Decision-Making on the Solution of a Stochastic Nonlinear Dynamical System of Kannan-Type in New Sequence Space of Soft Functions

Alsolmi, Meshayil M.; Bakery, Awad A.

doi:https://doi.org/10.1155/2022/9011506

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Unique and Non-Unique Fixed Points and their Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 9011506 | https://doi.org/10.1155/2022/9011506

Decision-Making on the Solution of a Stochastic Nonlinear Dynamical System of Kannan-Type in New Sequence Space of Soft Functions

Meshayil M. Alsolmi¹and Awad A. Bakery²

Academic Editor: Santosh Kumar

Received30 Jun 2022

Accepted01 Aug 2022

Published28 Aug 2022

Abstract

In this paper, we construct and investigate the space of weighted Gamma matrix of order in Nakano sequence space of soft functions. The idealization of the mappings has been achieved through the use of extended -soft functions and this sequence space of soft functions. This new space’s topological and geometric properties, the multiplication mappings that stand in on it, and the mappings’ ideal that correspond to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated prequasi ideal. Interestingly, several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of nonlinear difference equations of soft functions are introduced.

1. Introduction

Probability theory, fuzzy set theory, soft sets, and rough sets have all contributed substantially to the study of uncertainty. But there are drawbacks to these theories that must be considered. For more information and real-world examples, please refer to [1–10]. Numerous mathematicians have investigated potential expansions to the theorem and its applications in various contexts since the publication of the book [11] on the Banach fixed point theorem. The Banach contraction principle is an important part of nonlinear analysis, which uses it as a powerful tool [12–15]. Kannan [16] presented a collection of mappings with the same actions at fixed places as contractions. However, this collection is discontinuous. In Reference [17], an explanation of Kannan operators in modular vector spaces was once tried. Only this one try was ever made as [18–23] show that much attention has been paid to the -number mapping ideal and the multiplication operator hypothesis in functional analysis. Bakery and Mohamed [24] offered the idea of a prequasi norm on the Nakano sequence space with a variable exponent that fell somewhere in the range . They talked about the conditions that must be met to generate prequasi Banach and closed space when it is endowed with a specified prequasi norm and the Fatou property of various prequasi norms on it. They also determined a fixed point for Kannan prequasi norm contraction mappings on it, in addition to the ideal of prequasi Banach mappings derived from -numbers in this sequence space. Both of these ideals were established. In addition, several fixed point findings of Kannan nonexpansive mappings on generalized Cesàro backward difference sequence space of a nonabsolute type were discovered in [25]. Assume that is the set of real numbers and is the set of nonnegative integers. We denote the collection of all nonempty bounded subsets of by , and is the set of parameters. By and , we indicate the set of nonnegative and all soft real numbers (corresponding to ), where . The additive identity and multiplicative identity in are denoted by and , respectively. For more details on the arithmetic operations on , see [26]. Let , where , for all . Assume is defined by

Given that the proof of many fixed point theorems in a given space requires either growing the space itself or expanding the self-mapping that acts on it, both of these options are viable; we have constructed the space, , which is the domain of weighted Gamma matrix of order in Nakano soft sequence space since it is constructed by the domain of weighted Gamma matrix of order defined in , where the weighted Gamma matrix of order , , is defined as where is a positive integer, , for all and

In [27], Roopaei and Basar studied the Gamma spaces, including the spaces of absolutely -summable, null, convergent, and bounded sequences.

In this article, we have introduced a new general space called and the mappings’ ideal space of solutions for many stochastic nonlinear and matrix systems of Kannan contraction type. We have offered some geometric and topological structures of the soft function space, , multiplication operator acting on it, and its operators’ ideal. A fixed point of the Kannan contraction operator exists in this space, and its prequasi operator ideal is confirmed. Finally, we discuss many applications of solutions to nonlinear stochastic dynamical systems and illustrative examples of our findings.

2. Properties of and Its Operators’ Ideal

Some geometric and topological structures of the soft function space, , and its operators’ ideals are presented in this section.

By , , and , we denote the space of null, bounded, and -absolutely summable sequences of reals. We indicate the space of all bounded, finite rank linear mappings from an infinite-dimensional Banach space into an infinite-dimensional Banach space by and , and if , we write and . The space of approximable and compact bounded linear operators from into will be marked by and , respectively. The ideal of bounded, approximable, and compact mappings between every two infinite-dimensional Banach spaces will be denoted by , , and , respectively. Suppose is the class of all sequence spaces of soft reals.

Definition 1. If , is the space of all sequences of positive reals. The sequence space with the function is defined by

Lemma 2 (see [28]). If and , for all , and , then

Theorem 3. Suppose , then

Proof. Obviously, is a bounded sequence.

Theorem 4. The space is a nonabsolute type, whenever .

Proof. Clearly, since

Definition 5. Assume and , for all : where

Theorem 6. Suppose with hence .

Proof. Assume , as Then If we choose one gets and .
Suppose is a linear space of sequences of soft functions, and describes an integral part of the real number .

Definition 7. The space is said to be a private sequence space of soft functions if it satisfies the next setups: (a1)For all , then , where , while displays at the place(a2)If , and , with , then (a3), whenever

Definition 8 (see [29]). An -number is a function that gives all a holds the following conditions: (1)(2), for every , and , where and are arbitrary Banach spaces(3) for every and , (4)Assume and , then (5)If , then , for all (6) or , where indicates the unit mapping on the -dimensional Hilbert space Some examples of -numbers: (a)The th approximation number is defined as (b)The th Kolmogorov number is defined as

Notation 9 (see [30]).

Theorem 10. Assume the linear sequence space is a , then is an operator ideal.

Proof. (i)Assume and with , as for all and is a linear space, one has , for that then (ii)Suppose and then by Definition 7 condition (iii), one has and , as , by the definition of -numbers and is a decreasing sequence, we havefor each . In view of Definition 7 condition (ii) and is a linear space, one obtains , then (iii)If , , and , one has and as , by Definition 7 conditions (i) and (ii), one gets , hence Assume and is the space of finite sequences of soft numbers.

Definition 11. A subspace of the is called a premodular , if there is a function satisfies the next setups: (i)If , , and (ii)Assume and , one has so that (iii)There are so that , for all (iv)Assume , for all , then (v)One gets such that (vi)The closure of (vii)There are with

Definition 12. The is said to be a prequasi normed , if confirms the setups (i)-(iii) of Definition 11. The space is called a prequasi Banach , whenever is complete under .

Theorem 13. The space is a prequasi normed , whenever it is premodular . By and , we denote the space of all monotonic increasing and decreasing sequences of positive reals, respectively.

Theorem 14. is a prequasi Banach , if the next setups are confirmed: (f1) with (f2) or and there exists such that

Proof. First, we have to show that is a premodular . (i)Obviously, and (a1) and (iii) If , then hence (ii)Next, suppose , and as , we getwhere . So, .
As and , one obtains Therefore, , for every .
(a2) and (iv) If , for all and , then hence
(a3) and (v) Assume , with and we get where . Hence,
(vi) It is clear that the closure of
(vii)There are so that , for all and , if
By Theorem 13, the space is a prequasi normed . Second, to prove that is a Banach space, suppose is a Cauchy sequence in , hence for every , one has with , we have That implies As is a complete metric space. Therefore, is a Cauchy sequence in , for constant . So, it is convergent to . This implies , for every . Clearly, from condition (iii) that .
In view of Theorems 10 and 14, we have the next theorem.

Theorem 15. The space is an operator ideal, if the conditions of Theorem 14 are verified.

Theorem 16. If -type Assume is an operator ideal, one has the next setups: (a)-type (b)Suppose -type and -type , then -type (c)If and -type , one has -type (d)Suppose -type and , for all and , one gets -type , i.e., is a solid space

Proof. If is a mappings’ ideal.
(a)We have . Hence, for all , we have . This gives . Hence, -type
(b) and (c)The space is linear over . Hence, for each and , we have . That implies (d)If , , and , then . Therefore, since , then . Since . By using condition (c), if , we have . This means is solid
Some properties of type are presented in the next theorem according to Theorems 16 and 15.

Theorem 17. (a)-type (b)If -type and -type , then -type (c)Assume and -type , hence -type (d)-type is a solid space

Definition 18 (see [31]). A subclass of is said to be a mappings’ ideal, if every satisfies the following setups: (i), where indicates Banach space of one dimension(ii)The space is linear over (iii)If , , and , then

Definition 19 (see [32]). A function is said to be a prequasi norm on the ideal if the following conditions hold: (1)Assume , and , if and only if, (2)One has with , for all and (3)There are such that , for all (4)There are so that if , , and , then

Theorem 20 (see [32]). Every quasi norm on the ideal is a prequasi norm.
We have discussed some properties of the ideal constructed by our soft space and extended -numbers, supposing that the conditions of Theorem 14 are verified.

Theorem 21. The conditions of Theorem 14 are sufficient only for .

Proof. Clearly, from the linearity of the space and for all . Next, to show that . If one has . As , assume , we have so that , for some and Since , we get We get with rank and since , we have Therefore, one has Because of inequalities (5), (26), (27), (28), and (29), one gets On the other hand, one has a negative example as where for all and , but . This gives a negative answer to the Rhoades [33] open problem about the linearity of -type spaces.

Theorem 22. The class is a prequasi Banach ideal, where .

Proof. Evidently, is a prequasi norm on since is a prequasi norm on . Assume is a Cauchy sequence in . Since , we have then is a Cauchy sequence in . As is a Banach space, one has so that As for all . By Definition 11 conditions (ii), (iii), and (v), we have Hence, , so .

Theorem 23. If , and , for every , then

Proof. Let , then . One obtains then . Take with we have with Hence, and .
Clearly, . Take with Then and .
Recall that if and are infinite-dimensional, by Dvoretzky’s theorem [34], there are and operated onto through isomorphisms and such that and , for all . Assume is the quotient mapping from onto , is the identity operator on and is the natural embedding operator from into . Assume is the Bernstein numbers [18].

Theorem 24. The class is minimum, whenever

Proof. Assume , one has so that for all and We have Take , one has Therefore, for some , we obtain When , one has a contradiction. So, and both cannot be infinite-dimensional when .

Theorem 25. The class is minimum, whenever

Lemma 26 (see [19]). Suppose and , one has and with , for every .

Theorem 27 (see [19]). If is an infinite-dimensional Banach space, then

Theorem 28. If and , for every , then

Proof. Let and . In view of Lemma 26, there are and with . Therefore, for every , one has This contradicts Theorem 23; hence, .

Corollary 29. Suppose , and , for every , then

Proof. Evidently, as .

Definition 30 (see [19]). A Banach space is said to be simple whenhas a unique nontrivial closed ideal.

Theorem 31. The class is simple.

Proof. Let the closed ideal contain a mapping . By Lemma 26, there are so that . Therefore, . Hence, . Therefore, is a simple Banach space.

Theorem 32. Assumethen

Proof. Let , then and , for every . One has , for all , then for all . Hence , so .
Next, assume . Hence, . Therefore, one has Hence, If exists, for all . Then exists and bounded, for all . So, exists and bounded. Since is a prequasi ideal, one obtains One has a contradiction, as . Then, , for all . So, , for all . Therefore, .

3. Multiplication Mappings on

Under the conditions of Theorem 14, we have presented in this section some properties of the multiplication mapping acting on .

Let indicate the complement of . Let be the space of all sets with a finite number of elements. Assume is the space of bounded sequences of soft functions.

Definition 33. Suppose is a prequasi normed and . The mappingis said to be a multiplication mapping on, if, for all . The multiplication mapping is called constructed by , if .

Definition 34 (see [35]). A mapping is said to be Fredholm if , is closed and .

Theorem 35. (1)(2), for every , if and only if, is an isometry(3)(4)(5)(6), for every , if and only if, is closed(7), for all , if and only if, is invertible(8) is Fredholm operator, if and only if () and () , for all

Proof. (1)Suppose , one has with , for all . If , we haveTherefore, .
Next, if and . One has , for every with . Then, Hence, . So, . (2)Let and , for every . One obtainsthen is an isometry.
Next, if for some that one has When , so . Hence, , for every . (3)Assume , so . If . One has with . Let . We have be an infinite set in . For all , one getsHence, has not a convergent subsequence under . So, . Therefore, ; this is a contradiction. So, . Next, let . Hence, for every , we have . Therefore, for all , one gets . So, . If , for all , where Obviously, since , for all . According to with , we have Therefore, . This implies is a limit of finite rank mappings. (4)As , the proof follows(5)Since , where , one has and (6)Let the sufficient setups be verified. One has with , for every . We have to show that is closed; let be a limit point of . One has , for all with . Clearly, is a Cauchy sequence. Since , we havewhere Therefore, is a Cauchy sequence in . Since is complete. One has with . As , we have . As . So, . Then, , i.e., is closed. Next, suppose the necessary condition is satisfied. One has with and . Let , then for , we have which introduces a contradiction. So , we have for all . (7)First, assume so that . By Theorem 35 part (1), we have . One has . So, . Second, if is invertible. Then . Therefore, is closed. From Theorem 35 part (5), one has with , for all . Then, , when , where ; this implies , which is a contradiction, since is trivial. Then, , for all . As . From Theorem 35 part (1), one has with , for all . So , for all (8)First, if and , one has , for all . As ’s are linearly independent, we have ; this is a contradiction. Therefore, . The condition (g2) comes from Theorem 35 part (6). Next, assume the setups (g1) and (g2) are satisfied. According to Theorem 35 part (6), the setup (g2) gives that is closed. The condition (g1) implies that and . Therefore, is Fredholm

4. Fixed Points of Kannan Contraction Type

In this section, we offer the existence of a fixed point of Kannan contraction mapping acting on this new space under the conditions of Theorem 14 and its associated prequasi ideal. Interestingly, several numerical experiments are presented to illustrate our results.

Definition 36. A prequasi normed on confirms the Fatou property, if for every sequence so that and every , one has
Throughout the next part of this article, we will use the two functions and as for all .

Theorem 37. The function satisfies the Fatou property.

Proof. Assume so that Clearly, . For every , one has

Theorem 38. Suppose , then does not verify the Fatou property.

Proof. If so that Clearly, . For every , one has

Hence, does not satisfy the Fatou property.

Definition 39 (see [30]). A mapping is called a Kannan -contraction, if one has , with for all . When then is called a fixed point of .

Theorem 40. Suppose is Kannan -contraction operator, then has a unique fixed point.

Proof. If , one has . As is a Kannan -contraction, one has We get for all so that that

Therefore, is a Cauchy sequence in . As is prequasi Banach space. One has with . To show that . Since satisfies the Fatou property, one can see then . Therefore, is a fixed point of . To indicate the uniqueness of the fixed point. Let us have two different fixed points of . We have

Therefore,

Corollary 41. If is Kannan -contraction, then has a unique fixed point so that

Proof. By Theorem 40, one has a unique fixed point of . Hence,

Definition 42. If is a prequasi normed , and The mapping is called -sequentially continuous at , if and only if, when then .

Theorem 43. If , and . The element is the unique fixed point of , when the following conditions are confirmed: (i) is Kannan -contraction(ii) is -sequentially continuous at (iii)One has with has converges to

Proof. Assume is not a fixed point of , one has . According to conditions (ii) and (iii), we have As is Kannan -contraction, one has Take , one obtains a contradiction. Therefore, is a fixed point of . To explain the uniqueness of . Suppose we have two different fixed points of . Then So

Example 44. If and For all . If , we have For every , we have For every and , one has Hence, is Kannan -contraction, as satisfies the Fatou property. By Theorem 40, has a unique fixed point Assume so that where such that . As is continuous, one has So is not -sequentially continuous at . This implies is not continuous at .
For every . If , one has Let , one has For every and , one has Hence, is Kannan -contraction and

Evidently, is -sequentially continuous at and has a subsequence converges to . According to Theorem 43, the element is the only fixed point of .

Example 45. Let and As , we get For all , hence for all , we have For all and , one has Hence, is Kannan -contraction. Obviously, is -sequentially continuous at , and there is with such that the sequence of iterates includes a subsequence converges to . In view of Theorem 43, the operator has one fixed point . Note that is not continuous at .
For all . If , we have For all , hence for all , one has For all and , we have Therefore, the operator is Kannan -contraction. Since confirms the Fatou property. By Theorem 40, the operator has a unique fixed point .
In this part, we will use for every .

Definition 46. A function on satisfies the Fatou property if for all so that and all , one has

Theorem 47. The function does not verify the Fatou property.

Proof. Assume so that Clearly, . Hence, for every , we have

Therefore, does not satisfy the Fatou property.

Definition 48 (see [30]). A mapping is said to be a Kannan -contraction, assume there is with for all .

Definition 49. Assume and The mapping is called -sequentially continuous at , if and only if, when one has .

Theorem 50. If . The operator is the only fixed point of , when the following conditions are confirmed: (i) is Kannan -contraction(ii) is -sequentially continuous at (iii)One has with has converges to

Proof. Suppose is not a fixed point of , then . By conditions (ii) and (iii), one has As is Kannan -contraction operator, we get By , we have a contradiction. Then, is a fixed point of . To show the uniqueness of the fixed point . If one has two different fixed points of . So Therefore,

Example 51. Assume For all If , we have Suppose , one has Assume and , one gets Hence, is Kannan -contraction and Evidently, is -sequentially continuous at the zero operator and has a subsequence converges to . According to Theorem 50, the zero operator is the only fixed point of .
If with where so that . As is continuous, one has Therefore, is not -sequentially continuous at . This implies is not continuous at .

5. Applications on Stochastic Nonlinear Dynamical System

We investigate in this section a solution in to stochastic nonlinear dynamical system (106) under the conditions of Theorem 14. For every .

Consider the stochastic nonlinear dynamical system [36]:

and assume is constructed by

Theorem 52. The stochastic nonlinear dynamical system (106) has one and only one solution in if one has with and for every , one obtains

Proof. Let the conditions be established. Assume the mapping is defined by equation (11). Hence,

From Theorem 40, one has one and only one solution of (106) in

Example 53. Consider Suppose the stochastic nonlinear dynamical system: with , for all and suppose is defined by

Evidently, one has with and for every , we have

From Theorem 52, system (111) has one and only one solution in

Example 54. Suppose the sequence space Assume the stochastic nonlinear dynamical system: with , for all and suppose is defined by

Evidently, there is such that and for every , we have

According to Theorem 52, the stochastic nonlinear dynamical system (14) contains a unique solution in

Theorem 55. If is defined by (11) and . The stochastic nonlinear dynamical system (106) has a unique solution when the following conditions are satisfied: (1)If assume there is so that and for every , one has(2) is -sequentially continuous at (3)There is with has converging to

Proof. One has

By Theorem 43, one gets a unique solution of equation (106).

Example 56. Suppose the sequence space Consider the summable equation (111):
Let defined by (13). Assume is -sequentially continuous at and there is with has converging to . Evidently, there is such that and for all , one has By Theorem 57, the stochastic nonlinear dynamical system (111) has one and only one solution In this part, we search for a solution to nonlinear matrix equation (131) at the conditions of Theorem 14 are satisfied, and for every . Consider the stochastic nonlinear dynamical system: and suppose is defined by

Theorem 57. The stochastic nonlinear dynamical system (131) has one and only one solution if the following conditions are satisfied: (1) and for every , there is a positive real number so that , with(2) is -sequentially continuous at a point (3)There is so that the sequence of iterates has a subsequence converging to

Proof. Suppose the settings are verified. Consider the mapping defined by (132). We have In view of Theorem 50, one obtains a unique solution of equation (131) at .

Example 58. Assume the class , where Consider the stochastic nonlinear dynamical system: where and and let be defined as

Suppose is -sequentially continuous at a point , and there is so that the sequence of iterates has a subsequence converging to . It is easy to see that

By Theorem 57, the stochastic nonlinear dynamical system (18) has one solution .

6. Conclusion

In this article, we introduced a new general space called and the mappings’ ideal space of solutions for many stochastic nonlinear and matrix systems of Kannan contraction type. We have presented some topological and geometric properties of it, of the multiplication operators acting on it, of the mappings’ ideal, and of the spectrum of its mappings’ ideal. The existence of a fixed point in the Kannan contraction mapping on these spaces is explored. To put our findings to the test, we introduced several numerical experiments. In addition, various effective implementations of the stochastic nonlinear dynamical and matrix system are discussed. The ideal spectrum of mappings, multiplication operators, and the fixed points of any contraction mappings in this new soft functions space are investigated.

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.

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Copyright © 2022 Meshayil M. Alsolmi and Awad A. Bakery. 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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