mapping. Assume there exists such thatSuppose that the following conditions are satisfied:For and , we havewhere . Also, for each , we haveThen, is called double controlled contraction.

Theorem 3. *Let be a left complete double controlled quasi-metric type space with the functions and let be double controlled contraction. Then, has a unique fixed point .*

*Proof. *Let be an arbitrary element and be the sequence defined as above. If , then is a fixed point of . By (26), we haveNow,Combining (30) and the above inequality, we getContinuing in this way, we obtainNow, to prove that is a Cauchy sequence, for all natural numbers , we haveUsing (33), we getHence, we haveLet . By using (27), we have . By the ratio test, the infinite series is convergent, and let tend to infinity in (36), which yieldSo, the sequence is a left Cauchy. Since is a left complete double controlled quasi-metric type space, there must be exist some such thatWe claim that . By (26), we haveBy taking limit as tend to infinity together with (38), we getHence, , which is a contradiction. Now, we have to show that the fixed point of is unique for this let such that , so we haveBy taking limit as tend to infinity, we have . Hence, is a unique fixed point of .

*Example 3. *Take . Define byGiven as and . It is easy that is a double controlled quasi-metric type space. The given is not a controlled metric for the function . Indeed,Take , , and , and we observe the follows cases:(i)If and , we have If and , we get(ii)It is straightforward in the case when we take and .(iii)If and , we getSimilarly, in the case when we take and , that is, inequality (26) holds, we haveand . Now, let , and we have , , , which shows that (27) holds. In addition, for each , we haveThat is, (28) holds. All conditions of Theorem 3 are proved, and is the unique fixed point.

*Definition 7. *Let be a complete quasi-*b*-metric space. is called Chatterjee-type *b*-contraction if the following conditions are satisfied:for all , , and

Theorem 4. *Let be a complete quasi- b-metric space and be Chatterjee-type b-contraction. Then, has a unique fixed point.*

*Remark 2. *In the Example 3, is a quasi-*b*-metric with , but we cannot apply Theorem 4 because is not a Chatterjee-type *b*-contraction. Indeed, , for all .

#### 4. Conclusion

In the present paper, we have obtained sufficient conditions to ensure the existence of the fixed point for different types of contractive mappings in the setting of double controlled quasi-metric type spaces. Examples are given to demonstrate the variety of our results. New results in quasi-*b*-metric spaces, extended *b*-metric spaces, extended quasi-*b*-metric spaces, controlled metric spaces, and controlled quasi-metric spaces can be obtained as corollaries of our results. Also, results in right complete and dual complete double controlled quasi-metric type spaces can be obtained in a similar way. It is natural to ask, Are there other multivalued contraction mappings which can be applied to obtain more results in the double controlled quasi-metric type spaces? Is there interest to find serious applications to integral equations and dynamical systems?

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

Each author equally contributed to this paper, read, and approved the final manuscript.

#### Acknowledgments

Sabeena Kazi would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under the Project no. R-1441-137.

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Copyright © 2020 Abdullah Shoaib 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.