Kannan Nonexpansive Mappings on Nakano Sequence Space of Soft Reals with Some Applications

Bakery, Awad A.; Mohammed, Mustafa M.

doi:https://doi.org/10.1155/2022/2307519

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Advances in Nonlinear Analysis and Applications

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Research Article | Open Access

Volume 2022 | Article ID 2307519 | https://doi.org/10.1155/2022/2307519

Kannan Nonexpansive Mappings on Nakano Sequence Space of Soft Reals with Some Applications

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

Academic Editor: Reny George

Received22 Feb 2022

Accepted09 May 2022

Published27 May 2022

Abstract

We developed the operators ideal in this article by extending -soft reals and a particular space of sequences with soft real numbers. The criteria necessary for the Nakano sequence space of soft real numbers given with the definite function to be prequasi Banach and closed are investigated. This space’s () and normal structural features are illustrated. Fixed points have been introduced for Kannan contraction and nonexpansive mapping. Finally, we investigate whether the Kannan contraction mapping has a fixed point in the prequasi operator ideal with which it is linked. By examining some real-world instances and their applications, it is demonstrated that there exist solutions to nonlinear difference equations.

1. Introduction

The study of variable exponent Lebesgue spaces received additional impetus from the mathematical explanation of non-Newtonian fluids’ hydrodynamics (see [1, 2]). Electrorheological fluids have various applications in various fields, including military science, civil engineering, and orthopedics. Since the publication of the Banach fixed point theorem [3], there have been numerous developments in the field of mathematics. While contractions have fixed point actions, Kannan [4] illustrated a noncontinuous mapping. In Reference [5], a single attempt was made to explain Kannan operators in modular vector spaces, and this was the only one that worked. Mitrovi et al. [6] defined a cone -metric space over Banach algebra as a generalization of metric spaces, rectangular metric spaces, b-metric spaces, rectangular b-metric spaces, -generalized metric spaces, cone b-metric spaces over Banach algebra, and rectangular cone b-metric spaces over Banach algebra. They provided fixed point results for Banach and Kannan in cone -metric spaces over Banach algebra. Debnath et al. [7] showed the existence and uniqueness of common fixed points for pairs of self-maps of the Kannan, Reich, and Chatterjea types in a complete metric space. Younis et al. [8] used concepts from graph theory and fixed point theory to provide a fixed point result for Kannan-type mappings in the context of freshly published graphical b-metric spaces. They provided suitable examples of graphs that corroborated the existing theory. They demonstrated the anticipated results by applying them to several nonlinear issues encountered in engineering and research. Younis and Singh [9] discovered adequate conditions for the existence of solutions to certain classes of Hammerstein integral equations and fractional differential equations. They extended the concept of Kannan mappings in terms of F-contraction in the context of b-metric-like spaces and provided a series of novel and nontrivial instances, as well as computer simulations, to demonstrate the established results, therefore introducing the concept in a novel way. On the other hand, several unresolved issues are offered to enthusiastic readers. More information on Kannan’s fixed point theorems can be found here (see [10–15]). The mathematics underpinnings of fuzzy set theory, which were pioneered by Zadeh [16] in 1965 and have made significant progress, are well understood in fuzzy theory. The fuzzy theory has the potential to be applied to various real-world problems. The possibility theory, for example, has been developed by several researchers, including Dubois and Prade [17] and Nahmias [18]. The contribution of probability theory, fuzzy set theory, and rough sets to the study of uncertainty is critical. Yet, these theories have some limitations as well as advantages. The theory of soft sets, developed by Molodtsov [19], was introduced as a new mathematical strategy for dealing with uncertainties to overcome these characteristics. Soft sets have been widely used in various disciplines and technologies. In particular, Maji et al. [20, 21] studied several operations on soft sets and applied their findings to decision-making problems in the literature. Several writers, including Chen [22], Pei and Miao [23], Zou and Xiao [24], and Kong et al. [25], have discovered significant characteristics of soft sets. Soft semirings, soft ideals, and idealistic soft semirings were all investigated by Feng et al. [26]. Das and Samanta developed the ideas of a soft real number and a soft real set in [27] and discussed the characteristics of each concept. These principles served as the foundation for their investigation into the concept of “soft metrics” in “[28].” (See [29, 30] for a more in-depth examination.) Based on the idea of soft elements of soft metric spaces, Abbas et al. [31] developed the concept of soft contraction mapping, which they named “soft contraction mapping.” They focused on fixed points of soft contraction maps and obtained, among other things, a soft Banach contraction principle as a result of their efforts. In their paper, Abbas et al. [32] demonstrated that every complete soft metric induces an equivalent complete usual metric. They obtained in a direct way soft metric versions of various significant fixed point theorems for metric spaces, such as the Banach contraction principle, Kannan and Meir-Keeler fixed point theorems, and Caristi theorem, Kirk’s, among other things. In [33], Chen and Lin presented an extension of the Meir and Keeler fixed point theorem to soft metric spaces, which was previously published. Many researchers working on sequence spaces and summability theory were involved in introducing fuzzy sequence spaces and studying their many characteristics. When it comes to fuzzy numbers, Nuray and Sava [34] defined and explored the Nakano sequences of fuzzy numbers, equipped with a definite function. The following theories use operators’ ideals: fixed point theory, Banach space geometry, normal series theory, approximation theory, and ideal transformations. For additional evidence, see [35–37]. According to Faried and Bakery [38], prequasi operator ideals are broader than quasioperator ideals. This study is aimed at introducing a certain space of soft real number sequences, abbreviated (csss), under a pre-quasi-quasi function (csss). The structure of the ideal operators has been described using this space and -numbers. The conditions essential to generate prequasi Banach and closed (csss) supplied with the definite function are investigated. This space’s () and normal structure properties are illustrated. Fixed points have been introduced for Kannan contraction and nonexpansive mapping. Finally, we investigate whether the Kannan contraction mapping has a fixed point in the prequasi operator ideal with which it is linked. A few real-world examples and applications demonstrate the existence of solutions to nonlinear difference equations.

2. Definitions and Preliminaries

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.

Definition 1 (see [27]). A soft real set denoted by , or simply by , is a mapping . If is a single-valued mapping on taking values in , then is called a soft element of or a soft real number. If is a single-valued mapping on taking values in the set of nonnegative real numbers, then is called a nonnegative soft real number. We shall denote the set of nonnegative soft real numbers (corresponding to ) by . A constant soft real number is a soft real number such that for each , we have , where is some real number.

Definition 2 (see [39]). For two soft real numbers , , we say that (a) if , for all (b) if , for all (c) if , for all (d) if , for all

Note that the relation is a partial order on . 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

Let , where for all . Assume is defined by

Note that (1) is a complete metric space(2) for all (3), for all

Definition 3. A sequence of soft real numbers is said to be (a)bounded if the set of soft real numbers is bounded; i.e., if a sequence is bounded, then there are two soft real numbers such that (b)convergent to a soft real number if, for every , there exists such that , for all

By and , we indicate the spaces of bounded and -absolutely summable sequences of reals. Assume is the classes of all sequence spaces of soft reals. If , where is the space of positive real sequences, we introduce Nakano sequences of soft reals such as [34] and marked it by where The space , where and , for all , is a Banach space. Suppose , one has

Lemma 4 (see [40]). If and , for all , one gets where .

3. Some Properties of

We have investigated in this section the certain space of sequences of soft real numbers under definite function to form prequasi (csss). We present sufficient conditions of under definite function to construct prequasi Banach and closed (csss). The Fatou property of different prequasi norms on has been explained. We have explored the uniform convexity (UUC2), the property (), and this space’s -normal structure property.

Definition 5. The linear space is called a certain space of sequences of soft reals (csss), when (1), where , for marks at the place(2) is solid, i.e., if , , and , for all , one has (3), where indicates the integral part of , assume

Definition 6. A subclass of is said to be a premodular (csss), if one has holds the following conditions: (i)Suppose , with , where (ii)We have , the inequality holds, for all and (iii)One has , the inequality satisfies, for all (iv)When , for all , we have (v)The inequality verifies, for some (vi)Assume is the space of finite sequences of soft real numbers, one has the closure of (vii)We have with where , for every

Definition 7. If is a (csss). The function is said to be a prequasi norm on , if it satisfies the following settings: (i)Suppose , with , where (ii)One has , the inequality verifies, for all and (iii)We have , the inequality satisfies, for all

Evidently, by the last two definitions, one has the following two theorems.

Theorem 8. Assume is a premodular (csss), then it is prequasi normed (csss).

Theorem 9. is a prequasi normed (csss), when it is quasinormed (csss).

Definition 10. (a)The function on is called -convex, when for all and (b) is -convergent to , if and only if, If the -limit exists, then it is unique(c) is -Cauchy, if (d) is -closed, if for every -converges to , one has (e) is -bounded, assume (f)The -ball of radius and center , for all , is denoted by (g)A prequasi norm on verifies the Fatou property, if for all sequence with and every , we have

Recall that the Fatou property gives the -closedness of the -balls. We will indicate the space of all increasing sequences of reals by .

Theorem 11. , where , for every , is a premodular (csss), if with .

Proof. (i) Clearly, and .
(1-i) Assume . Then, Hence, .
(ii) We have with , for every .
(1-ii) Suppose and , one has Since . By parts (1-i) and (1-ii), we have is linear. And for every as
(iii) One has with , for every and .
(2) If , for every and . Then then .
(iv) Evidently, from (24).
(3) Assume , one has so . (v) From (25), there are .
(vi) Clearly the closure of .
(vii) One gets , for or , for with

Theorem 12. Assume with , one has which is a prequasi Banach (csss), where , for all .

Proof. From Theorems 11 and 8, the space is a prequasi normed (csss). If is a Cauchy sequence in , then for all , we have such that for every , we obtain Therefore, Since is a complete metric space, so is a Cauchy sequence in , for constant . Then, , for fixed . So , for all . As Then, .

Theorem 13. If with , we have a prequasi closed (csss), where , for all .

Proof. By Theorems 11 and 8, the space is a prequasi normed (csss). When and , one has for every , there is such that for every , one gets Therefore, Since is a complete metric space, so is a convergent sequence in , for constant . Then, , for fixed . As We have .

Theorem 14. The function verifies the Fatou property, when so that , for every .

Proof. Assume with As is a prequasi closed space, we have . For every , then

Theorem 15. The function does not satisfy the Fatou property, for every , if and , for every .

Proof. Assume with As is a prequasi closed space, we have . For all , one can see

Example 16. For , the function is a norm on .

Example 17. The function is a prequasi norm (not a norm) on .

Example 18. The function is a prequasi norm (not a quasinorm) on .

Example 19. The function is a prequasi norm, quasi norm, and not a norm on , for .

Definition 20. (1)[41] If and . Mark For , let Suppose , we take (2)[41] The function holds (UUC2) when for all and , one has such that (3)[42] The function is strictly convex, (SC), when for every with and one gets

Lemma 21. (i)[43] If and for every , one has (ii)[44] Assume and for all with , one obtains

In the next part of this section, we will use the function as , for all .

Theorem 22. If so that , one has is (UUC2).

Proof. Suppose and . If with By using the definition of , one can see then Assume and . For all one has Therefore, or Let first In view of Lemma 21, part (i), one gets then Since by summing inequalities 2 and 3, and from inequality 1, one can see This implies After, assume Put Since and the power function is convex. Hence, As one has For all one obtains In view of Lemma 21, part (ii), one gets So then As by summing inequalities 5 and 6, we have As by summing inequalities 7 and 8, and from inequality 1, then So Evidently, From inequalities 4 and 9, and Definition 20, when we take Therefore, we have so is (UUC2).

Definition 23. The space verifies the property (), if and only if, for every decreasing sequence of -closed and -convex nonempty subsets of so that for some then

By denoting a nonempty -closed and -convex subset of .

Theorem 24. Suppose so that , we have (i)if such that One has a unique with (ii) satisfies the property ().

Proof. To prove (i), if as is -closed, we have . Then, for every , we have so that . Assume is not -Cauchy. There is a subsequence and so that for all Also, we obtain for every As for all , one has So for every . By choosing we have This is a contradiction. Hence, is -Cauchy. Since is -complete, one has -converges to some . For every , we have -converges to . As is -closed and -convex, we have As -converges to one gets Suppose and from Theorem 14, as verifies the Fatou property, we get So As is (UUC2), then it is (SC), which explains the uniqueness of . To prove (ii), if , for some As is increasing, take . If , otherwise, , for every . From (i), one has one point so that , for all . A similar proof will show that -converges to some . Since are -convex, decreasing, and -closed, we have

Definition 25. verifies the -normal structure property, if and only if, for every nonempty -bounded, -convex, and -closed subset of not decreased to one point, then so that

Theorem 26. Suppose so that , then satisfies the -normal structure property.

Proof. Theorem 22 implies that is (UUC2). Suppose is a -bounded, -convex, and -closed subset of not decreased to one point. Then, Put If with , then For all we have and Since is -convex, we have . Since for every We get

4. Kannan Contraction Mapping on

In this section, we have constructed with distinct so that one has a unique fixed point of Kannan contraction mapping.

Definition 27. A mapping is called a Kannan -contraction, when we have so that for every . The mapping is said to be Kannan -nonexpansive, if .

A vector is said to be a fixed point of , if

Theorem 28. Suppose so that and is Kannan -contraction mapping, where , for every , then has a unique fixed point.

Proof. Let , we have . Since is a Kannan -contraction mapping, then Hence, for every so that , then Therefore, is a Cauchy sequence in . Since the space is prequasi Banach space, we have so that . To show that , as holds the Fatou property, we get so . Hence, is a fixed point of . To prove the uniqueness, assume are two not equal fixed points of . Then, Hence,

Corollary 29. Assume so that , and is Kannan -contraction mapping, where , for every , then has unique fixed point with

Proof. By Theorem 28, we have a unique fixed point of . Then,

Example 30. If , where , for all and Since for all so that , we have For every with , we get For each with and , one has Therefore, is Kannan -contraction. Since holds the Fatou property, by Theorem 28, we have that holds unique fixed point

Definition 31. If is a prequasi normed (csss), and The mapping is said to be -sequentially continuous at , if and only if, assume one has .

Example 32. If , where , for all and is obviously both -sequentially continuous and discontinuous at .

Example 33. Suppose is defined as in Example 30. If with where so that .
Since the prequasi norm is continuous, one obtains Hence, is not -sequentially continuous at .

Theorem 34. Assume so that , and , where , for every . If (1) is Kannan -contraction mapping(2) is -sequentially continuous at (3)One has so that has converging to Then, is the only fixed point of

Proof. Suppose is not a fixed point of , we have . By using conditions (24) and (25), one has Since is Kannan -contraction, then Since , this gives a contradiction. So is a fixed point of . To prove the uniqueness, assume is two not equal fixed points of . We have Therefore,

Example 35. If is defined as in Example 30. Suppose , for every . As for every so that , we have For every such that , then For every with and , we have Then, is Kannan -contraction and

Clearly, is -sequentially continuous at and verifies converges to . From Theorem 34, the element is the only fixed point of .

5. Kannan Nonexpansive Mapping on

The enough setups of , where , for all , so that the Kannan nonexpansive mapping on it has a fixed point are presented.

By letting a nonempty -bounded, -convex, and -closed subset of .

Lemma 36. Suppose verifies the () property and the -quasinormal property. If is a Kannan -nonexpansive mapping, for put . Let Hence, , -convex, -closed subset of and

Proof. As , one has . Since the -balls are -convex and -closed, one gets is a -closed and -convex subset of . To prove that , let If we have Otherwise, when , let From the definition of , we have Hence, so By taking , we have so that . Then, Since is an arbitrary positive, we have so . As , we have then is -invariant. To prove that As for every If We get The definition of implies Hence, Then, for all this implies

Theorem 37. Assume verifies the -quasinormal property and the () property. If is a Kannan -nonexpansive mapping, so has a fixed point.

Proof. Put and for all By the definition of we have , for all If is defined as in Lemma 36, it is obvious that is a decreasing sequence of nonempty -bounded, -closed, and -convex subsets of . The property () holds that Put then for every If one has then Hence, Therefore, . Otherwise, then fails to have a fixed point. Put as defined in Lemma 36. Since fails to have a fixed point and is -invariant, so has more than one point, then . By the -quasinormal property, we have so that for every In view of Lemma 36, one has By definition of then We have which contradicts the definition of . So which gives that any point in is a fixed point of .

In view of Theorems 24, 26, and 28, we have the following.

Corollary 38. If so that and is a Kannan -nonexpansive mapping. One has that holds a fixed point.

Example 39. Suppose so that where and , for all . From Example 35, is Kannan -contraction. Therefore, it is Kannan -nonexpansive. From Corollary 38, then has a fixed point in .

6. Kannan Contraction and Structure of Operators Ideal

The structure of the operators ideal by under definite function , where , for all , and -soft reals has been offered. Finally, we study the idea of Kannan contraction mapping in its linked prequasi operator ideal. Also, the existence of a fixed point of Kannan contraction mapping has been offered. We mark the space of all bounded, finite rank linear operators from a Banach space into a Banach space by , and and if , we indicate and .

Definition 40 (see [45]). An -number function is which gives all a holds the next conditions: (a), for every (b) for every and , (c), for all , and , where and are arbitrary Banach spaces(d)Suppose and , one has (e)If , then , for all (f) or , where marks the unit map on the -dimensional Hilbert space

Definition 41 (see [37]). Suppose is the class of all bounded linear operators between any arbitrary Banach spaces. A subclass of is called an operator ideal, when every holds the next setups: (i), where marks Banach space of one dimension(ii)The space is linear over (iii)If , and , one has

Notations 42. for every

Theorem 43. If is a (csss), one has an operator ideal.

Proof. (i)Suppose and , for every , since , for every , and is a linear space, then ; for that then (ii)If and so by Definition 5 condition (25) one has and , as , by the definition of -numbers and is decreasing, we have for all . By Definition 5 part (2) and is a linear space, we get ; hence, (iii)Assume , , and , then and since , from Definition 5 parts (1) and (2), then , then

In view of Theorems 11 and 43, we have the following theorem.

Theorem 44. If so that , then is an operator ideal.

Definition 45 [38]. A function is said to be a prequasi norm on the ideal , when the next setups are verified. (1)If , , and , if and only if, (2)One has so as to , for all and (3)One has with , for all (4)One has for to if , , and , one has

Theorem 46 (see [38]). is a prequasi norm on the ideal , whenever is a quasinorm on the ideal .

Theorem 47. Suppose so that ; hence, the function is a prequasi norm on , with , for every .

Proof. (1)If , and if and only if, for all if and only if, (2)One has with , for every and (3)There are with for , we have (4)There are , assume , and , one has

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

Theorem 48. If so that , then is a prequasi Banach operator ideal.

Proof. Assume is a Cauchy sequence in . Since , we have Then, is a Cauchy sequence in . Since is a Banach space, one has with and as , for every and is a premodular (csss). Then, one gets , then .

Theorem 49. Assume so that , then is a prequasi closed operator ideal.

Proof. If , for every and . Since , we have Hence, is convergent in ; i.e., and as , for every and is a premodular (csss). Then, We obtain ; hence, .

Definition 50. A prequasi norm on the ideal holds the Fatou property if for all with and , then

Theorem 51. If so that , then does not satisfy the Fatou property.

Proof. Let so that As is a prequasi closed ideal, one has . So for all , then

Definition 52. An operator is called a Kannan -contraction, if there is so that for every .

Definition 53. An operator is called -sequentially continuous at , where if and only if, .

Example 54. Assume , where , for all and Clearly, is -sequentially continuous at the zero operator . Suppose so that where with . As the prequasi norm is continuous, we have Hence, is not -sequentially continuous at .

Theorem 55. If so that and . Suppose (i) is Kannan -contraction mapping(ii) is -sequentially continuous at a vector (iii)we have so that the sequence of iterates has a converging to

Then, is the unique fixed point of .

Proof. Assume is not a fixed point of , one has . By using conditions (ii) and (iii), one has As is Kannan -contraction, we get Since , we have a contradiction. Therefore, is a fixed point of . To prove the uniqueness of the fixed point , assume there are two not equal fixed points of . We get So,

Example 56. According to Example 54, as for every with , then For every with , then For each with and , one gets Therefore, is Kannan -contraction and

Clearly, is -sequentially continuous at and has a subsequence that converges to . According to Theorem 55, is the only fixed point of .

7. Applications

In this section, some successful applications to the existence of solutions of nonlinear difference equations of soft functions are introduced.

Theorem 57. Assume the summable equations which are considered by many authors [46–48], and let , where with and , for all , given by The summable equation (92) has a unique solution in when and for all , suppose

Proof. We have In view of Theorem 37, there is a unique solution of equation (92) in

Example 58. If , where , for every . Assume the summable equations so that and . Let . Clearly, is a nonempty, -convex, -closed, and -bounded subset of . Suppose is defined as Evidently, According to Theorem 57 and Corollary 38, the summable equations (96) have a solution in .

Example 59. If , where , for every , let the nonlinear difference equations so that , , for every , and suppose explained by It is clear that In view of Theorem 57, the nonlinear difference equations (99) have a unique solution in .

8. Conclusion

The site we discussed was a “pre-quasinormed” place rather than a “quasinormed” location. In the prequasi Banach space, the concept of a fixed point of the Kannan prequasi norm contraction mapping is introduced (csss). Both () and the pre-quasinormal structure are supported. The occurrence of a fixed point in the Kannan nonexpansive mapping was studied in this study. A fixed point of Kannan contraction mapping in the prequasi Banach operator ideal formed by Nakano (csss) and the -soft real numbers has also been investigated for a fixed point of Kannan contraction mapping. Finally, we have demonstrated how the results can be applied to a problem by presenting a few examples of how this has happened. Under a wide range of flexible conditions, the presence of a sequence can be established using the Nakano sequence space. Specifically, when it comes to the variable exponent in the previously described space, our key conclusions have helped to strengthen several well-established ideas.

Data Availability

No data were used to support this study.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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 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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