Making Decisions about How to Solve Nonlinear Dynamical Systems of the Kannan Type in a New Complex Function Space of Formal Power Series

Alsolmi, Meshayil M.; Mustafa, Arafa O.; Mohammed, Mustafa M.; Bakery, Awad A.

doi:https://doi.org/10.1155/2023/9672865

Journal of Mathematics

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Research Article | Open Access

Volume 2023 | Article ID 9672865 | https://doi.org/10.1155/2023/9672865

Making Decisions about How to Solve Nonlinear Dynamical Systems of the Kannan Type in a New Complex Function Space of Formal Power Series

Meshayil M. Alsolmi,¹Arafa O. Mustafa,^2,3Mustafa M. Mohammed ,^1,4and Awad A. Bakery^1,5

Academic Editor: Ji Gao

Received18 Aug 2022

Revised07 Sept 2022

Accepted15 Dec 2022

Published31 Jan 2023

Abstract

The sufficient settings of the space generated by absolute type-weighted gamma matrices of rank in the Nakano complex functions of the formal power series, as well as its associated prequasioperators’ ideal equipped with definite functions, are defined and explained in this paper. Assorted prequasinorms are shown to have the Fatou characteristic. Some of its geometric and topological features and the prequasioperators’ ideal that goes with it are talked about. These structures have a relationship with the fixed points of the Kannan contraction and nonexpansive mappings. By looking at real-world examples and how they are used, it is shown that there are solutions for nonlinear dynamical systems of the Kannan type.

1. Introduction

Since the book [1] on the subject came out, many researchers have looked into possible extensions to the Banach fixed point theorem (BFPT) and how it can be used. The nonlinear analysis places significant weight on the Banach contraction principle (BCP), a potent instrument for nonlinear analysis. For more interesting articles on various real-world applications of the BCP, see the following: Diening et al. [2] discussed Lebesgue and Sobolev spaces with variable exponents, Ruẑiĉka [3] gave a mathematical description of electrorheological fluids, the types of approaches to impartial probabilistic functional differential equations that are motivated by pure leaps were described by Mao et al. [4], Younis et al. [5] presented generalized contractions, Guo and Zhu [6] investigated the fixed point approach of stochastic Volterra–Levin equations with Poisson jumps, Ahmad et al. [7] provided dual partial metric type spaces, as well as converging findings, Beg [8] applied FPT to ordered uniform convexity in ordered convex metric spaces, Yang and Zhu [9] examined the kind of solutions to stochastic neutral functional differential equations of Sobolev-type, and Beg et al. [10] discussed about the polytopic fuzzy sets to multiple-attribute decision-making problems. For more interesting articles on different real-world applications of fractional calculus, see the following: Rezapour et al. [11] discussed a mathematical analysis of a system of Caputo–Fabrizio fractional differential equations for the anthrax disease model in animals, Mohammadi et al. [12] investigated a theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to the Mumps virus with optimal control, Khan et al. [13] suggested a case study of a fractal-fractional tuberculosis model in China: existence and stability theories along with numerical simulations, Etemad et al. [14] studied some novel mathematical analyses on the fractal-fractional model of the AH1N1/09 virus and its generalized Caputo-type version, Matar et al. [15] proposed using generalized Caputo fractional derivatives to investigate the -Laplacian nonperiodic nonlinear boundary value problem, Baleanu et al. [16] gave a new study on the mathematical modeling of the human liver with Caputo-Fabrizio fractional derivative, Tuan et al. [17] presented a mathematical model for COVID-19 transmission by using the Caputo fractional derivative, Baleanu et al. [18] used a new fractional derivative approach to analyze the model of HIV-1 infection of T-cells, and Kannan [19] illustrated a new category of discontinuous contraction operators. For the acting of Kannan mappings on modular vector spaces, prequasi-normed spaces, and prequasi-normed spaces of nonabsolute, see [20] and [21, 22], respectively. As a generalisation of Cesàro sequence space of the variable exponent, we have defined the space generated by absolute type-weighted gamma matrices of rank in the Nakano complex functions of the formal power series , , where is defined as follows:where is a positive integer, , for all , where , and .

In [23], Roopaei and Başar examined , where and . When , so that , and , for all .

The space is denoted by the following equation:where is the set of complex numbers and . For further information on formal power series spaces and their associated behaviors, see the following: Shields [24] investigated weighted shift operators, Hedayatian [25] discussed cyclicity in the space , Emamirad and Heshmati [26] examined chaotic weighted shifts in Bargmann spaces, and Faried et al. [27] offered -numbers of shift mappings. In functional analysis, the mappings’ ideal hypothesis has earned much respect. The application of -numbers is a fundamental method, see [28, 29], [30], [31], and [32].

Lemma 1 (see [33]). If and , where is the set of real numbers, for every , then where .

Since proving several more FPTs in a particular space necessitates either increasing the space itself or expanding rapidly the self-mapping that works on it, both of these options are good. In this article, we have constructed vast spaces of solutions to a variety of nonlinear summable and difference equations. The following is an outline of the objectives of this research: In Section 2, we investigate sufficient setups of equipped with definite function to form prequasi-Banach (ssfps). Some topological and geometric structures of are discussed, which are connected with the fixed point property. Section 3 explains the geometric and topological structure of the operators’ ideal generated by this function space and -numbers. In Section 4, we look at how to configure with different and its operators ideal so that there is only one fixed point of Kannan contraction mapping (Kcm). We give some illustrative examples to clarify our results. Specifically, in Section 5, we present the requirements of this function space with definite for the Kannan nonexpansive mapping to have a fixed point on it, and we do this by introducing it. Numerous examples and discussions of our findings, including their practical implementations, can be found in Section 6.

2. Structure of

In this section, we have examined some topological and geometric structures of , which are connected with the fixed points of Kcm.

The bounded sequence space of is indicated by .

Theorem 1. If , then

Proof. Obviously, is a bounded sequence.
Assume is the zero function of and

Definition 1 (see [34]). A mapping is called modular (m), where is a vector space if the following setups are satisfied:(a)Assume , then and (b)Suppose and , then (c)If and , then

Definition 2 (see [35]). The space is named a ssfps, when the next setups are confirmed:(1)Suppose and , with ; then, (2), for all , so that (3), if , where , and marks the integral part of

Definition 3 (see [35]). The function is named premodular (p-m) on , when the next parts are confirmed:(i)Assume , then and (ii)If and , one has with (iii)Suppose , one has with (iv)Assume , for all , then (v)One gets such that (vi)the closure of (vii)One obtains so that , for all Also, the space is a p-mssfps. If is complete, then is said to be a premodular Banach ssfps (p-mBssfps). P-m vector spaces are a more general idea than m vector spaces. Examples of p-m vector space and m vector space are as follows.

Example 1. The space , whereAn example of premodular vector space but not modular vector space is given.

Example 2. The space , where. As for every , one has

Definition 4 (see [35]). A subspace of the ssfps is called a prequasi-normed ssfps (p-qNssfps), if the function verifies the setups (i), (ii), and (iii) of Definition 2.4.
If is complete, then is called a p-qBssfps.

Theorem 2 (see [36]). All qNssfps are a p-qNssfps.

Example 3. The function is a qN and not a N on , for .

Example 4. The function is a p-qN (not a qN) on

Example 5. For , the function is a N on .

Theorem 3 (see [36]). Every p-mssfps is a p-qNssfps.

Definition 5. (i)Suppose , then is said to be -bounded(ii)A prequasinorm on verifies the Fatou property (Fa-p), whenever for all under and all , one has (iii)Denote for the -ball of radius and center Assume and are the space of all increasing and decreasing sequences of real numbers, respectively.

Theorem 4. , where , for all , is a p-mssfps, if the next setups are confirmed: (b1) with (b2) or and one has with

Proof. (i)Obviously, and . (1-i) Assume . One has with then .(ii)One gets with , for all . (1-ii) Assuming and , we obtain As , by setups (1-i) and (1-ii), one can see that is linear. Clearly, , for all , since .(ii)There is so that , for all and .(2)Assume , for all and . One finds then .(iv)It is clear from the proof part (2)(3)Considering and , we get then .(v)By using (3), one can find .(vi)It is obvious that the closure of .(vii)One finds , for or , for with .

Theorem 5. If the conditions of Theorem 4 are satisfied, then is a p-qBssfps, where , for every .

Proof. According to Theorems 4 and 3, the space is a p-qNssfp. Suppose is a Cauchy sequence (CS) in , then for every , one has . We have for every thatThis implies . Fix , then is a CS in . Hence, . One has , for all . As , then .

Theorem 6. The function satisfies the Fa-p, whenever the parts of Theorem 4 are satisfied, for all .

Proof. Let with . Since is a prequasi-closed space, we have . For every , one gets

Theorem 7. The function does not verify the Fa-p, for every , if the conditions of Theorem 4 are satisfied with .

Proof. Assume so that . Since is a prequasi-closed space, one gets . For every , we obtainWe will use that , for every , in the next part of this section. By and , we mean the unit sphere and the unit ball of , respectively.

Definition 6 (see [37]). A function ; assume for every , one has and withIf , for every so that depending on a, then .

Theorem 8 (see [37]). Assume , then for every and , there exists so that , , with and .

Theorem 9. Suppose the conditions of Theorem 4 are satisfied, then for every and we have so that , for every , with and .

Proof. By using Theorems 4 and 8, one has .

Definition 7 (see [38]). The function is strictly convex, (SC), whenever every with and , then .

Definition 8 (see [39]). A sequence , is called separated for some , whenever

Definition 9 (see [39]). A Banach space (BS) is named -nearly uniformly convex (-NUC), where , whenever one finds with any , so that one has. This meansNote that -NUC gives reflexivity.

Theorem 10. Suppose the conditions of Theorem 4 are satisfied with , then is-NUC.

Proof. Assume and , where so that . For all , suppose , where . As for all , . In view of the diagonal method, one gets of so that converges for all , . We obtain an increasing sequence of positive integers with . Hence, we have a sequence of positive integers with , such thatfor every . Fix , if from Theorem 9 one gets withIf and , as , for all , we have such that and . Define . According to inequality (18), we have . Suppose for and . From inequalities (18) and (19), is convex, for all ; then,So, is -NUC.
By fixing a -closed, -bounded, and -convex subset of .

Definition 10. The space verifies the R-pr, if and only if, for every decreasing sequence of nonempty -closed and -convex subsets of with , where , for some , one has.

Theorem 11. If the conditions of Theorem 4 are satisfied with , one has(i)Suppose with . There is a unique so that .(ii) holds the R-pr.

Proof. For (i), assume as is -closed. We find . Hence, for all , then with . If is not -Cauchy, one gets a and with , for any . Because offor any . Since with , one finds is strictly convex, for any . Therefore, the space is strictly convex,Then,for all . By putting , one has a contradiction. So is -Cauchy. As is -complete, then -converges to some . For all , one gets -converges to . Since is -closed and -convex, then . Since -converges to , then . Given Theorem 6 and assume , as verifies the Fa-p, one has the following:Then, . Since is (SC), the uniqueness of is established. To prove (ii), if , for some . Because of is increasing. Put . Suppose . Else , with . By (i), one finds a unique point with , for every . Clearly, -converges to some . As are -convex, decreasing and -closed, one has .

Definition 11 (see [40]). If is a BS, thenwhere

Definition 12. satisfies the -normal structure-property (NS-pr), if and only if, for any is not decreased to one point, then with .

Theorem 12 (see [41]). A reflexive BS with verifies the NS-pr.

Theorem 13. If the conditions of Theorem 4 are satisfied with , then satisfies the -NS-pr.

Proof. Suppose is an asymptotic equidistant sequence with and . Assume . One has with . As coordinate-wise, we have so that , for ; putting , we have so that . As coordinate-wise, we get so that . For , by induction, there exists a subsequence of so thatTake for So,For all so that , we have:which gives . Take , for Then, On the other hand, , for every with . Hence,Since , by using equations (30)–(32), we get , such that and is asymptotic equidistant.
Fix large enough with , where . One gets, for ,that is, . Note thatTherefore,for and . Hence, and, since this is true for all , we have . From Theorem 10 and Theorem 12, then the functions space has the -NS-pr.

3. Structure of Mappings’ Ideal

In this section, we have examined some topological and geometric structures of mappings’ ideal (MI) constructed by , where , for every and numbers, which are connected with the fixed points of Kcm.

Notations 1. (1) and are two infinite dimensional BSs. and are arbitrary BSs.(2) is the space of all bounded linear mappings from into . If , we write .(3) is the space of finite rank linear mappings from into .(4) is the space of approximable bounded linear mappings from into .(5) is the space of compact bounded linear mappings from into .(6) is the space of every bounded linear operators between arbitrary two BSs.(7) is the unit mapping on the -dimensional Hilbert space .

Definition 13 (see [42]). An operator is called an -number if it maps every to a unique that validates the following settings:(a)(b), for all and ,(c), for all ,, and (d)If and , then (e)Let rank , then , for each (f)orSome -number examples are as follows:(i)The -th approximation number is (ii)The -th Kolmogorov number is

Definition 14 (see [29]). A sub class of is called a MI, whenever any verifies the next parts:(1), where marks BS of one dimension(2)Assume ,, and , hence (3)The space is linear

Notations 2 (see [36]).

Theorem 14 (see [35]). The class is a MI whenever is an ssfps.
By Theorems 4 and 14, we have the following theorem.

Theorem 15. If the conditions of Theorem 4 are satisfied, then is a MI.

Definition 15 (see [43]). A function is called a p-qN on the ideal , if it verifies the following conditions:(1)If, and , if and only if, (2)We have so as to , for all and (3)There are so that , for each (4)One can find , when , and ; hence,

Theorem 16 (see [44]). Every qN on the ideal is a p-qN on the same ideal.

Theorem 17. If the conditions of Theorem 4 are satisfied, then is a p-qN on , so that , where and .

Proof. (1)Suppose , and , if and only if, , for every , if and only if, .(2)One has with , for every and .(3)One has so that for , hence there are with and . Therefore, for , we have so that(4)One can find , if , and , hence there is with . Then, for , one has the following equation:

Theorem 18. If the conditions of Theorem 4 are satisfied, one has , which is a p-qB MI.

Proof. Assume is a CS in . As ; hence, there is with for every ; then,where is a CS in . As is a BS, so there exists so that and since , for all and is a p-mssfps, hence one can seewe obtain ; hence, .

Theorem 19. , if the conditions of Theorem 4 are satisfied. But the converse is not necessarily true.

Proof. Because of , for all and the linearity of . Assume and rank , for , one gets with , one has . Therefore, the closure of . Assume , we have . As , assume , then there is with , for some , where . Since is decreasing, we haveHence, there is so that rank andSince the conditions of Theorem 4 are satisfied, we haveTherefore, one hasAs , hence , where . In view of inequalities (43-46), one hasTherefore, . Contrarily, one has a counter example as , but is not verified. That indicates the nonlinearity of type spaces, see open problem of Rhoades [45].

Theorem 20. Suppose the conditions of Theorem 4 are satisfied with , and , for all , hence

Proof. Let , hence , where . One getsThen, , this implies . After, if we choose with , we have such thatThen, and .
Clearly, . Next, if we put with . We have such that .

Theorem 21. Assume the conditions of Theorem 4 are satisfied with , hence is minimum.

Proof. Let , one has so that , where , for every . In view of Dvoretzky’s theorem [46], with , we get quotient spaces and subspaces of which can be transformed onto by isomorphisms and with and . If is the identity mapping on , is the quotient map from onto and is the natural embedding map from into . Suppose is the Bernstein numbers [28], one hasfor . Then, we haveThen, we obtain , henceIf , we have a contradiction. Hence, and both cannot be infinite dimensional if . □
We can show the next theorem.

Theorem 22. If the conditions of Theorem 4 are satisfied with, hence is minimum.

Lemma 2 (see[29]). If and , then and with , for .

Theorem 23 (see [29]). In general, we have

Theorem 24. Let the conditions of Theorem 4 be satisfied with , and , for all , hence

Proof. Assume and . By using Lemma 2, we have and so that , for , thenThat fails Theorem 20. So, .

Corollary 1. Let the conditions of Theorem 4 be satisfied with , and , for all , hence

Proof. Evidently, .

Definition 16. (see [29]). A BS is called simple if there is a unique nontrivial closed ideal in .

Theorem 25. If the conditions of Theorem 4 are satisfied, hence is simple.

Proof. Let and . From Lemma 2, there exist with , which gives that . Then, , hence is a simple BS. □

Notations 3.

Theorem 26. Assume the conditions of Theorem 4 are satisfied with , hence

Proof. Let , hence , where and , with . We have , for all , sowith . One gets , hence . Next, suppose . Hence, , one gets the following equation:Then, . If exists, with . Then, exists and bounded, for all . So, exists and bounded. Since is a prequasi-MI, we have the following equation:where . This gives a contradiction, as . Therefore, , with , which explains .

4. Kannan Contraction Mapping

We look at how to configure with different and its operators ideal so that there is a unique fixed point of Kcm in this section. We also give some illustrative examples to clarify our results.

Definition 17. A mapping is called a Kannan -contraction (K-c), if there is such that , for every . The mapping is named Kannan -nonexpansive (K-NE), when .
A function is named a fixed point of , if .

Theorem 27. If the conditions of Theorem 4 are satisfied, and is K-cm, where ,for all , then has a unique fixed point.

Proof. If , one has . As is a K -cm, we haveFor with , thenThen, is a CS in . Since is prequasi-BS, we obtain with to show that . As has the Fa-p, we getThen, . So, is a fixed point of . To prove the uniqueness, suppose are two not equal fixed points of . We haveSo, .

Corollary 2. Assume the settings of Theorem 4 are satisfied, and is K-cm, where , for all , one has as the unique fixed point so that .

Proof. By Theorem 27, we get a unique fixed point of . Therefore,

Example 6. Assume , where , for every andAs for each with , one hasFor all with , one hasFor all with and , we getSo, is K -c because satisfies the Fa-p. From Theorem 27, one has that holds one fixed point .

Definition 18. Assume is a prequasi-normed (ssfps), and . The mapping is called -sequentially continuous (-SC) at , if and only if, when , then .

Example 7. Suppose , where , for every and is -SC and discontinuous at.

Example 8. Assume is defined as in Example 6. Suppose is such that , where with .
As the p-qN is continuous, we haveSo, is not -SC at .

Theorem 28. Suppose the parts of Theorem 4 are satisfied under and , where , for all . Suppose(1) is K-cm(2) is -SC at (3)There is with has converging to Then, is the unique fixed point of

Proof. If is not a fixed point of , one has . By parts (2) and (3), one can seeSince is K -c, we getSince , we obtain a contradiction. Then, is a fixed point of . To prove the uniqueness. Suppose are two not equal fixed points of . We getHence, .

Example 9. Assume is defined as in Example 6. Let , for all . Since for all with , one getsFor all with , one getsFor all with and , one getsSo, is K -c and
Obviously, is -SC at and holds converges to .By Theorem 28, the point is the only fixed point of .
Here, the construction of under definite function is presented such that there is a unique fixed point of Kcm.

Definition 19. A p-qN on the ideal verifies the Fa-p if for all with and , one gets

Theorem 29. Suppose the conditions of Theorem 4 are satisfied, then does not hold the Fa-p.

Proof. Assuming that with . Since is a prequasi-closed ideal, then . So for every , one has

Definition 20. An operator is said to be -sequentially continuous at , where , if and only if, .

Example 10. If , where , for every andObviously, is at the zero operator . Let be such that , where with . As is continuous, we haveHence, is not at .

Definition 21. An operator is called a Kannan -contraction , if one has , so that , for all .

Theorem 30. Assume the conditions of Theorem 4 are satisfied and , where , for every . The vector is the unique fixed point of , if the next setups are fulfilled:(a) is mapping(b) is at a point (c)There is such that the sequence of iterates has a subsequence that converges to

Proof. Let enough setups be satisfied. Assume is not a fixed point of , then . Because of conditions (b) and (c), one hasAs is K -cm, we getFor , we have a contradiction. Hence, is a fixed point of . To prove that the fixed point is unique, assume we have two distinct fixed points of . Therefore, one getsThen, .

Example 11. Suppose , where , for each andSince for all with , we haveFor all with , one hasFor all with and , one getsTherefore, the operator is and .
Evidently, is at and has a subsequence converges to . According to Theorem 30, the zero operator is the only fixed point of . Assume is such that , where with . Since the p-qN is continuous, we haveHence, is not at . Therefore, the operator is not continuous at .

5. Kannan Nonexpansive Mapping on

We introduce the sufficient conditions of , where , for every , such that the KNEM on it has a fixed point.

Lemma 3. Suppose verifies the R-pr and the -qNSP. Let be a K-NEM. When , with , putThen, , -convex, -closed subset of ,

Proof. Since , then . As the -balls are -convex and -closed, then is a -closed and -convex subset of . To show that , assume . When , one has . Else, assume . PutAccording to the definition of , one gets . Therefore, , then . Let . One has with . So,As is an arbitrary positive, one obtains , then . Since , one gets , so is -invariant. To show that , sincefor all , let . Then, . The definition of gives . Therefore, . One has , for all , so .

Theorem 31. If holds the -qNSP and the R-pr. Let be a K-NEM. Then has a fixed point.

Proof. Let and , for all . By the definition of , one gets , with . Suppose is defined as in Lemma 3. Clearly, is a decreasing sequence of nonempty -bounded, -closed, and -convex subsets of . The R-pr investigates that . Considering, one has , with . Letting , hence , then . So, . Hence, . Else, , so fails to have a fixed point. Suppose is defined in Lemma 3. Since fails to have a fixed point and is -invariant, hence has more than one point, then . From the -qNSP, one has withfor all . From Lemma 3, we get . From definition of , so . Then,which contradicts the definition of . Then, which implies that any point in is a fixed point of .

According to Theorems 11, 13 and 31, we conclude the following.

Corollary 3. If the conditions of Theorem 4 are satisfied with , and is a K-NEM, then has a fixed point in .

Example 12. Assume with where and , for every . By using Example 6, is Kannan -contraction. So it is -NEM. By Corollary 3, holds a fixed point in .

6. Existence of Solutions of Nonlinear Difference Equations

In this section, we explore a solution in to summable equations say (96), defined in [47], where the conditions of Theorem 4 are satisfied and , for all .

We examine the summable equations:

If is defined as follows:

Theorem 32. Summable equation (96) holds an unique solution in , when ,,,, assume there is so that and for all , we have

Proof. Let the conditions be established. Assume the mapping is defined by equation (97). Hence,By Theorem 27, we have a unique solution of (96) in .

Example 13. Consider , where, for all .
Assume the nonlinear difference equations:with , and supposeis defined byEvidently, there is such that and for all , one has:According to Theorem 32, the nonlinear difference (100) contains a unique solution in .

Example 14. Assume the nonlinear difference equations (102) and is defined as equation (102), where and , for all . Clearly, is a nonempty -bounded,-convex, and-closed subset of. By using Example 13, is K-c. So it is K-NEM. By Corollary 3, has a fixed point in .

Theorem 33. Consider the summable equations (96), and assume is defined by (97),where the conditions of Theorem 4 are satisfied with and , for all . The summable (96) has a unique solution , if the following conditions are satisfied:(1)Suppose ,,,, one has with and for all , one has(2) is -SC at .(3)There is with has converging to .

Proof. One hasBy Theorem 28, one gets a unique solution of equation (96).

Example 15. Consider , where , for all .
Let summable (100), defined by (101). Assume is -SC at , and there is with has converging to . Evidently, there is such that and for all , one hasBy Theorem 34, the summable equations (100) have a unique solution .
It is in this part that the solution of nonlinear matrix (107) at is investigated. Suppose the conditions of Theorem 4 are satisfied, and , for every . Consider the summable equationsIf is defined as follows:

Theorem 34. The summable equations (107) have a unique solution , if the following conditions are satisfied:(a),,,, and for all , one has with , and(b) is at .(c)We have so that has a converging to .

Proof. If defined by (111). We haveIn view of Theorem 30, we have a unique solution of (107) at .

Example 16. Assume the class , where , for all . Consider the nonlinear difference equations:where and and let be defined as follows:If is at , and one obtains with having a converging to , obviously,By Theorem 34, the nonlinear difference (111) has one solution .

7. Conclusion

We defined and presented sufficient conditions of the new complex function space equipped with the definite function to be prequasi-Banach in this paper. The Fa-p of various prequasinorms has been presented. Some topological and geometric structures in the new complex function space and their related prequasioperator ideals, which are linked to the fixed points of the contraction and nonexpansive of the Kannan mappings are offered. We introduce many possible solutions for various nonlinear dynamical and matrix systems. Our findings on the variable exponent in the previously outlined space have bolstered several well-known theories. As a future effort, we will extend this newly constructed complex function space to its nonabsolute type and solve several fractional nonlinear dynamical and matrix systems of Kannan type in this newly constructed complex function space and their associated prequasioperator ideals.

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

MA, AM, and MM conceptualized the study; MA, AM, and MM developed methodology; AB and MM validated the study; formal analysis was performed by MM and MA; MA and AM validated the study; resources were brought by MM and AM; MM and MA curated the data; MA, AM, and MM wrote the original draft; AB reviewed and edited the manuscript; MM, AM, and MA visualized the study; supervision was performed by AB and MM; and project administration was performed by MA, AM, and MM. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under grant no. UJ-20-111-DR. The second author, therefore, acknowledges the University of Jeddah for the technical and financial support.

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