On New Banach Sequence Spaces Involving Leonardo Numbers and the Associated Mapping Ideal

Yaying, Taja; Hazarika, Bipan; Mohamed, O. M. Kalthum S. K.; Bakery, Awad A.

doi:https://doi.org/10.1155/2022/8269000

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Volume 2022 | Article ID 8269000 | https://doi.org/10.1155/2022/8269000

On New Banach Sequence Spaces Involving Leonardo Numbers and the Associated Mapping Ideal

Taja Yaying,¹Bipan Hazarika,²O. M. Kalthum S. K. Mohamed,^3,4and Awad A. Bakery^3,5

Academic Editor: Muhammed nar

Received13 Sept 2021

Accepted29 Mar 2022

Published14 Apr 2022

Abstract

In the present study, we have constructed new Banach sequence spaces and where is a regular matrix defined by for all , where is a sequence of Leonardo numbers. We study their topological and inclusion relations and construct Schauder bases of the sequence spaces and Besides, -, - and -duals of the aforementioned spaces are computed. We state and prove results of the characterization of the matrix classes between the sequence spaces and to any one of the spaces , and Finally, under a definite functional and a weighted sequence of positive reals , we introduce new sequence spaces and . We present some geometric and topological properties of these spaces, as well as the eigenvalue distribution of ideal mappings generated by these spaces and -numbers.

1. Introduction and Preliminaries

Let denote the set of all real- or complex-valued sequences. A linear subspace of is called a sequence space. Some of the well-known examples of sequence spaces are the space of absolutely -summable sequences, the space of null sequences, the space of convergent sequences, and the space of bounded sequences, denoted by , and respectively. Here and afterwards, unless stated otherwise. Let and denote the spaces of all bounded and convergent series, respectively. A Banach sequence space with continuous coordinates is called a -space. The spaces and are -spaces equipped with the supremum norm and the norm respectively, where is the set of nonnegative integers and is any one of the spaces , or .

Let be an infinite matrix over the complex field The -transform of a sequence is a sequence provided that the series exists, for each . In addition, if and are two sequence spaces and , for every sequence then the matrix is said to define a matrix mapping from to The notation represents the family of all matrices that map from to Furthermore, the matrix is called a triangle if and , for For any define Then, is a sequence space and is called the matrix domain of in the space It is well known that if is a -space and is a triangle, then, the matrix domain is also a -space under the norm We refer to [1–13] for papers related to theory of sequence spaces and summability.

1.1. Some Special Integer Sequences and the Associated Sequence Spaces

We shall briefly highlight the literature concerning special integer sequences and the construction of the associated sequence spaces.

Let be the sequence of Fibonacci numbers defined by the recurrence relation with and Several authors constructed different types of sequence spaces involving Fibonacci numbers. For instance, Kara [3] studied the Fibonacci sequence spaces and and examined certain topological and geometrical structures of these Banach sequence spaces, where is a double band matrix of Fibonacci numbers defined by

Besides, Baarir et al. [7] studied the sequence spaces and where The studies on Fibonacci sequence spaces are further strengthened by Kara and Baarir [4] by introducing the matrix domain where represents any one of the sequence spaces , or and is a regular matrix of Fibonacci numbers defined by for all Furthermore, another regular matrix of Fibonacci numbers is defined by Debnath and Saha [1] as follows: for all By using this matrix, Debnath and Saha [1] and Ercan and Bektas [2] defined and studied the matrix domains , and More studies concerning construction of Banach sequence spaces involving Fibonacci numbers can be tracked in the literature that are generalization or extension of any one of the above discussed Fibonacci sequence spaces. We refer to [5, 6, 8, 9], for such studies.

The number sequence defined by the recurrence relation with and is called tribonacci sequence. Recently, Yaying and Hazarika [10] introduced tribonacci sequence spaces and where is an infinite matrix of tribonacci numbers defined by for all Quite recently, Yaying and Kara [11] studied the matrix domains and Moreover, Yaying et al. [12] studied Banach sequence spaces defined by the sequence of Padovan numbers . Besides, A. M. Karakas and M. Karakas [13] also constructed -sequence spaces defined by using Lucas numbers .

1.2. Leonardo Numbers

The number sequence is termed as Leonardo sequence. Let denote the Leonardo number. Then, the Leonardo numbers are defined by the following recurrence relation:

It is believed that Leonardo sequence is invented by Leonardo de Pisa, also known as Leonardo Fibonacci. But not much studies related to Leonardo numbers can be traced in the literature due to scarcity of research related to this integer sequence. Leonardo sequence has a very close relationship with the well-known Fibonacci sequence and the Lucus sequence :

Quite recently, Catarino and Borges [14] studied basic properties of Leonardo numbers and established several interesting identities, some of which are listed below:

Besides, Alp and Koçer [15] also established interesting relationships between Fibonacci, Lucus, and Leonardo numbers. Vieira et al. [16] worked in the matrix form of the Leonardo numbers and established several interesting relations. Moreover, Shannon [17] also worked on the extension and generalization of the Leonardo numbers.

Inspired by the above studies, we define an infinite matrix involving Leonardo numbers and construct sequence spaces and We study their topological and inclusion properties and obtain Schauder bases of the sequence spaces , and In Section 3, -, -, and -duals of these new spaces are determined. In Section 4, matrix classes from the space to any one of the spaces , and are characterized. In Sections 5 and 6, we introduce new sequence spaces and under a definite functional and weighted sequence of positive reals and discuss certain geometric and topological properties of and the eigenvalue distribution of mappings ideally generated by these spaces and -numbers are presented.

2. Leonardo Sequence Spaces

Define an infinite matrix by for all Equivalently,

The inverse of the matrix is given by the matrix defined by for all

Now, we define the following sequence spaces: where the sequence defined by for each which is known as the -transform of the sequence In what follows, the sequences and are related by (12). It is trivial that the above defined sequence spaces can be expressed in the form where represents any one of the spaces , and That is, is the domain of the matrix in the sequence space

We observe by the definition of the matrix that That is, and . Additionally, for each Thus, we conclude that the matrix is regular.

Theorem 1. The following inclusion relations hold: (i) where is any one of the spaces or (ii)(iii) for

Proof. (i)The inclusion part is trivial. Assume that and consider the sequence We observe that Howeverwhich converges. Thus, In the similar manner strictness can be established for the other inclusions. (ii)It is known that the matrix is regular and the inclusion holds. These imply that the inclusion part holds. Now, consider the sequence . Then, for all Thus, That is, This verifies the strictness of the inclusion In the similar fashion, strictness of other inclusions can be established.(iii)Assume that Since is regular and the inclusion holds, therefore the desired inclusion holds. To prove the strictness part, we consider a sequence Define a sequence by Then, we getfor each where the terms with negative subscripts are considered to be zero. Thus, we deduce that which implies Thus there exists at least one sequence that is contained in but not in Hence, the desired inclusion is strict. This completes the proof.

Theorem 2. We have the following results: (i)The sequence spaces and are -spaces equipped with the bounded norm (ii)The sequence space is a -space equipped with the norm

Proof. The proof is a routine exercise and so omitted.

Theorem 3. where is any one of the spaces , or

Proof. We present the proof for the space Define the mapping by for all We observe that the mapping is linear and injective.
In view of the relation (12), we write for each and Then, Thus, Thus, , and this implies that is surjective and norm preserving. Thus, In the similar manner, we can prove the existence of isomorphism between other given spaces. This completes the proof.

Let us consider the following sequences:

Observe that and Since is linear, so and With some elementary calculation, we deduce that

Thus, we realize that the norm violates the parallelogram identity for This immediately allows us to write the following result.

Theorem 4. The sequence space is not a Hilbert space for

Proof. The proof is immediate from the above discussion.

We are well awarded that a matrix domain where is a triangle, has a basis, if and only if, has a basis (cf. [18]). Thus, in the light of Theorem 3, we have the following result:

Theorem 5. Define the sequence byfor each fixed Then, (i)The sequence is the Schauder basis of the sequence spaces and and every in or is expressed uniquely in the form where is the -transform of the sequence (ii)The sequence is the Schauder basis of the sequence space and every in is expressed uniquely in the form where and is the unit sequence(iii)The sequence space has no Schauder basis

Corollary 6. The sequence spaces and are separable spaces.

3. -, -, and -Duals

In this section, we obtain the -, -, and -duals of the sequence spaces and Before proceeding, we recall the definitions of -, -, and -duals. Define the multiplier sequence space by

In particular, if is or then, the sets are, respectively, termed as -, -, and -dual of the sequence space

We present Lemma 7 which is essential to compute the dual spaces. In what follows, we denote the collection of all finite subsets of by and .

Lemma 7 (see [19]). The following statements hold: (i), if and only if, (ii), if and only if (iii), if and only if, (22) holds with (iv), if and only if, (v), if and only if, (vi), if and only if (23) holds with

Theorem 8. Consider the following sets: Then, where

Proof. We observe that for each and where the matrix is defined by for all . We notice that the sequence whenever , if and only if, the sequence whenever the sequence We realize that the sequence , if and only if, Thus, by employing Part (iii) of Lemma 7, we get that Moreover, for any (27) holds, if and only if, Consequently,
In the similar manner, -dual of the other sequence spaces can be obtained by employing Part (i), Part (ii), and Part (iii) of Lemma 7.

Theorem 9. Consider the following sets: where Then, , and where

Proof. Let Then, we have for each By employing Theorem 2 and Corollary 1 of Malkowsky and Sava [20], we get This completes the proof.

Theorem 10. We have the following results: (i)(ii) where (iii)

Proof. It can be obtained by using relation (30) and Parts (iv), (v), and (vi) of Lemma 7, respectively.

4. Characterization of Matrix Classes

Let be an infinite matrix over the field of complex numbers. Denote

Now, we state the following result:

Lemma 11. Let denote either of the spaces or Then, , if and only if, for each and where and are defined in (32) and (33), respectively.

Proof. It follows straightly from ([20], Theorem 3]).

Lemma 12. , if and only if,

Proof. Assume that Then, which immediately shows the necessity of condition (35). Also, it is known that Hence, by employing Lemma 11, we get that Thus, in the light of (30), we get that It is clear from the assumptions that and These together yield
Conversely, we assume that conditions (34), (35), and (36) hold. We realize that conditions (34) and (35) together imply that Again condition (34) implies (37). By condition (35), , for all This together with condition (36) implies that , for all This proves that

Now, using Lemmas 11 and 12 together with the properties and we deduce the following results:

Corollary 13. The following statements hold: (i), if and only if (ii)Let Then , if and only if, (38) holds, and (iii), if and only if, (38) holds, and (40) holds with (iv), if and only if, (34) and (40) hold with , and also holds(v), if and only if, and (40) holds with

Corollary 14. The following statements hold: (i), if and only if, (38) and (39) hold, and also holds(ii)Let Then, , if and only if, (38), (40), and (43) hold(iii), if and only if, (38) and (40) hold with , and (43) also holds(iv), if and only if, (34), (40) with and (43) hold, and also holds(v), if and only if, (42) holds, and also holds

Corollary 15. The following statements hold: (i), if and only if, (38), (39), and also holds(ii)Let Then , if and only if, (38), (40) and (46) holds(iii), if and only if, (38) holds, (40) with and (46) hold(iv), if and only if, (34), (40) with and (46) hold, and also holds(v), if and only if, (42) and (46) hold, and also holds

Corollary 16. The following statements hold: (i), if and only if, (38) holds and also holds(ii)Let Then , if and only if, (38) holds, and also holds(iii), if and only if, (38) holds, and (50) holds with (iv), if and only if, (34) and (50) hold with , and also holds(v), if and only if, (42) and (50) hold with

5. Mapping Ideal

In this section, we construct -type mapping ideals on Leonardo sequence spaces and By we denote the class of all bounded linear mappings between any two Banach spaces. In particular, denote the class of all bounded linear mappings acting from Banach space to Banach space We note down certain notations and definitions before moving to our results:

Definition 17 (see [21, 22]). Let represent the set of nonnegative real sequences. Then, -number is a mapping that satisfies the following settings: (i) for each (ii) for each and (iii) for all and where and are any two Banach sequence spaces(iv)Let and Then, (v)If then for all (vi) for or for where denotes the identity mapping on the -dimensional Hilbert space In an assorted illustration of -numbers, we intimate the next settings: (1)The -th Kolmogorov number, denoted by , is defined as (2)The -th approximation number, denoted by , is defined as

Definition 18 (see [23]). Let and denote Then, is known as a mapping ideal if it satisfies the following settings: (i) where is a Banach sequence space of one dimension(ii) is a linear space over (iii)If and then where and are any two normed spaces

Definition 19 (see [24]). A prequasi norm on the ideal is a mapping satisfying the following settings: (i) and if and only if for all (ii)There exists such that for all (iii)There exists such that for all (iv)There exists such that whenever and

Definition 20 (see [24]). The subspace is said to be a private sequence space (or in short ) if it satisfies the following settings: (i), for each where denotes the sequence with in the position and elsewhere(ii)If and for then (iii) whenever where denotes the integral part of

Definition 21 (see [24]). A subspace of the is said to be a premodular , if there is a function satisfying the following conditions: (i)For every , , and , with is the zero vector of (ii)If and , then there are with (iii) holds for some , with (iv)Assume , , we have (v)The inequality, verifies, for (vi), where denotes the closure of the space of all sequences with infinite zero coordinates(vii)We have such that , with

Definition 22 (see [24]). The is said to be a prequasi normed , if confirms the setups (i)-(iii) of Definition 21. If is complete equipped with , then is called a prequasi Banach .

Lemma 23 (see [24]). Every premodular is a prequasi normed
In what follows, we will use the following inequality: where and For detailed studies concerning -numbers and mapping ideals, we refer to [23–28].

Definition 24. We define the following sequence spaces: where and .
By and , we will denote the space of all monotonic increasing and decreasing sequences of positive reals, respectively.

Theorem 25. is a , whenever or , and there exists such that .

Proof. (i)Let , we obtain Thus,
Assume that and Then, we have Thus is a linear space. Moreover This implies , for each (ii)Assume that for all and Then, we have This concludes that (iii)Let , and there exists such that . Then, we have Thus,
This completes the proof.

Theorem 26. is a , whenever or , and there exists such that .

Proof. (i)Let By using (54), we obtain Thus,
Assume that and Then, we have Thus, is a linear space. Moreover, This implies for each (ii)Assume that , for all and Then, we have This concludes that (iii)Let , , and there exists such that . Then, we have Thus,
This completes the proof.

Define the sets , , and by where and are any two Banach sequence spaces. We denote , , and , respectively.

Lemma 27 (see [24]). Let the linear sequence space be a Then, is a mapping ideal.

Theorem 28. Let or , and there exists such that . Then, is a mapping ideal.

Proof. It follows straightly from Lemma 27.

Theorem 29. Let or , and there exists such that . Then, is a mapping ideal.

Proof. It follows straightly from Lemma 27.

Theorem 30. Let or , and there exists such that . Then, is a premodular

Proof. (i)Clearly, for all that and , if and only if, (ii)For any Then , for all and (iii)Observe that , for all (iv)We have , whenever (see Proof Part (ii), Theorem 25).(v)It is immediate from Proof Part (iii) of Theorem 25 that with . (vi)We have, when then , for and when then This completes the proof.

Theorem 31. Let or , and there exists such that . Then, is a premodular

Proof. (i)Clearly, for all that and , if and only if, (ii)Let Then, , for all and (iii)Observe that , for all (iv)We have , whenever (see Proof Part (ii), Theorem 26).(v)It is immediate from Proof Part (iii) of Theorem 26 that with (vi)We have, when then , for , and when then This completes the proof.

Theorem 32. Assume that or , and there exists such that . Then, is a prequasi Banach

Proof. In view of Theorem 30 and Lemma 23, it is enough to prove that every Cauchy sequence in is convergent in We assume that is a Cauchy sequence in Then, for all there exists such that for all This implies that , for all Thus, is a Cauchy sequence in Since is complete, , for a fixed This yields, by using (69), that , for all Besides, we have This concludes that Thus, is a prequasi Banach

Theorem 33. Assume that or, and there exists such that . Then, is a prequasi Banach

Proof. In view of Theorem 31 and Lemma 23, it is enough to prove that every Cauchy sequence in is convergent in We assume that is a Cauchy sequence in Then, for all there exists such that for all This implies that , for all Thus, is a Cauchy sequence in Since is complete, , for a fixed This yields, by using (70), that , for all Besides, we have This concludes that Thus is a prequasi Banach

Theorem 34 (see [27]). Suppose type If is a mapping ideal, then the following conditions are verified: (1)(2)Suppose and ; then (3)Assume and ; then (4)The sequence space is solid; i.e., if and , for all and ; then In view of Theorem 34, we construct the next properties of the and the .

Theorem 35. Let The next conditions are established: (1)One has (2)Suppose and ; then (3)Assume and ; hence (4)The is solid

Theorem 36. Let The next conditions are established: (1)One has (2)Suppose and ; then (3)Assume and ; hence (4)The is solid

6. Characteristics of the Prequasi Ideal

The conventions listed below will be followed throughout the article; if the species is preowned, we will give it to you.

Conventions 1. Please see the following conventions:
the ideal of finite rank mappings between any arbitrary Banach spaces
the ideal of approximable mappings between any arbitrary Banach spaces
the ideal of compact mappings between any arbitrary Banach spaces
the space of finite rank mappings from a Banach space into a Banach space
the space of finite rank mappings from a Banach space into itself
the space of approximable mappings from a Banach space into a Banach space
the space of approximable mappings from a Banach space into itself
the space of compact mappings from a Banach space into a Banach space
the space of compact mappings from a Banach space into itself

Lemma 37 (see [28]). If and , then there are operators and so that , for all .

Definition 38 (see [28]). A Banach space is called simple if the algebra includes one and only one nontrivial closed ideal.

Theorem 39 (see [28]). Suppose is a Banach space with ; then

In this section, firstly, we introduce the enough setups (not necessary) on and such that and . This investigates a negative answer of Rhoades [29] open problem about the linearity of and spaces. Secondly, for which conditions on and , are and closed and complete? Thirdly, we explain the enough setups on and such that and are strictly contained for different weights and powers. We offer the setups so that is minimum. Fourthly, we introduce the conditions so that the Banach prequasi ideal and are simple Banach spaces. Fifthly, we investigate the enough conditions on and such that the space of all bounded linear operators which sequence of eigenvalues in and equal and , respectively.

6.1. Finite Rank Prequasi Ideal

Theorem 40. ; suppose the setups or , and there exists such that are satisfied. But the converse is not necessarily true.

Proof. To investigate that , as for every , is a linear space. Let , one gets . To explain that , assume we obtain . Since , let ; hence, there is with , for some . Since , we get Hence, there is so that rank and we have Therefore, one has In view of inequalities (72)–(75), and , one gets On the opposite side, one has a negative example as where . This shows the proof.

Theorem 41. ; suppose the setups or, and there exists such that are confirmed. But the converse is not necessarily true.

Proof. To investigate that , as for every , is a linear space. Let ; one gets . To explain that , assume ; we obtain . Since , let , hence, there is with , for some , where Since , we get Hence, there is so that rank and Since , we have Therefore, one has In view of inequalities (54), (77)–(80), and , one gets On the opposite side, one has a negative example as where . This shows the proof.

6.2. Banach and Closed Prequasi Ideal

Theorem 42 (see [24]). The function is a prequasi norm on , where , for every , if is a premodular .

Theorem 43. If the setups or, and there exists such that with are satisfied, then is a prequasi Banach ideal, where .

Proof. As is a premodular , hence from Theorem 42, is a prequasi norm on . Suppose is a Cauchy sequence in . As , one obtains Hence, is a Cauchy sequence in . Since is a Banach space, then there is with Since every . According to Definition 21 setups (ii), (iii), and (v), one gets Therefore, ; then .

Theorem 44. If the setups or, and there exists such that with are confirmed; then is a prequasi Banach ideal, where .

Proof. As is a premodular , hence from Theorem 42, is a prequasi norm on . Suppose is a Cauchy sequence in . As , one obtains Hence, is a Cauchy sequence in . Since is a Banach space, then there is with Since every . According to Definition 21 setups (ii), (iii), and (v), one gets Therefore, ; then .

Theorem 45. Assume , are normed spaces; the setups or ; and there exists such that with are satisfied; then is a prequasi closed ideal, where .

Proof. As is a premodular , by using Theorem 42, is a prequasi norm on . Assume , every and . As , we have Hence, is a convergent sequence in . Since for every . In view of Definition 21 setups (ii), (iii), and (v), one has We get , so .

Theorem 46. Assume , are normed spaces; the setups or ; and there exists such that with are satisfied; hence, is a prequasi closed ideal, where .

Proof. As is a premodular , by using Theorem 42, is a prequasi norm on . Assume , for every and . As , we have Hence, is a convergent sequence in . Since every . In view of Definition 21 setups (ii), (iii), and (v), one has We get , so .

6.3. Minimum Prequasi Ideal

Theorem 47. Suppose and are Banach spaces with , and the setups or , and there exists such that are confirmed with for all , hence

Proof. Let ; then . One obtains Then, . Next, if we choose with and , one gets such that and .
Clearly, . Next, if we put such that . We have such that . This explains the proof.

Theorem 48. Suppose and are Banach spaces with , and the setups or, and there exists such that are confirmed with and for all ; hence,

Proof. Let ; then . One obtains Then . Next, if we choose with , one gets such that Therefore, and .
Clearly, . Next, if we put such that . We have such that . This explains the proof.

Theorem 49. Let and be Banach spaces with , and the setups or , and there exists such that are established with ; then, is minimum.

Proof. Suppose the enough setups are confirmed; then , where is a prequasi Banach ideal. Suppose ; hence, there is with for every . According to Dvoretzky’s Theorem [23], for every , one obtains quotient spaces and subspaces of which can be mapped onto by isomorphisms and with and . Let be the identity operator on and be the quotient operator from onto , and is the natural embedding operator from into . Suppose is the Bernstein numbers [26]; then for . We have Hence, for some , one gets Therefore, we have a contradiction, if . Then, and both cannot be infinite dimensional if . This shows the proof.

By the same manner, we can easily conclude the next theorem.

Theorem 50. Let and be Banach spaces with , and the setups or , and there exists such that are established with ; then is minimum.

6.4. Simple Banach Prequasi Ideal

Theorem 51. Suppose and are Banach spaces with , and the setups or , and there exists such that are confirmed with , for all ; then

Proof. Let and . In view of Lemma 37, there are and with . Therefore, for every , we get This contradicts Theorem 47. Then , which finishes the proof.

Corollary 52. Assume and are Banach spaces with , and the setups or , and there exists such that are established with , for all ; then

Proof. Clearly,.

Theorem 53. Suppose and are Banach spaces with , and the setups or , and there exists such that are confirmed with and, for all ; then

Proof. Let and . In view of Lemma 37, there are and with . Therefore, for every , we get This contradicts Theorem 48. Then , which finishes the proof.

Corollary 54. Assume and are Banach spaces with , and the setups or , and there exists such that are established with and , for all ; then

Proof. Clearly,.

Theorem 55. Suppose and are Banach spaces with , and the setups or , and there exists such that are confirmed with and , for all , then

Proof. Let and . In view of Lemma 37, there are and with . Therefore, for every , we get This contradicts . Then , which finishes the proof.

Theorem 56. Let and be Banach spaces with , and the setups or , and there exists such that are satisfied; hence, is simple.

Proof. Assume the closed ideal includes an operator . In view of Lemma 37, we have with . This gives that . Then, . Hence, is simple Banach space.

Theorem 57. Let and be Banach spaces with , and the setups or , and there exists such that are satisfied; hence, is simple.

Proof. Assume the closed ideal includes an operator . In view of Lemma 37, we have with . This gives that . Then, . Hence, is simple Banach space.

6.5. Eigenvalues of S-Type Operators

Conventions 2. Please see the following conventions:

Theorem 58. Let and be Banach spaces with , and the setups or , and there exists such that are verified with ; then

Proof. Let ; hence, and , for all . We have , for all ; hence, , for every . Therefore, ; then .
Secondly, suppose . Then . Hence, we have Therefore, Assume exists, for every . Hence, exists and bounded, for every . Then, exists and bounded. As is a prequasi operator ideal, we get So we have a contradiction, since . Hence, , for every . This gives . This shows the proof.

Theorem 59. Let and be Banach spaces with , and the setups or , and there exists such that are satisfied with ; then

Proof. Assume ; hence, and , for all . We have , for all ; hence, , for every . Therefore, ; then .
Secondly, suppose . Then . Hence, we have Therefore, Assume exists, for every . Hence, exists and bounded, for every . Then, exists and bounded. As is a prequasi operator ideal, we get So we have a contradiction, since . Hence, , for every . This gives . This shows the proof.

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 to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

The research of the first author (T. Yaying) is supported by SERB, DST, New Delhi, India, under grant number EEQ/2019/000082. This work was funded by the University of Jeddah, Saudi Arabia, under grant number UJ-21-DR-92). The third and fourth authors, therefore, acknowledge with thanks the university’s technical and financial support.

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Copyright © 2022 Taja Yaying 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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