Exciting Fixed Point Results on a Novel Space with Supportive Applications

Hammad, Hasanen A.; Aydi, Hassen; Gaba, Yaé Ulrich

doi:https://doi.org/10.1155/2021/6613774

Journal of Function Spaces

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Abstract Introduction Examples Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6613774 | https://doi.org/10.1155/2021/6613774

Exciting Fixed Point Results on a Novel Space with Supportive Applications

Hasanen A. Hammad,¹Hassen Aydi,^2,3,4and Yaé Ulrich Gaba^4,5,6

Academic Editor: V. Vijayakumar

Received10 Dec 2020

Revised29 Dec 2020

Accepted05 Jan 2021

Published28 Jan 2021

Abstract

The goal of this paper is to present a new space, a complex valued controlled rectangular -metric space (for short, -metric space). Some examples and topological properties of -metric spaces are given. Also, some related common fixed point results are discussed. Our results generalize a lot of works in this direction. Moreover, we apply the theoretical results to find a unique solution of a complex valued Atangana-Baleanu fractional integral operator and a system of complex linear equations. Finally, a numerical example to find the current that passes through the RLC circuit is illustrated.

1. Introduction

In 2011, the concept of a complex valued metric space has been initiated by Azam et al. [1]. Later, Al-Mezel et al. [2] emphasized that this space is just a particular case of cone metric spaces [3, 4]. Several fixed point results have been investigated on complex valued metric spaces; see [5–9]. The notion of a -metric space, as a generalization of metric spaces, was introduced by Bakhtin [10] and Czerwik [11]. Combining the above concepts, Rao et al. [12] initiated the notion of complex valued -metric spaces. For related results on this setting, see [13, 14]. Going in the same direction, recently, Ullah et al. [15] presented complex valued extended -metric spaces to extend the idea of extended -metric spaces [16].

On the other hand, in the last few years, the fixed point theory became an essential tool to resolve several problems in nonlinear analysis, including fractional calculus. The contribution of our paper is to solve a complex valued Atangana-Baleanu fractional integral operator and a system of complex linear equations. The idea is to introduce the concept of a new space, the so-called complex valued controlled rectangular -metric space (for short, -metric space). We also provide a numerical example illustrating the fact that the current passes through the RLC circuit.

2. Statements about -Metric Spaces

Let be the set of complex numbers and . Define the below partial order on as

Hence, if one of the below statements holds:

Also, we can write if and one of , , and holds. By following the same pattern, we write if only is fulfilled. This implies that

This important part is devoted to present new definitions and examples of a -metric space.

Definition 1. A mappingis called a-metric on a nonempty set, if it satisfies the following hypotheses, for all

there is a function such that (the so-called controlled rectangular inequality), for all distinct points

Then, is called a -metric space.

Example 1. Assume thatis a complex valued extended rectangular-metric with coefficientandis a strictly increasing continuous function with, for alland. Suppose thatfor all. For any two distinct points, one can obtain that

Thus, the pair is a -metric space with .

Example 2. Let. Define, by, and, , respectively. The conditionsandare clear and the conditionfollows immediately by the Jensen inequalityfor all . This implies that is a -metric space with . Note that in general is not continuous.

Example 3. Let. Define, by, and, respectively. The conditionsandare trivial. We only show the condition. For all, we have

Thus, is a -metric space with .

Example 4. Let, whereand, and choose a functionfor all. Define a functionby

It is clear that the pair is a -metric space with . If we take the sequence , it is easy to prove that this sequence converges to both 0 and 2, so it is not a Cauchy sequence. Moreover, is not continuous, since

This means that there exists a sequence in which is not convergent to a unique point in . Hence, is not metrizable.

Example 5. Assume that. Defineby

Define by

Now, we examine the condition in the following cases: (i)When then the result follows immediately(ii)When and , then(iii)When and , then(iv)When and , then(v)When and , then(vi)When and , then(vii)When and , then

Note that

Thus, according to the above cases, is not a complex valued rectangular -metric for the function , but is a -metric (conditions and are trivial).

Example 6. Letand define, where, thenis a complete-metric space with the function.

Verification: initially, we show that is a -metric space by discussing the following conditions. Assume that and , where

Since , one writes . So that .

Let , then for .

If , then for each , .

.

Since for all , one gets

Taking the maximum on two sides for all , we have

This leads to . From the above condition, we conclude that is a -metric space.

Finally, we show the completeness property of . It means that for every Cauchy sequence in is convergent. Let be a Cauchy sequence in , then for every , there exists so that

That is, for , it satisfies

Hence, for every , is a Cauchy sequence in . Since is complete, for . Now, define so that . Passing in (19), we have , for , i.e., . This implies that , so is complete.

Now, we introduce topological properties of a -metric space.

Definition 2. Assume thatis a-metric space,is a sequence inand. For everywith, there existssuch that(i)for all , , then is called convergent, converges to , or we say that is a limit point of and write or (ii)for all , , , then is called a Cauchy sequence(iii)if every Cauchy sequence in is convergent, then the pair is called a complete -metric space

Lemma 1. Assume thatis a-metric space andis a sequence in, thenis a Cauchy sequence iff, where.

Proof. Let be a Cauchy sequence in , then for every , so that

Since , there are such that . Consider , and take the absolute value in two sides of (20), we get . From definition of the limit, we get as .

Let as . It follows that , for every . Since , there exists such that . It holds that , for every . Thus, there is so that for , is a complex valued Cauchy sequence.

Lemma 2. Assume thatis a-metric space andis a sequence inthenis the limit point ofiff.

Proof. Passing in the proof of Lemma 1, we get the result.

Remark 1. If we take, for all, then an-metric space reduces to a complex valued rectangular-metric space.

The below lemma is very important in the equal.

Lemma 3. Letbe a-metric space andbe a Cauchy sequence insuch that, whenever. Then,converges at most to one point

Proof. Suppose that the sequence has two limit points , that is, . Since is Cauchy, from , for , whenever , we can write

It follows that , i.e., . Thus, converges at most to one point.

The following diagram shows the latest generalizations of complex valued metric spaces. However, the inverse implications need not be true. where c.v.m.s. refers to a complex valued metric space.

3. Exciting Fixed Point Results on -Metric Spaces

This section is aimed at introducing two fixed point theorems in the context of -metric spaces.

Theorem 1. Letbe a complete-metric space and let the mappingssatisfywhere , and are nonnegative real numbers with .

Assume also and for all exists and is finite. Then, and have a common fixed point (cfp).

Proof. Let be an arbitrary point such that the sequence in satisfies the conditions of theorem and iterates below as follows: Assume that This implies that

Similarly,

Now, with , we can write

Now, we want to show that is a Cauchy sequence. So, we distinguish two cases.

Case 1. Let be odd, that is, for any . Now, by for any , we get

This implies that

As, in view of (24), we get ; therefore, by the ratio test, we conclude the convergence of the series for all . Consider

It follows by (30) that

Passing in (33), we can write

Case 2. Let be even, that is, for any . Then,

Similar to Case 1, we conclude that

Applying the limit on (36) as , we have

It follows from (34) and (37) that , , which illustrates that the sequence is Cauchy in . The completeness of leads that there exists an element such that is complex valued convergent to , i.e.,

By using controlled rectangular inequality , we can write

Passing the limit as in (39) and using (38), we have

Now, let us demonstrate that , that is, is a fixed point of ; otherwise, , and for , by (23), we have

Taking the limit in (41) as and applying (38) and (40), we conclude that , a contradiction. Hence, . Similarly, we can prove . Observe that, in view of Lemma 3, the sequence converges uniquely at point .

To prove is the unique cfp of and , suppose that is a distinct cfp of and , then . So from (23), we get

This implies that , a contradiction. Thus, . Hence, and have a unique cfp. This finishes the proof.

The following theorem is more general than Theorem 1.

Theorem 2. Letbe a complete-metric space and let the mappingssatisfyfor all and , where If , where and for all exists and is finite. Then, and have a unique cfp.

Proof. Let be an arbitrary point. Define a sequence as follows:

Consider

If , we can write

This is a contradiction. Hence, or

Similarly, we can obtain

In general, we have

Using the same scenario as Cases 1 and 2 of Theorem 1, we conclude that the sequence is Cauchy in . Since is complete, there is an element such that

By Lemma 3, a sequence converges uniquely at point . By the condition , we get

Passing the limit as in (52) and using (51), we have

Now, we show that is a fixed point of ; otherwise, , and for , by (43), we can get

After entering the limit as in (54) and using (51) and (53), we conclude that , a contradiction. Hence, . Similarly, we can prove .

The uniqueness comes easy from the condition (51), and it is similar to Theorem 1. Therefore, the mappings and have a unique cfp in . This ends the proof.

4. Examples and Some Sequences on the Main Theorems

This part is devoted to obtain some sequences of Theorems 1 and 2.

If we consider in Theorem 1, we have the below result.

Corollary 1. Letbe a complete-metric space and let the mappingsatisfywhere and are nonnegative real numbers with .

Assume also where and exists and is finite for all . Then, has a unique fixed point.

Assume that and in Corollary 1, we get the important below result.

Corollary 2. Letbe a completemetric space and let the mappingsatisfywhere is a non-negative real number with . Assume also where and exists and is finite for all . Then, has a unique fixed point.

Remark 2. If we takein Theorem2, we have the following immediate consequences.

Corollary 3. Letbe a complete-metric space and let the mappingsatisfyfor all and , where Assume that where and for all exists and is finite. Then, has a unique fixed point.

Corollary 4. Letbe a complete-metric space and let. Suppose that there existssuch thatfor all and , where Assume that , where and for all exists and is finite. Then, possesses a unique fixed point.

Proof. By Corollary 3, we have such that

So

Therefore, the result follows.

The below examples support Theorems 1 and 2.

Example 7. Assume that all requirements of Example3hold. It is clear thatis a complete-metric space with. Define the mappingsbyand . We Only prove the contractive condition ((23)) of Theorem1withand. Indeed, for all

Hence, condition (23) of Theorem 1 is satisfied with and such that and the point is the unique cfp of and

Example 8. Suppose that the statements of Example2are fulfilled. Definebyandfor allIt is clear thatis a complete-metric space with Now, we will go to fulfill condition ((43)) forand. We have

Hence, all required assumptions of Theorem 2 are fulfilled with and is the unique cfp of and

Remark 3. (i) Theorems1and2extend some previous works in the different spaces. Some of them are as follows.

In Theorem 1(i)when and , we have the work of Ege [17] in complex valued rectangular -metric spaces, where (ii)when and we have the exciting results of Azam [1] in complex valued metric spaces(iii)if we reduce a complete -metric space to a complete complex valued -metric space with we get directly the work of Mukheimer [13](iv)when and are two mappings in a complete complex valued metric space, we have the results of Sintunavarat et al. [5]

In Theorem 2(i)if we reduce a complete -metric space to a complete complex valued metric space with and we obtain the results of Hammad [6](ii)the conclusion of Theorems 1 and 2 is still valid by using one of the following contractive conditions:

(⋆₁) for all and are nonnegative real numbers with and .

(⋆₂) for all and are nonnegative real numbers with and .

(⋆₃) for all and are nonnegative real numbers with and , where is a Lebesgue integrable mapping on a compact subset of and for all This direction involves fixed point results of integral type.

(⋆₄) for all and , where , where is a Lebesgue integrable mapping on a compact subset of and for all This direction concerns fixed point results of exponential type.

5. Solving a Complex Valued Atangana-Baleanu Fractional Integral Operator

Let be the set of all complex analytic functions and be defined as for all . Also, define by Then, is a complete -metric space. We will use the form to express the complex valued Atangana-Baleanu integral (AB-fractional integral), where is a fixed complex number, is a complex function which is analytic on an open star-domain centered at and It is defined by where refers to the Reimann-Lioville fractional integral equation for a complex analytical function

Another form of the complex valued Atangana-Baleanu integral is presented by Fernandez [18] as follows: where and refers to the Hankel contour [19].

Define the operator by

Now, we will seek for the existence and uniqueness of a fixed point for the operator (72), which is considered as the unique solution for the operator (70) under the following condition:

Consider

Thus, the conditions of Corollary 2 are satisfied. Hence, has a unique fixed point, which concludes that the fractional integral (70) has a unique solution.

6. Solving a System of Complex Linear Equations

In this part, we shall try to find a unique solution to a system of complex linear equations and apply it to solve the RLC circuit. Consider the below system: in linear equations and variables, where and is the complex coefficients matrix. Assume that Define by and , respectively. According to Example 6, the pair is a complete -metric space.

The following theorem is the main result of this section.

Theorem 3. Suppose thatis a complete-metric space, then the system (75) has only one solution iffor

Proof. Define the linear mapping by where and If we can prove that is a contraction mapping, the proof is over. Consider and then we have where then is a contraction mapping. Thus, by Corollary 3, the system (75) has a unique solution.

To illustrate the usefulness of Theorem 3, we give the following example.

Example 9. Consider the RLC circuit containing three as follows.
The symbols , and refer to the resistance, capacity, voltage, and current, respectively. Also, let denote the imaginary number with current. Let us take the values and and It is obvious that the RLC circuit forms a complex valued linear equation system that has one solution.
Verification: from Kirchhoff voltage law “the sum of all voltage in a closed loop is zero”; in this example, we observe it in three loops (Figure 1). From our simple physical information, we know that where and (i)From loop 1, we have and(ii)From loop 2, we have and(iii)From loop 3, we have and ; hence,

So we have the below complex linear equation system:

By testing the sum of the coefficients where and we have (i)for (ii)for (iii)for

Hence, according to Theorem 3, the system (81) has a unique solution in the form of , , and .

7. Conclusion

In this paper, we ensured the existence of a unique common fixed point for several contraction type self-mappings in the class of complex valued controlled rectangular -metric spaces. Our presented results complement and generalize several significant known fixed point theorems in literature. As applications, we resolve a complex valued Atangana-Baleanu fractional integral operator and a system of complex linear equations, and we presented a related numerical example. We think our paper will be the source of motivation for researchers to improve and extend our results for new applications.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

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.

Acknowledgments

The third author (Y.U.G.) wishes to acknowledge that this work was carried out with the aid of a grant from the Carnegie Corporation provided through the African Institute for Mathematical Sciences.

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Copyright © 2021 Hasanen A. Hammad 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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