Research Article | Open Access

Ehsan Lotfali Ghasab, Hamid Majani, Erdal Karapinar, Ghasem Soleimani Rad, "New Fixed Point Results in -Quasi-Metric Spaces and an Application", *Advances in Mathematical Physics*, vol. 2020, Article ID 9452350, 6 pages, 2020. https://doi.org/10.1155/2020/9452350

# New Fixed Point Results in -Quasi-Metric Spaces and an Application

**Academic Editor:**Emilio Turco

#### Abstract

The main goal of the present paper is to obtain several fixed point theorems in the framework of -quasi-metric spaces, which is an extension of -metric spaces. Also, a Hausdorff -distance in these spaces is introduced, and a coincidence point theorem regarding this distance is proved. We also present some examples for the validity of the given results and consider an application to the Volterra-type integral equation.

#### 1. Introduction and Preliminaries

In the last century, nonlinear functional analysis has experienced many advances. One of these improvements is the introduction of various spaces and is the proof of fixed point results in these spaces along with its applications in engineering science. One of these spaces is function weighted metric space introduced by Jleli and Samet [1]. This is a generalization of metric spaces.

*Definition 1 [1, 2]. *A function is named called a nondecreasing function if for every . Also, is said to be logarithmic-like when every positive sequence satisfies iff .

In the sequel, we apply for the set of all nondecreasing functions that are logarithmic-like.

In 2019, some of researchers such as Alqahtani et al. [2], Aydi et al. [3], and Bera et al. [4] discussed on the structure of this space and on the fixed points of mappings satisfying in various contractive conditions.

*Definition 2 [1]. *Consider a mapping , a constant and a so that

() for all ;

() for all ; and

() implies that for every with , for every , and for all with .

Then, the function is named as a function weighted metric or a -metric on , and the pair is called a -metric space.

*Definition 3 [5]. *Consider a mapping satisfies the properties and from the definition of a -metric. Then, is named a -quasi-metric on and is called a -quasi-metric space.

In [5], Karapinar et al. showed that generally induces other -quasi-metrics such as defined by and . Regarding the discussion above, we conclude that any quasi-metric is a -quasi-metric by choosing for the axiom with .

*Definition 4 [5]. *Let be a -quasi-metric space. For , the right (left) centered ball at and of radius is the set ().

*Definition 5 [5]. *Consider a -quasi-metric space with a sequence therein. Then, is said to be aright-convergent sequence (left-convergent sequence) to if (). Further, is said to be a biconvergent sequence (in summary, convergent sequence) when it is both right-convergent and left-convergent.

Proposition 6 [5]. *Consider a -quasi-metric space with a sequence therein. Also, let for every . Then .*

*Definition 7 [5]. *Consider a -quasi-metric space with a sequence therein. Then, is a right-Cauchy sequence (a left-Cauchy sequence) if (). With this interpretation, is bi-Cauchy sequence (in summary, Cauchy sequence) if it is both left-Cauchy sequence and right-Cauchy sequence.

Now, a -quasi-metric space is named right-complete (left-complete) if every right-Cauchy sequence (left-Cauchy sequence) in is a right-convergent sequence (left-convergent sequence) in . Further, is bicomplete (in short, complete) if it is both left-complete and right-complete.

*Example 8. *Define by
for every . Evidently, is a bicomplete -quasi-metric space.

On the other hand, Bhaskar and Lakshmikantham [6] defined the notion of coupled fixed point and presented several coupled fixed point propositions for a mixed monotone mapping in partially ordered matric spaces. Also, they studied the existence and uniqueness of a solution to a periodic boundary value problem. For more details on coupled, tripled, and -tupled fixed point assertions, one can see [7] and references therein.

*Definition 9 [8, 9]. *Let and be two optional mappings. An element is said to be a coupled coincidence point of and if and . Further, an element is named acommon fixed point of and if and .

Note that if is the identity mapping, then is called a coupled fixed point of [6].

*Definition 10 [9]. *Let and be two optional mappings. Then and is said to be commutative if for every .

In this paper, we introduce several common fixed point and common coupled fixed point theorems in such spaces and prove them. In Section 2, we prove a common fixed point theorem and a common coupled fixed point result in this space. In Section 3, we obtain a coincidence point result for single-valued and multivalued mappings regarding a Hausdorff -distance. Ultimately, as an application of these results, the existence of solution of the Volterra-type integral equation is investigated in Section 4.

#### 2. **-**Quasi-Metric Space and Fixed Point Theory

Theorem 11. *Let be a bicomplete -quasi-metric space. Also, let be two arbitrary mappings so that and are commutative, , is closed, and
for every , where . Then and contain a unique common fixed point in .*

*Proof. *Due to , we select a point such that for a given . By continuing this process, we can construct a sequence in by for . First, note that and possess a unique coincidence point. On the contrary, assume that are two different coincidence points of and . Then, with and . Now, by (2), we get
which is a contradiction.

Assume so that (_{3}) is complied. For an arbitrary and because of (_{3}), there exists such that
Now, let be a sequence in . Without loss of totality, suppose that . Otherwise, is a coincidence point of and . Now, using (2), we obtain
which implies by induction that
for every in . Hence, for every and in so that , we get
Since
there is some so that for every . Hence, from (4) and (_{1}), we observe that
for . Employing (_{3}) together with (9), we obtain
It follows that . Therefore, is right-Cauchy. Similarly, by changing the order of the pairs in the above process, we conclude that is also a left-Cauchy sequence. Hence, it is a Cauchy sequence. Now, since is a bicomplete space, there exists such that is convergent to . Since and is closed, we have . As a next step, we show that is a coincidence point of and . On the contrary, consider . Then we have
As in the inequality above, we obtain
which is a contradiction. Hence, ; that is, is a unique coincidence point of and . Therefore, and contian a unique point of coincidence . By commutativity of the mapping and , we have . Hence, is another point of coincidence of and . Now, by the uniqueness of the point of coincidence of and , we have ; that is, and contain a unique common fixed point. This completes the proof.

In the sequel, denote for simplicity by , where is a nonempty set and .

Lemma 12. *Consider a -quasi-metric space . Then, the following assertions hold:
*(1)* is a -quasi-metric space with
*(2)*The mapping and contain an-tuple common fixed point iff the mapping and defined by
possess a common fixed point in .*(3)* is bicomplete iff is bicomplete.*

*Proof. *Clearly, satisfies in (_{1}). We show that satisfies in (_{3}). For every for and , consider . Suppose that
Then, we have
where and . Therefore, we obtain

The proofs of (2) and (3) are straightforward and left to the reader.

Remember that Lemma 12 is a two-way relationship. Consequently, we can establish -tuple fixed point propositions from fixed point assertions and conversely. Now, set in Lemma 12. Then, we have the following theorem.

Theorem 13. *Let be a bicomplete -quasi-metric space. Also, let and be two arbitrary mappings so that and are commutative, , is closed, and
for all and in , where . Then, and contain a unique common coupled fixed point in .*

*Proof. *Let us define by for all . Further, we consider by and by for all . Using Lemma 12, is a bicomplete -quasi-metric space. Also, is a common coupled fixed point of and iff it is a common fixed point of and . On the other hand, from (18), we have either
or
Now, by Theorem 11, and have a common fixed point and by Lemma 12, and have a common coupled fixed point.

*Example 14. *Let . Define by
for every . Evidently, is a bicomplete -quasi-metric with and . Consider by and define by . Clearly and are commutative. Also, we have
Therefore, by letting , all of the hypotheses of Theorem 13 hold. Thus, and possess a common coupled fixed point in .

#### 3. Fixed Point Theorem and Hausdorff -Distance

Let us start with the following definition:

Consider a -quasi-metric space , and denote thefamily of all nonempty bounded closed subsets of by . Then, is said to be a Hausdorff -distance on , if where .

*Definition 15 [10]. *Let be a nonempty set, be a single-valued mapping, and be a multivalued mapping. Also, let for some . Then is said to be a point of coincidence of and , and is said to be a coincidence point of and .

Theorem 16. *Let be a bicomplete -quasi-metric space. Also, let be a single-valued mapping and be a multivalued mapping so that , is closed, and is continuous. Assume that there exists such that
for all . Then and have a coincidence point in .*

*Proof. *Due to , we select a point such that for a given . By continuing this procedure, we can construct a sequence in such that for . Suppose that so that (_{3}) holds. For an arbitrary and due to (_{3}), there exists such that
Consider the sequence . Now, without loss of generality, suppose that . Otherwise, is a coincidence point of and . Now, from (24), we have
Hence, we have for all . Now, let and be two natural numbers with . Then, we have
On the other hand, since , there exists so that
for . Hence, by (25) and (_{1}), we have
for all . Employing (_{3}) together with (29), we obtain
Now, by (_{1}), we have . This proves that is right-Cauchy. Similarly, by changing the order of the pairs in the above process, we conclude that is also a left-Cauchy sequence. Therefore, it is a Cauchy sequence. Note that is bicomplete and is closed. Thus, there exists such that . Now, we shall show that . For this purpose, using (24), we have
Thus,
Hence, . Consequently, and possess a point of coincidence.

*Example 17. *Consider . Define by and by . Also, define by
Clearly, is a bicomplete -quasi-metric with and . Also, evidently, and is closed. First, let . Then
Therefore, we may consider and are not zero. Without loss of totality, suppose that . Then
Due to , we have . For every , we have Also, for every , we obtain
This yields that
We deduce that
Obviously, all other hypotheses of Theorem 16 hold. Hence, and possess a coincidence point in .

#### 4. An Application

As an application of our results, we consider the following Volterra-integral equation: where , , and .

Consider a Banach space of all real continuous functions defined on () with norm for every . Also, let be the space of all continuous functions defined on . On the other hand, the Banach space can be dedicated with the Bielecki norm for all and , and the derived metric for all . Define by

Also, define by

Theorem 18. *Consider a bicomplete -quasi-metric space with . Also, let be an operator from into with , and let . Assume that is an operator such that
*(i)* is continuous;*(ii)* for all is increasing; and*(iii)*for every and in , and and in , we have
*

Then, the integral equation (39) possesses an answer in .

*Proof. *By definition of , we have

Now, we consider that the function for every , , and . Therefore, all assertions of Theorem 11 hold. As a result, Theorem 11 confirms the existence of fixed point of so that this fixed point is the answer of the integral equation.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no competing interests.

#### Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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#### Copyright

Copyright © 2020 Ehsan Lotfali Ghasab 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.