Kannan Nonexpansive Operators on Variable Exponent Cesàro Sequence Space of Fuzzy Functions

Bakery, Awad A.; Mohammed, Mustafa M.

doi:https://doi.org/10.1155/2022/1992684

Journal of Function Spaces

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

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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 1992684 | https://doi.org/10.1155/2022/1992684

Kannan Nonexpansive Operators on Variable Exponent Cesàro Sequence Space of Fuzzy Functions

Awad A. Bakery^1,2and Mustafa M. Mohammed^1,3

Academic Editor: Santosh Kumar

Received30 May 2022

Accepted13 Aug 2022

Published27 Aug 2022

Abstract

In general, we have constructed the operator ideal generated by extended -fuzzy numbers and a certain space of sequences of fuzzy numbers. An investigation into the conditions sufficient for variable exponent Cesàro sequence space of fuzzy functions furnished with the definite function to create pre-quasi-Banach and closed is carried out. The and the normal structural properties of this space are shown. Fixed points for Kannan contraction and nonexpansive mapping have been introduced. Lastly, we explore whether the Kannan contraction mapping has a fixed point in its associated pre-quasioperator ideal. The existence of solutions to nonlinear difference equations is illustrated with a few real-world examples and applications.

1. Introduction

Probability theory, fuzzy set theory, soft sets, and rough sets have contributed substantially to the study of uncertainty. But there are drawbacks to these theories that must be considered. After Zadeh [1] established the concept of fuzzy sets and fuzzy set operations, many researchers adopted the concept of fuzziness in cybernetics and artificial intelligence as well as in expert systems and fuzzy control. For more information and real-world examples, some comparable fixed point results were discussed by Javed et al. [2] to ensure that a fixed point exists and is unique in -fuzzy -metric spaces. The viability of the proposed methodologies was demonstrated through a challenging case study. There was no doubt about the superiority of the findings delivered. For the first type of Fredholm-type integral equation, an application was described. In [3], Al-Masarwah and Ahmad defined and investigated the -Polar -Fuzzy Ideals in BCK/BCI-Algebras and explored some pertinent properties. There are many other orthogonal fuzzy metric spaces; however, Javed et al. [4] expanded the orthogonal image fuzzy metric space concept. In the context of the newly specified structure, they displayed some fixed point outcomes. Fuzzy sequence spaces were introduced, and their various features were studied by many workers on sequence spaces and summability theory. Nuray and Sava [5] defined and studied the Nakano sequences of fuzzy numbers, equipped with the function . The operator ideal is very important in fixed point theory, Banach space geometry, normal series theory, approximation theory, and ideal transformations. See [6–8] for further proof. Pre-quasioperator ideals are more extensive than quasioperator ideals, according to Faried and Bakery [9]. The learning about the variable exponent Lebesgue spaces obtained impetus from the mathematical description of the hydrodynamics of non-Newtonian fluids (see [10, 11]). There are numerous uses for electrorheological fluids, which include military science, civil engineering, and orthopedic. There have been many developments in mathematics since the Banach fixed point theorem [12] was first published. While contractions have fixed point actions, Kannan [13] cited an example of a type of mapping that is not continuous. In Reference [14], the only attempt was made to explain Kannan operators in modular vector spaces. For more details on Kannan’s fixed point theorems, see [15–20]. 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 options are viable. Hence, we have constructed the Cesàro sequence spaces of fuzzy functions and have presented the solutions of a fuzzy nonlinear dynamical system in this newly created space. This work is aimed at introducing the certain space of sequences of fuzzy numbers, in short (cssf), under a certain function to be pre-quasi (cssf). This space and -numbers have been used to describe the structure of the ideal operators. We explain the sufficient conditions of variable exponent Cesàro sequence space of fuzzy functions, which is denoted by , equipped with the definite function to be pre-quasi-Banach and closed (cssf). The and the normal structure property of this space are shown. Fixed points for Kannan contraction and nonexpansive mapping have been introduced. Lastly, we explore whether the Kannan contraction mapping has a fixed point in its associated pre-quasioperator ideal. The existence of solutions to nonlinear difference equations is illustrated with a few real-world examples and applications.

2. Definitions and Preliminaries

As a reminder, Matloka [21] presented the notion of ordinary convergence of sequences of fuzzy numbers, where he introduced bounded and convergent fuzzy numbers, explored some of their features, and proved that any convergent fuzzy number sequence is bounded. Nanda [22] studied the sequences of fuzzy numbers and showed the set of all convergent sequences of fuzzy numbers from a complete metric space. Kumar et al. [23] investigated the notion of limit points and cluster points of sequences of fuzzy numbers. Assume is the set of all closed and bounded intervals on the real line . For and in , suppose

Define a metric on by

Matloka [21] showed that is a metric on and is a complete metric space. Also, the relation is a partial order on .

Definition 1. A fuzzy number is a fuzzy subset of , i.e., a mapping which verifies the following four settings: (a) is fuzzy convex, i.e., for and , (b) is normal, i.e., there is such that (c) is an upper semicontinuous, i.e., for all , for all is open in the usual topology of (d)the closure of is compactThe -level set of a fuzzy real number , indicated by is defined as The set of every upper semicontinuous, normal, convex fuzzy number, and is compact is denoted by . The set can be embedded in , if we define by The additive identity and multiplicative identity in are denoted by and , respectively.
The arithmetic operations on are defined as follows: The absolute value of is defined by Suppose and the -level sets are , , and . A partial ordering for any as follows: , if and only if , for all . Then, the above operations can be defined in terms of -level sets as follows: Assume is defined by
Recall that (1) is a complete metric space(2)(3).(4)

Definition 2. A sequence of fuzzy numbers is said to be (a)bounded if the set of fuzzy numbers is bounded, i.e., if a sequence is bounded, then there are two fuzzy numbers such that (b)convergent to a fuzzy real number if for every , there exists such that , for all

Lemma 3 (see [24]). Suppose and , for every , then where .

3. Main Results

3.1. Some Properties of

In this section, we have introduced the certain space of sequences of fuzzy numbers or in short (cssf), under the definite function to form pre-quasi (cssf). We explain the sufficient setting of equipped with the definite function to construct pre-quasi-Banach and closed (cssf). The Fatou property of various pre-quasinorms on has been investigated. We have presented this space’s -nearly uniformly convex, the property , and the -normal structure-property, which are connected with the fixed point theorem.

By and , we denote the spaces of bounded and -absolutely summable sequences of real numbers, respectively. Let denote the classes of all sequence spaces of fuzzy real numbers. Suppose , where is the space of positive real sequences. The variable exponent Cesàro sequence space of fuzzy functions is denoted by the following: , when If , then

Definition 4 (see [25]). The linear space is said to be a certain space of sequences of fuzzy numbers (cssf), if (1), where , while displays at the place(2)suppose , and , for all , then (3), where marks the integral part of , if

Definition 5 (see [25]). A subclass of is called a premodular (cssf), if there is satisfies the next settings: (i)If , with , where (ii)There is , and the inequality holds, for every and (iii)There is , and the inequality holds, for every (iv)If , for every , one has (v)The inequality holds, for some (vi)Let be the space of finite sequences of fuzzy numbers; then, the closure of (vii)There is with , where

Definition 6 (see [25]). Suppose is a (cssf). The function is called a pre-quasinorm on , if it satisfies the following conditions: (i)If , with , where (ii)There is , and the inequality satisfies, for every and (iii)There is , and the inequality holds, for each

Clearly, from the last two definitions, we conclude the following two theorems:

Theorem 7 (see [25]). If is a premodular (cssf), then it is pre-quasinormed (cssf).

Theorem 8 (see [25]). is a pre-quasinormed (cssf) if it is quasinormed (cssf).

Definition 9. (a)The function on is named -convex, if for every and . (b) is -convergent to , if and only if When the -limit exists, then it is unique(c) is -Cauchy, if (d) is -closed, when for all -converges to , then (e) is -bounded, if (f)The -ball of radius and center , for every , is described as follows: (g)A pre-quasinorm on satisfies the Fatou property, if for every sequence under and all , one has Note that the Fatou property implies the -closed of the -balls. We will denote the space of all increasing sequences of real numbers by .

Theorem 10. , where , for all , is a premodular (cssf), when with .

Proof. (i) Evidently, and
(1-i) Let . One has and then, .
(iii) One gets with , for all
(1-ii) Assume and , and we obtain As . Hence, from conditions (1-i) and (1-ii), one has is linear. Also, , for all , since
(ii) There is with , for all and
(2) Assume , for all and . One finds and then, .
(iv) Obviously, from (2)
(3) Let , and we get and then, .
(v) From (4), we obtain
(vi) Evidently the closure of
(vii) There is , for or , for with

Theorem 11. If with , then is a pre-quasi-Banach (cssf), where , for every .

Proof. In view of Theorem 10 and Theorem 7, the space is a pre-quasinormed (cssf). Assume is a Cauchy sequence in . Hence, for every , one has such that for all , one gets That implies As is a complete metric space. Then, is a Cauchy sequence in , for fixed , which implies , for constant . Hence, , for every , since So .

Theorem 12. Suppose with , then is a pre-quasiclosed (cssf), where , for every .

Proof. In view of Theorem 10 and Theorem 7, the space is a pre-quasinormed (cssf). Assume and ; then, for all , there is such that for all , we obtain which implies As is a complete metric space, therefore, is a convergent sequence in , for fixed . So, , for fixed . Since , one has .

Theorem 13. The function verifies the Fatou property, when with , for all .

Proof. Let such that Since is a pre-quasiclosed space, one has . For all , one gets

Theorem 14. The function does not satisfy the Fatou property, for all , when and , for all .

Proof. Let so that Since is a pre-quasiclosed space, one gets . For every , we obtain

Example 1. For , the function is a norm on .

Example 2. The function is a pre-quasinorm (not a norm) on .

Example 3. The function is a pre-quasinorm (not a quasinorm) on .

Example 4. The function is a pre-quasinorm, quasinorm, and not a norm on , for .
In the next part of this section, we will use the function as , for every .

Definition 15 [26]. The function is said to be strictly convex, (SC), if for all such that and , we get

Definition 16 [27]. A sequence is said to be -separated sequence for some , if

Definition 17 (see [27]). Let be an integer, and a Banach space is called -nearly uniformly convex (-NUC), if for any , there exists such that for any sequence , with , there are , such that

Definition 18 (see [28]). A function is said to satisfy the -condition (), if for any , there exists a constant and such that , for each , with
If satisfies the -condition for any with depending on , we say that satisfies the strong -condition ().

The following known results are very important for our consideration.

Theorem 19 (see [28], Lemma 2.1). If , then for any and , there exists such that , where , with and .

Theorem 20. Pick an with ; then, for any and , there exists such that , for all , with and .

Proof. Since is bounded, it is easy to see that . Hence, the proposition is obtained directly from Theorem 19.

Theorem 21. Suppose with ; then, is -NUC, for any integer .

Proof. Let and with , for each , and let . Since for each , is bounded, and by using the diagonal method, we can find a subsequence of such that converges for each , . Therefore, there exists an increasing sequence of positive integers such that . Hence, there is a sequence of positive integers with , such that for each . For fixed integer , let ; then, by Theorem 20, there exists such that whenever and . Since , for any , then there exist positive integers with such that . Define . By inequality (1), we have . Let for and . Then, in virtue of inequality (1), inequality (2), and convexity of the function for any , we have Therefore, is -NUC.

Recall that -NUC implies reflexivity.

Definition 22. The space satisfies the property , if and only if, for all decreasing sequence of -closed and -convex nonempty subsets of with for some one has
By fixing a nonempty -closed and -convex subset of .

Theorem 23. If with , one has the following: (i)Suppose with There is a unique so that (ii) verifies the property .

Proof. To prove (i), assume as is -closed. One has . Hence, for all , one has with . If is not -Cauchy, one gets a subsequence and with , for every , since for every . Since with , then the function is strictly convex, for any . Therefore, the space is strictly convex; hence, 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 Let , and from Theorem 13, since satisfies the Fatou property, one has Then Since is (SC), this implies the uniqueness of . To prove (ii), assume , for some Since is increasing, put , when . Otherwise, , for all . According to (i), there is one point with , for every . A similar proof will prove that -converges to some . As is -convex, decreasing, and -closed, one has

Definition 24. The space verifies the -normal structure-property, if and only if, for all nonempty -bounded, -convex and -closed subset of not decreased to one point, and one has with

Definition 25 (see [29]). is a real Banach space, and is the unit sphere of . The weakly convergent sequence coefficient of , denoted by , is defined as follows: where

Theorem 26 (see [30]). A reflexive Banach space with has normal structure-property.

Theorem 27. If with , then holds the -normal structure-property.

Proof. Take any and an asymptotic equidistant sequence with and put . There exists such that Since coordinate-wise, there exists such that , whenever . Take ; then, there is such that Since coordinate-wise, there exists such that , whenever . Continuing this process in such a way by induction, we get a subsequence of such that Put for Then, Moreover, for any with , we have This means that Put for Then, On the other hand, for any with . Therefore, By the arbitrariness of , we have from the relations (35), (36), and (38) that such that Take large enough such that , where We have for that that is, . Note that Therefore, for any with . Therefore, , and by the arbitrariness of , we obtain . From Theorem 21 and Theorem 26, the sequence space has the -normal structure-property.

4. Kannan Contraction Mapping on

In this section, we look at how to configure with different so that there is only one fixed point of Kannan contraction mapping.

Definition 28. An operator is said to be a Kannan -contraction, if one gets with , for all . The operator is called Kannan -nonexpansive, when .
An element is called a fixed point of when

Theorem 29. If with , and is Kannan -contraction mapping, where , for all , then has a unique fixed point.

Proof. If , one has . As is a Kannan -contraction mapping, one gets So for all with , one gets Then, is a Cauchy sequence in . As the space is pre-quasi-Banach space, one has with . To prove that , since has the Fatou property, one obtains and then, . So is a fixed point of . To show the uniqueness. Let be two not equal fixed points of . One has So,

Corollary 30. If with , and is Kannan -contraction mapping, where , for all , one has has unique fixed point so that

Proof. In view of Theorem 29, one has a unique fixed point of . So

Example 5. Assume , where , for every and As for each with , one has For all with , one has For all with and , we get Hence, is Kannan -contraction. As satisfies the Fatou property, from Theorem 29, one has holds one fixed point

Definition 31. Pick up be a pre-quasinormed (cssf), , and The operator is called -sequentially continuous at , if and only if when , then .

Example 6. Suppose , where , for every and is clearly both -sequentially continuous and discontinuous at .

Example 7. Assume is defined as in Example 5. Suppose such that , where with .
As the pre-quasinorm is continuous, we have Therefore, is not -sequentially continuous at .

Theorem 32. If with , , where , for all . Suppose (1) is Kannan -contraction mapping(2) is -sequentially continuous at (3)there is with has converging to Then, is the only fixed point of

Proof. Assume is not a fixed point of , and one has . From parts (2) and (4), we get As is Kannan -contraction, one obtains As , one has a contradiction. Then, is a fixed point of . To show the uniqueness, let be two not equal fixed points of . One obtains Hence,

Example 8. Assume is defined as in Example 5. Let , for all . Since for all with , one gets For all with , one gets For all with and , one gets So is Kannan -contraction and
Obviously, is -sequentially continuous at , and holds converges to . By Theorem 32, the point is the only fixed point of .

5. Kannan Nonexpansive Mapping on

We introduce the sufficient conditions of , where , for every , such that the Kannan nonexpansive mapping on it has a fixed point, by fixing a nonempty -bounded, -convex, and -closed subset of .

Lemma 33. If verifies the property and the -quasinormal property. Assume is a Kannan -nonexpansive mapping. For let . Put Then, , -convex, -closed subset of , and

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 Put From 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 , since for all Let Then, The definition of gives Therefore, One has for all so

Theorem 34. If satisfies the -quasinormal property and the property, let be a Kannan -nonexpansive mapping. Then, has a fixed point.

Proof. Let and , for every By the definition of , one gets , for every Assume is defined as in Lemma 33. Clearly, is a decreasing sequence of nonempty -bounded, -closed, and -convex subsets of . The property investigates that Let , and one has , for all Suppose ; then, , so Therefore, Then, . Else, ; then, fails to have a fixed point. Let be defined in Lemma 33. As fails to have a fixed point and is -invariant, then has more than one point, so . By the -quasinormal property, one has with for all From Lemma 33, we get From definition of , Then, which contradicts the definition of . Then, which gives that any point in is a fixed point of .

According to Theorems 23, 27, and 34, we conclude the following:

Corollary 35. Assume with , and is a Kannan -nonexpansive mapping. Then, has a fixed point.

Example 9. Assume with where and , for every . By using Example 8, is Kannan -contraction. So it is Kannan -nonexpansive. By Corollary 35, has a fixed point in .

6. Kannan Contraction and Structure of Operator Ideal

The structure of the operator ideal by equipped with the definite function , where , for every , and -numbers has been explained. Finally, we examine the idea of Kannan contraction mapping in its associated pre-quasioperator ideal. As well, the existence of a fixed point of Kannan contraction mapping has been introduced. We indicate the space of all bounded, finite rank linear operators from a Banach space into a Banach space by , and , and if , we inscribe and .

Definition 36 (see [31]). An -number function is which sorts every a verifies the following settings: (a)(b) for all and , (c), for all , , and , where and are arbitrary Banach spaces(d)If and , then (e)Suppose , and then, , for each (f) or , where denotes the unit map on the -dimensional Hilbert space

Definition 37 (see [8]). (i) is the class of all bounded linear operators within any two arbitrary Banach spaces. A subclass of is said to be an operator ideal, if all verifies the following conditions: , where denotes Banach space of one dimension(ii)The space is linear over (iii)Assume , , and , then

Notation 38. ,where where

Theorem 39. Suppose is a (cssf); then, is an operator ideal.

Proof. (i)Assume and for all ; as for all and is a linear space, one has; for that then . (ii)Suppose and , then by Definition 4 condition (33), one has and , as ; by the definition of -numbers and is a decreasing sequence, one gets , for each . In view of Definition 4 condition (23) and is a linear space, one obtains ; hence, .(iii)Suppose , , and , one has , and as , by Definition 4 conditions (22) and (23), one gets, and then, .

According to Theorems 10 and 39, one concludes the following theorem.

Theorem 40. Let with , and one has is an operator ideal.

Definition 41 (see [9]). A function is called a pre-quasinorm on the ideal if the next conditions hold: (1)Let , , and , if and only if (2)We have so as to , for every and (3)We have so that , for each (4)We have for , , and ; then, .

Theorem 42 (see [9]). is a pre-quasinorm on the ideal if is a quasinorm on the ideal .

Theorem 43. If with , then the function is a pre-quasinorm on , with , for all .

Proof. (1)When , and , if and only if , for all , if and only if (2)There is with , for all and (3)One has so that for , one can see (4)We have , if , , and , and then, .

In the next theorems, we will use the notation , where , for all .

Theorem 44. Suppose with , and one has is a pre-quasi-Banach operator ideal.

Proof. Suppose is a Cauchy sequence in . As , one has Hence, is a Cauchy sequence in . is a Banach space, so there exists so that and since , for all , and is a premodular (cssf). Hence, one can see We obtain , and hence, .

Theorem 45. If with , one has is a pre-quasiclosed operator ideal.

Proof. Suppose , for all and . As , one has So is convergent in . i.e., , and since , for all and is a premodular (cssf). Hence, one can see We obtain , and hence, .

Definition 46. A pre-quasinorm on the ideal verifies the Fatou property if for every so that and , one gets

Theorem 47. Suppose with , then does not satisfy the Fatou property.

Proof. Assume with Since is a pre-quasiclosed ideal, then . So for every , one has

Definition 48. An operator is said to be a Kannan -contraction, if one has with , for all .

Definition 49. An operator is said to be -sequentially continuous at , where , if and only if .

Example 10. ,
where , for every and Evidently, is -sequentially continuous at the zero operator . Let be such that , where with . Since the pre-quasinorm is continuous, one gets Therefore, is not -sequentially continuous at .

Theorem 50. Pick up with and . Assume (i) is Kannan -contraction mapping(ii) is -sequentially continuous at an element (iii)there are such that the sequence of iterates has a converging to Then, is the unique fixed point of .

Proof. Let be not a fixed point of ; hence, . By using parts (ii) and (iii), we get Since is Kannan -contraction, one obtains As , there is a contradiction. Hence, is a fixed point of . To prove the uniqueness of the fixed point , suppose one has two not equal fixed points of . So, one gets Then,

Example 11. Given Example 10, since for all with , we have For all with , we have For all with and , we have Hence, is Kannan -contraction and
Obviously, is -sequentially continuous at , and has a subsequence converges to . By Theorem 50, is the only fixed point of .

7. Applications

Theorem 51. Consider the summable equation which presented by many authors [32, 33, 34], and assume , where with and , for all , is defined by The summable equation (83) has a unique solution in , if , , , and ; assume there is such that , and for all , let

Proof. One has By Theorem 29, one gets a unique solution of equation (83) in

Example 12. Suppose , where , for all . Consider the summable equation with and . Suppose . Indeed, is a nonempty, -convex, -closed, and -bounded subset of . Let be defined by Obviously, By Corollary 35 and Theorem 51, the summable equation (87) has a solution in .

Example 13. Suppose , where , for every . Consider the following nonlinear difference equation: with , , for all , and assume is defined by Evidently, By Theorem 51, the nonlinear difference equation (90) has a unique solution in .

8. Conclusion

Rather than simply referring to a “quasi-normed” place, we used the term “prequasi-normed.” It is the concept of a fixed point of the Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach variable exponent Cesàro sequence spaces of fuzzy functions (cssf). Pre-quasinormal structure and are supported. The Kannan nonexpansive mapping’s presence of a fixed point was investigated. The presence of a fixed point of Kannan contraction mapping in the pre-quasi-Banach operator ideal produced by variable exponent Cesàro sequence spaces of fuzzy functions (cssf) and -fuzzy numbers has also been examined. To put our findings to the test, we introduce several numerical experiments. In addition, various effective implementations of the stochastic nonlinear dynamical system are discussed. The fixed points of any Kannan contraction and nonexpansive mappings on this new fuzzy functions space, its associated pre-quasi-ideal, and a new general space of solutions for many stochastic nonlinear dynamical systems 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.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under grant No. UJ-21-DR-75. The authors, therefore, acknowledge with thanks the university for the technical and financial support.

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Copyright © 2022 Awad A. Bakery and Mustafa M. Mohammed. 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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