On a New Stochastic Space of Solution for the Volterra-Type Summable Equations of Fuzzy Functions

Mohammed, Mustafa M.; Bakery, Awad A.

doi:https://doi.org/10.1155/2023/4573332

Journal of Mathematics

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

Research Article | Open Access

Volume 2023 | Article ID 4573332 | https://doi.org/10.1155/2023/4573332

On a New Stochastic Space of Solution for the Volterra-Type Summable Equations of Fuzzy Functions

Mustafa M. Mohammed¹and Awad A. Bakery¹

Academic Editor: Ammar Alsinai

Received07 Feb 2023

Revised16 Aug 2023

Accepted11 Oct 2023

Published06 Dec 2023

Abstract

This article explains the sufficient requirements for a newly constructed stochastic space by Poisson-like matrices and weighted variable exponent sequence spaces of fuzzy functions, as well as the ideals of their operators, for the Kannan contraction operator to have a unique fixed point. Moreover, we investigate some examples and the numerous applications of solutions to Volterra-type summable equations of fuzzy functions.

1. Introduction

Summable equations come up in many situations in the critical point theory for nonsmooth energy functionals, mathematical physics, control theory, biomathematics, difference variational inequalities, fuzzy set theory [1], probability theory [2], and traffic problems, to mention but a few. In particular, Volterra-type summable equations are known to be of great importance in investigating dynamical systems [3] and stochastic processes [4, 5]. Some instances are in the fields of granular systems, sweeping processes, oscillation problems, control problems, decision-making problems [6], and so on. The solution of summable equations is contained in a certain sequence space. So, there is a great interest in mathematics to construct new sequence spaces, see [7]. Mursaleen and Noman [8] examined some new sequence spaces of nonabsolute type related to the spaces and , and Mursaleen and Başar [9] constructed and investigated the domain of the matrix in some spaces of double sequences. Mustafa and Bakery [10] introduced the concept of the private sequence space of fuzzy functions . Suppose is the set of real numbers and is the set of nonnegative integers. We have constructed the space equipped with a definite function by the domain of Poisson-like operator defined in , where the Poisson-like matrix, , is defined as follows:where and , for all .

For , Matloka [11] introduced the -level set of a fuzzy real number as follows:

Let us denote to the space of every is normal, compact, fuzzy convex, and upper semicontinuous. For , one has

If , , , for every . Consider the following Volterra-type summable equations of fuzzy functions [12]:and presume , for certain functional , is defined as follows:

Mustafa and Bakery [10] investigated the unique solution of (4) of Kannan-type (5) in the operators’ ideal (or in short O.I) formed by a weighted binomial matrix in the variable exponent sequence space of extended s-fuzzy functions. Bakery and Mohammed [13] explained Kannan nonexpansive mappings on the variable exponent Cesàro sequence space of fuzzy functions. We can use the newly constructed stochastic space to explore more spaces of solutions for the fuzzy fractional evolution equations; see the following interesting articles: Abuasbeh et al. [14], Niazi et al. [15], and Iqbal et al.’s [16] studies. In this paper, geometric and topological properties are used to define the space of fuzzy functions and the ideal space of its operators. The fixed point for the Kannan contraction operator is demonstrated, and its prequasi operator ideal is proven to be confirmed in this space. In the final section of this article, we discuss the various applications of solutions to Volterra-type summable equations of fuzzy functions and demonstrate the practical relevance of our findings.

2. Structures of and Its O.I

A few topological and geometric characteristics of and the corresponding O.I have been examined here.

Notation 1. (1) and : infinite-dimensional Banach spaces.(2): the space of all bounded linear operators from into .(3): the space of all bounded linear operators from into itself.(4): the space of finite rank linear operators from into .(5): the space of approximable operators from into .(6): the space of compact operators from into .(7): the ideal of bounded operators between any two Banach spaces.(8): the ideal of finite rank operators between any two Banach spaces.(9): the ideal of approximable operators between any two Banach spaces.(10): the ideal of compact operators between any two Banach spaces.(11): the space of all sequences of fuzzy reals.(12): the space of all sequences of positive reals.(13): the linear space of sequences of fuzzy functions.(14): the integral part of the real number .(15).(16): the unit operator on the -dimensional Hilbert space .(17), where and are the multiplicative and additive identity in , respectively, while marks at the place.(18).(19): the space of finite sequences of fuzzy numbers.(20): the space of all monotonic increasing sequences of positive reals.(21): the space of all monotonic decreasing sequences of positive reals.(22): the space of each monotonic decreasing sequence of fuzzy functions.(23): the Banach space of one dimension.

Definition 2. Presume .
and

Lemma 3 (see [17]). where and , for every .

Theorem 4. Presume , then

Proof. Since , the proof follows.

Theorem 5. Presume , then is a nonabsolute type.

Proof. Evidently, since

Definition 6. Presume and , for all .where .

Theorem 7. Suppose so that and , then .

Proof. If , thenThen, . By taking , then and .

Definition 8 (see [10]). The space is called a if it verifies the next conditions. (c1) Presume , then , (c2) if , and , for all , then , (c3) if , then .

Definition 9 (see [18]). The sequence , for all , verifies the following conditions:(a), for every ,(b)Suppose and are arbitrary Banach spaces. If , , and , then ,(c),(d)Presume and , then ,(e), whenever ,(f) or .

Notations 10 (see [10]).

Theorem 11 (see [10]). The space is an O.I, if is a .

Definition 12 (see [19]). A subspace of the is said to be a premodular (p-m-) if one has a function [) which satisfies the following parts: (d1) If , and , (d2) presume and , then with , (d3) there are such that , for every , (d4) if , for every , then , (d5) one has under , (d6) the closure of , (d7) one obtains with .

Definition 13 (see [19]). The is called a prequasi normed (p-qN-) if holds the parts (d1)-(d3) of Definition 12. The space is called a prequasi Banach (p-qB-); assume is complete under .

Theorem 14 (see [10]). The space is a p-qN-; assume it is p-m-.

Theorem 15. is a p-qB-, if the next parts are established. (f1). (f2).

Proof. First is to show that is a p-m-.
The condition (d1): obviously, and .
The conditions (c1) and (d3): presuming , one hashence .
The condition (d2): suppose , , and as , thenwhere . Hence, .
As , henceSo, .
The conditions (c2) and (d4): presuming , for all and , one can see thathence .
The conditions (c3) and (d5): if , with and , thenwhere . Hence, .
The condition (d6): the closure of .
The condition (d7): there are with , for all and , if .
From Theorem 14, the space is a p-qN-. If is a Cauchy sequence in , hence for all , we have with , one hasHence, , as is a complete metric space. Therefore, is a Cauchy sequence in , for fixed . So, it is convergent to . So, , for each . Clearly, part (d3) that implies that is a Banach space.
By Theorems 11 and 15, we obtain the following theorem.

Theorem 16. The space is an O.I, when the parts of Theorem 15 are established.

Theorem 17 (see [10]). If type, . Presume is an O.I, then the next parts are established.(a) type .(b)When type and type , then type .(c)Presume and type , then type .(d)Let type and , for every and , then type . i.e., is a solid space.

Some properties of type are explained in the next theorem in view of Theorems 16 and 17.

Theorem 18. (a) type .(b)If type and type , then type .(c)Presume and type , then type .(d) type is a solid space.

Definition 19. (see [20]). A subclass of is called an O.I, if each holds the following parts:(1).(2)The space is linear over .(3)Assume , , and , then .

Definition 20 (see [21]). A function is said to be a p-qN on the ideal when the next parts are established.(1)Presume , and ,(2)there are with , for all and ,(3)one has such that , for every ,(4)there are such that if , , and , then .

Theorem 21. (see [21]). Every qN on the ideal is a p-qN.

We have offered some properties of the ideal generated by our fuzzy space and extended numbers in this part, presuming that the parts of Theorem 15 are satisfied.

Theorem 22. The conditions of Theorem 15 are sufficient only for .

Proof. Obviously, by the linearity of and , for all . After that, to show , presume , hence . As , let , there exists such that , for some and . Since , henceThen, such that rank andsince , we getas one hasGiven inequalities (18)–(21), one obtainsNext, one concludes a negative example for the problem in [22] since , where and , but .

Theorem 23. The class is a p-qB ideal, where .

Proof. Obviously, is a p-qN on since is a p-qN on . Presume is a Cauchy sequence in . As , we getso is a Cauchy sequence in . As is a Banach space, then with . As , for all . From Definition 12 parts (d2), (d3), and (d5), henceSo, , then .

Theorem 24. If and , for each , then

Proof. Suppose , hence . One obtainshence . Put with , then withHence, and .
Clearly, . Fix with . So, and .
From Dvoretzky’s theorem [23], one has and mapped onto through isomorphisms and such that and , for each . If is the quotient operator from onto , is the identity operator on and is the natural embedding operator from into . Presume is the Bernstein numbers [24].

Theorem 25. Suppose , then is minimum.

Proof. Presuming , one obtains under , for all and . We obtainFixing , one getsSo, for some , thenAs , one has a contradiction. So, and both cannot be infinite-dimensional whenever .

Theorem 26. The class is minimum, whenever .

Lemma 27 (see [25]). Presume and , then and so that , for each .

Theorem 28 (see [25]). If is an infinite-dimensional Banach space, then

Theorem 29. Presume and , for each , then

Proof. If and. By Lemma 27, then and such that . Therefore, for every , thenwhich contradicts Theorem 24, hence .

Corollary 30. Presume and , for each , then

Proof. Obviously, since .

Definition 31. (see [25]). A Banach space is defined as simple if is a unique nontrivial closed ideal.

Theorem 32. The class is simple.

Proof. Suppose has . By Lemma 27, then there are such that . So,
. Hence, . Therefore, is a simple Banach space.

Theorem 33. Presume , then .

Proof. If , then and , for each . We obtain , for all , sofor all . Hence, , so .
After that, assume . Then, . So, one getsHence, . Assume exists, for all . Hence, bounded and exists, for each . Hence, bounded and exists. From which is a prequasi ideal, one hasOne has a contradiction, as . So, , for every . Hence, , for all . So, .

3. Kannan’s Contraction Fixed Points

Supposing that the parts of Theorem 15 are established, the existence of a fixed point of the Kannan contraction operator acting on this new space and its associated p-q ideal are presented with some numerical examples to show our results.

The Banach fixed point theorem, as presented in reference [26], provided mathematicians with a means to extend the applicability of contraction operators by generalization, for instances, the Kannan contraction operator [27], Kannan operators in modular vector spaces [28], and Kannan p-qN contraction operator [29].

Definition 34. (see [13]). A p-qN- on holds the Fatou property (or in short FPr); presume for every such that and , then .

We will use the following notations:for all .

Theorem 35. The function holds the FPr.

Proof. Presume so that . Clearly, . For each , hence

Theorem 36. If , then does not hold the FPr.

Proof. Suppose so that . Obviously, . For every , one obtainsSo, does not verify the FPr.

Definition 37 (see [29]). A mapping is said to be a Kannan -contraction, if there is [) with , for every . Presume , then is named a fixed point of .

Theorem 38. The space has a unique fixed point, whenever is the Kannan -contraction operator.

Proof. Presume , hence . As is a Kannan -contraction, one getsHence, for all so that , thenSo, is a Cauchy sequence in , as is a p-qB space. Hence, with to show that . Since verifies the FPr, one obtainshence . So, is a fixed point of . To show the uniqueness of the fixed point, let one has two different fixed points of . So,Therefore, .

Corollary 39. If is Kannan -contraction, then has a unique fixed point such that .

Proof. Given Theorem 38, we obtain a unique fixed point of . Hence,

Definition 40. (see [13]). Presume is a p-qN-,, and . The operator is called -sequentially continuous (or in short -s.c) at , if and only if, when , then .

Theorem 41. Presume and . The vector is the unique fixed point of , whenever the following parts are established: (g1) is Kannan -contraction, (g2) is - s.c at , (g3) one has such that has which converges to .

Proof. Let be not a fixed point of , then . From parts (g2) and (g3), we getAs is Kannan -contraction, thenBy , there is a contradiction. Then, is a fixed point of . For the uniqueness of , if one gets two different fixed points of . So,Hence, .

Example 1. Consider andIf . Assume , thenPresume , thenSuppose and , henceSo, is Kannan -contraction, as verifies the FPr. By Theorem 38, has a unique fixed point . Presume such that , where so that . As is continuous, one hasSo, is not -s.c at , which explains that is not continuous at .
Consider . If , thenIf , henceSuppose and , thenSo, is Kannan -contraction and
Evidently, is -s.c at and has a which converges to . From Theorem 41, the vector is the only fixed point of .

Example 2. Consider andAs , thenIf , then for every , we obtainSuppose and , thenHence, is Kannan -contraction. Obviously, is - s.c at and one has such that with has a which converges to . According to Theorem 41, has a unique fixed point . Note that is not continuous at .
Presume . When , thenSuppose , then for all , one can seeIf and , thenSo, is Kannan -contraction. Since satisfies the FPr. In view of Theorem 38, has a unique fixed point .

We will use in this part , for all .

Definition 42. (see [10]). A function on verifies the FPr if for every such that and all , then .

Theorem 43. The function does not hold the FPr.

Proof. Presume such that . Evidently, . Hence, for each , thenTherefore, does not hold the FPr.

Definition 44. (see [29]). A mapping is named a Kannan -contraction; assume we have with , for all .

Definition 45. (see [10]). Presume and . The operator is said to be - s.c at , if and only if, when , then .

Theorem 46. If . The operator is the unique fixed point of , if the following parts are confirmed: (h1) is Kannan -contraction, (h2) is -s.c at , (h2) there is such that has which converges to .

Proof. Let be not a fixed point of , then one gets . From the parts (h2) and (h3), one getsAs is Kannan -contraction operator, thenWe get a contradiction as . Then, is a fixed point of . For the uniqueness of the fixed point , assume one gets two different fixed points of . So,Therefore, .

Example 3. ConsiderPresume . If , thenIf , thenSuppose and , one can seeHence, is Kannan -contraction and
Clearly, is -s.c at the zero operator and has a which converges to . From Theorem 46, is the unique fixed point of .
Presume with , where
with . As is continuous, thenSo, is not -s.c at . Hence, is not continuous at .

4. Applications

In this section, a solution in to (1) is examined such that the parts of Theorem 15 are established.

Theorem 47. System (1) holds one and only solution in , if , there is such that and for each , then

Proof. Suppose is defined by equation (2). By Theorem 38 andThis completes the proofs.

Example 4. Consider .
Suppose the following nonlinear uncertainty equation of fuzzy functions:with and , for all , and ifis defined asObviously, one obtains such that and for every , thenFrom Theorem 47, system (8) has a unique solution in .

Theorem 48. Consider is defined by (2) and . System (1) holds one and only solution , if the following parts are established:(1)Suppose and , if there is with and for each , then(2) is - s.c at ,(3)one has such that has converging to .

Proof. From Theorem 41 andThis completes the proofs.

Example 5. Let .
Consider the Volterra-type summable equations (8).
Suppose is defined by (9). When is -s.c at and one has
such that has converging to . Clearly, there is such that and for all , thenBy Theorem 49, the nonlinear uncertainty equation of fuzzy functions (8) has a unique solution .

In this section, we propose a solution to system (10) at . Supposing that the parts of Theorem 15 are established and , for each . Consider the following nonlinear uncertainty equation of fuzzy functions:and if is defined as

Theorem 49. System (10) holds one and only one solution , if the following parts are confirmed:(1), , , , and for every , one has such that and(2) is -s.c at an element .(3)There is such that has a converging to .

Proof. Consider as defined by (11). From Theorem 46 andThis finishes the proof.

Example 6. Let , where , for all .
Consider the following nonlinear uncertainty equation of fuzzy functions:where and and if is defined asSupposing that is -s.c at a point and one has with has a converging to . By Theorem 49 andThis completes the proof.

5. Conclusion

We explained a few topological and geometric properties of of the class and of the class in this article. The Kannan contraction operator on these spaces is analyzed, and the possibility of a fixed point is considered. We ran many numerical experiments to ensure our theories were correct. Fuzzy functions with nonlinear uncertainty equation implementations are also investigated. This novel fuzzy function space is used to investigate the fixed points of all contraction operators, providing a new universal solution space for a wide variety of stochastic nonlinear dynamical systems.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant no. UJ-22-DR-89. The authors, therefore, acknowledge with thanks the University of Jeddah for its technical and financial support.

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Copyright © 2023 Mustafa M. Mohammed 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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