Kannan Contraction Maps on the Space of Null Variable Exponent Second-Order Quantum Backward Difference Sequences of Fuzzy Functions and Its Pre-Quasi Ideal

Bakery, Awad A.; Mohamed, OM Kalthum S. K.

doi:https://doi.org/10.1155/2022/5339667

Discrete Dynamics in Nature and Society

On this page

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

Research Article | Open Access

Volume 2022 | Article ID 5339667 | https://doi.org/10.1155/2022/5339667

Kannan Contraction Maps on the Space of Null Variable Exponent Second-Order Quantum Backward Difference Sequences of Fuzzy Functions and Its Pre-Quasi Ideal

Awad A. Bakery^1,2and OM Kalthum S. K. Mohamed^2,3

Academic Editor: Ewa Pawluszewicz

Received25 Mar 2022

Revised23 May 2022

Accepted23 Jun 2022

Published27 Aug 2022

Abstract

In this paper, we construct and investigate the space of null variable exponent second-order quantum backward difference sequences of fuzzy functions, which are crucial additions to the concept of modular spaces. The idealization of the mappings has been achieved through the use of extended fuzzy functions and this sequence space of fuzzy functions. This new space’s topological and geometric properties and the mappings’ ideal that corresponds to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi ideal. To demonstrate our findings, we give a number of numerical experiments. There are also some significant applications of the existence of solutions to nonlinear difference equations of fuzzy functions.

1. Introduction

We assume that is the set of non-negative integers. Yaying et al. [1] defined quantum second-order backward difference operator, , where , for , and , for all and . Note that the operator reduces to when , which defined and studied in Reference [2]. They proved that the spaces and are Banach spaces linearly isomorphic to and , respectively, and obtained their Schauder bases and , , and duals. They determined the spectrum, the point spectrum, the continuous spectrum, and the residual spectrum of the operator over the Banach space of null sequences. It is clear to see that

For the strict inclusion, we have and , and and .

The mappings’ ideal theory is well regarded in functional analysis. Fixed-point theory, Banach space geometry, normal series theory, approximation theory, and ideal transformations all use mappings’ ideal. Using -numbers is an essential technique. For more background details, see Pietsch [3], Constantin [4], and Tita [5]. Pre-quasi mappings’ ideals are more extensive than quasi mappings’ ideals, according to Faried and Bakery [6]. Bakery and Elmatty [7] explained a note on Nakano generalized difference sequence space under premodular. Since the booklet of the Banach fixed-point theorem [8], many mathematicians have worked on many developments. For more background and recent works on applicative approach of fixed-point theory, see Ruẑiĉka [9], Mao et al. [10], and Younis et al. [11–14]. Kannan [15] gave an example of a class of mappings with the same fixed-point actions as contractions, though that fails to be continuous. The only attempt to describe Kannan operators in modular vector spaces was once made in Reference [16]. Bakery and Mohamed [17] explored the concept of the pre-quasi-norm on Nakano sequence space such that its variable exponent belongs to . They explained the sufficient conditions on it equipped with the definite pre-quasi-norm to generate pre-quasi Banach. They examined the Fatou property of different pre-quasi-norms on it. Moreover, they showed a fixed point of Kannan pre-quasi-norm contraction maps on it and on the pre-quasi Banach operator ideal constructed by -numbers that belong to this sequence space.

Zadeh [18] established the concept of fuzzy sets and fuzzy set operations, and many researchers adopted the concept of fuzziness in cybernetics and artificial intelligence as well as in expert systems and fuzzy control. We refer the reader to the following exciting works dealing with Kannan mappings and fuzzy concepts with different applications, see Reference [19–25]. Many researchers in sequence spaces and summability theory studied fuzzy sequence spaces and their properties. In Reference [26], the Nakano sequences of fuzzy integers were defined and analyzed. Bakery and Mohamed [27] introduced 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. They defined and studied the weighted Nakano sequence spaces of fuzzy functions. They constructed the ideal generated by extended fuzzy functions and the sequence spaces of fuzzy functions. They presented some topological and geometric structures of this class of ideal and multiplication mappings acting on this sequence space of fuzzy functions. Moreover, the existence of Caristi’s fixed point was examined. Many fixed-point theorems are effective when applied to a given space because they either enlarge the self-mapping acting on it or expand the space itself. Specifically, in this study, we construct and investigate the space of null variable exponent second-order quantum backward difference sequences of fuzzy functions, which are crucial additions to the concept of modular spaces. The idealization of the mappings has been achieved through the use of extended fuzzy functions and this sequence space of fuzzy functions. The topological and geometric properties of this new space and the mappings’ ideal that corresponds to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi 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 fuzzy functions are introduced.

2. Definitions and Preliminaries

It is worth mentioning that Matloka [28] introduced bounded and convergent fuzzy numbers, investigated some of their properties, and demonstrated that any convergent fuzzy number sequence is bounded. Nanda [29] researched fuzzy number sequences and demonstrated that the set of all convergent fuzzy number sequences forms a complete metric space. Kumar et al. [30] presented the concept limit points and cluster points of sequences of fuzzy numbers. If is the set of all closed and bounded intervals on the real line , then we assume and in , let

Clearly, the relation is a partial order on . We define a metric on by

Matloka [28] proved that is a metric on and is a complete metric space.

Definition 1. A fuzzy number is a fuzzy subset of , that is, a mapping that verifies the four conditions:(a) is fuzzy convex; that is, for , and , .(b) is normal; that is, there is such that .(c) is an upper-semicontinuous, that is, for all , , for all , is open in the usual topology of .(d)The closure of is compact.The -level set of a fuzzy real number , denoted by , is defined asThe set of all upper semicontinuous, normal, convex fuzzy number, and is compact, is marked by . The set can be embedded in , if we define byThe additive identity and multiplicative identity in are denoted by and , respectively. We assume that and the -level sets are , , and . A partial ordering for any is as follows: , if and only if, , for all .
We assume that is defined by
We recall that(1) is a complete metric space(2) for all (3)(4), for all By , , and , we denote the space of null, bounded, and -absolutely summable sequences of real numbers, respectively. 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 when , we inscribe and . The space of approximable and compact bounded linear mappings from into will be denoted by and , and if , we mark and , respectively.

Definition 2 (see [31]). An -number function is a mapping that gives all a holds the following conditions:(a), for every .(b), for every and , .(c), for every , , and , where and are arbitrary Banach spaces.(d)Assume and , then .(e)If , then , for all .(f) or , where indicates the unit mapping on the -dimensional Hilbert space .We give here some examples of -numbers:(1)The th Kolmogorov number, denoted by , is marked by .(2)The -th approximation number, indicated by , is marked by .

Definition 3 (see [32]). Let be the class of all bounded linear operators within any two arbitrary Banach spaces. 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 .

Notations 1 (see [27]). where

Definition 4 (see [6]). A function is said to be a pre-quasi-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 1 (see [6]). is a pre-quasi-norm on the ideal , whenever is a quasi-norm on the ideal .

Lemma 1 (see [33]). If and , for all , then where .

3. Some Characteristics of

This section is devoted to provide sufficient criteria for the space of null variable exponent second-order quantum backward difference sequences of fuzzy numbers, , endowed with definite function , to be pre-quasi Banach. We have examined some algebraic and topological properties such as completeness, solidness, symmetry, and convergence-free. The Fatou property of various pre-quasi-norms on has been presented.

Let denote the classes of all sequence spaces of fuzzy real numbers. If , where is the space of positive reals. The space of null variable exponent second-order quantum backward difference sequences of fuzzy numbers is defined as follows:.

Theorem 2. If , then

Proof. It is clear to see that if , thenFor the strict inclusion, we have and , and and .
For , a given sequence denotes the set of all permutation of the elements of ; that is, .

Definition 5. (1)A sequence space of fuzzy numbers is said to be symmetric if , for all (2)A sequence space of fuzzy numbers is said to be convergence-free if whenever and implies

Theorem 3. If , then the space is not symmetric.

Proof. consider. Then, . Now, if is the rearrangement of defined by , then . Therefore, the space is not symmetric.

Theorem 4. If , then the space is not convergence-free.

Proof. consider the sequence . Then, . Again if , then clearly, . Hence, the space is not convergence-free.
Let us mark the space of all functions by .

Definition 6 (see [34]). is a vector space. A function is said to be modular if the following conditions hold:(a)Assume , with , where (b) verifies for every and (c)The inequality holds for every and

Definition 7 (see [27]). The linear space is called a certain space of sequences of fuzzy numbers (cssf), when(1), where , while displays at the place(2) is solid; that is, if , , and for every , then (3), where denotes the integral part of , if

Definition 8 (see [27]). A subclass of is said to be a premodular (cssf), if there is , which satisfies the following conditions:(i)Assume , with , where .(ii)One has , the inequality holds for all and .(iii)One has , the inequality verifies, for all .(iv)Suppose , for all , then .(v)The inequality, , verifies, for some .(vi)If is the space of finite sequences of fuzzy numbers, then the closure of .(vii)One has with , whereWe note that the notion of premodular vector spaces is more general than modular vector spaces. There are some examples of premodular vector spaces but not modular vector spaces.

Example 1. The function on the vector space . As for every , one has

Example 2. The function on the vector space . As for every , one hasSome examples of premodular vector spaces and modular vector spaces are as follows:

Example 3. The function on the vector space . As for every , one has

Example 4. The function is a premodular (modular) on the vector space .

Definition 9 (see [27]). is a cssf. The function is said to be a pre-quasi-norm on , if it verifies the following settings:(i)Suppose , with , where .(ii)We have , the inequality holds, for all and .(iii)One has , the inequality verifies, for all .

Theorem 5 (see [27]). We suppose that is a premodular (cssf), then it is pre-quasi-normed (cssf).

Theorem 6 (see [27]). is a pre-quasi-normed (cssf), if it is quasi-normed (cssf).

Definition 10. (a)The function on is called -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-quasi-norm on satisfies the Fatou property, if for every sequence under and all , one has .We recall that the Fatou property gives the -closedness of the -balls. We will denote the space of all increasing sequences of real numbers by .

Theorem 7. , where , for every , is a premodular (cssf), if the following conditions are satisfied:(a) with .(b) is an absolute nondecreasing; that is, if , for all , then .

Proof. Clearly, and .(i)Assume . We have Then, .(ii)There are with , for every .(iii)If and , one has So, . From parts (1-i) and (1-ii), we have is linear. Also, , for every , as (iv)One has with , for every and .(v)If , for every and . We obtain Then, .(vi)Evidently, from Reference (2).(vii)Assume , one can see Then, . (v) From (3), one has .(viii)Clearly, the closure of .(ix)One gets , for or , for with

Theorem 8. If the conditions of Theorem 7 are satisfied, then is a pre-quasi Banach (cssf), where , for all .

Proof. According to Theorem 7 and Theorem 5, the space is a pre-quasi-normed (cssf). If is a Cauchy sequence in , hence for all , then such that for every , we haveTherefore, Since is a complete metric space, is a Cauchy sequence in , for fixed . This gives , for fixed . Then, , for all . As Then, .

Theorem 9. The function satisfies the Fatou property, when the conditions of Theorem 7 are satisfied.

Proof. Let such that . Since is a pre-quasi closed space, we have . For every , then

Theorem 10. The function does not satisfy the Fatou property, for all , if the conditions of Theorem 7 are satisfied with .

Proof. Assume such that . As is a pre-quasi closed space, we have . For all , then

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

Example 6. The function is a pre-quasi-norm (not a norm) on .

Example 7. The function is a pre-quasi-norm (not a quasi-norm) on .

4. Structure of Mappings’ Ideal

The structure of the mappings’ ideal by , where , for all , and extended fuzzy functions have been explained. We study enough setups on such that the class is complete. We investigate conditions (not necessary) on such that . This gives a negative answer of Rhoades [34] open problem about the linearity of type spaces. We explain enough setups on such that is strictly contained for different powers, weights, and backward generalized differences, the class is simple, and the space of every bounded linear mappings is which sequence of eigenvalues in equals .

Theorem 11 (see [27]). If is a (cssf), then is a mappings’ ideal.
In view of Theorem 7 and Theorem 11, one has the following theorem:

Theorem 12. If the conditions of Theorem 7 are satisfied, then is a mappings’ ideal.

Theorem 13. If the conditions of Theorem 7 are satisfied, then the function is a pre-quasi-norm on , with , for every .

Proof. (1)Suppose , and , if and only if, , for all , if and only if, .(2)One has with , for all and .(3)For , we have(4)There are , if , and , thenIn the next theorems, we will use the notation , where , for all .

Theorem 14. assume that the conditions of Theorem 7 are satisfied, then is a pre-quasi Banach mappings’ ideal.

Proof. Let be a Cauchy sequence in . Since , thenThis implies that is a Cauchy sequence in . Since is a Banach space, one has such that and as , for every , and is a premodular (cssf). Then, we haveHence, one has , then .

Definition 11. A pre-quasi-norm on the ideal satisfies the Fatou property if for all such that and , then

Theorem 15. If the conditions of Theorem 7 are satisfied, then does not satisfy the Fatou property.

Proof. Let with . Since is a pre-quasi closed ideal, then ; hence, for all , we have

Theorem 16. , if the conditions of Theorem 7 are satisfied. But the converse is not necessarily true.

Proof. As , for all and is a linear space. If , one has . Then, . Suppose , one has . Since , if , one has so that . As is decreasing, one getsThen, one has such that rank andAs with , we takeAccording to inequalities (1–3), thenThis implies . Contrarily, one has a counterexample as , but is not satisfied.

Theorem 17. assume the conditions of Theorem 7 are satisfied with , for every , then

Proof. suppose that , then . We haveThen, . Next, if we take , one has so thatTherefore, and .
Evidently, . After, if we choose , then . One has such that .

Lemma 2 (see [3]). If we suppose and , then and with , with .

Theorem 18 (see [3]). In general, one has

Theorem 19. If the conditions of Theorem 7 are satisfied with , for all , then

Proof. Let and
. In view of Lemma 2, one has
and so that , and then, with , we haveThis contradicts Theorem 18. As .

Corollary 1. suppose that the conditions of Theorem 7 are satisfied with , for every , then

Proof. Obviously, since .

Definition 12. [3] A Banach space is said to be simple, if there is only one nontrivial closed ideal in .

Theorem 20. assume that the conditions of Theorem 7 are verified, then is simple.

Proof. Let and . From Lemma 2, there exist with . This implies . If , then is a simple Banach space.

Notations 2.

Theorem 21. If the conditions of Theorem 7 are satisfied, then

Proof. Let , then and , for all . Therefore, . One has , for every , sofor every . Hence, , and then, . After, we assume . Hence, . We haveAs is continuous, then If exists, with , then exists and bounded, for every . As exists and bounded. As is a pre-quasi mappings’ ideal, one getsWe have a contradiction, since . Then, , with , which proves that .

Theorem 22. For type, If is a mappings’ ideal, then the following conditions are verified:(1) type .(2)Assume type and type , then type .(3)If and type , then type .(4)The sequence space is solid; that is, if type and , for all and , then type .

Proof. If is a mappings’ ideal.(i)We have . Hence, for all , we have . This gives . Hence, type .(ii)The space is linear over . Hence, for each and , we have . This implies(iii)If , and , then , where and are arbitrary Banach spaces. Therefore, since , then . Since . By using condition 3, if , we have . This means is solid.In view of Theorem 12 and Theorem 23, we conclude the following properties of the space.

Theorem 23. If , then the following conditions are verified:(1) type .(2)Assume type and type , then type .(3)If and type , then type .(4)The sequence space is solid; that is, if type and , for all and , then type .

Theorem 24. The space is not mappings’ ideal, if the conditions (a) and (c) of Theorem 7 are satisfied

Proof. If we choose , , , for or , otherwise, for all . We have , for all , and . Hence, the space is not solid.

5. 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. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi ideal. Interestingly, several numerical experiments are presented to illustrate our results.

Definition 13. An operator is said to be a Kannan -contraction, if one gets with , for all .
An element is called a fixed point of , when .

Theorem 25. If the conditions of Theorem 7 are satisfied, 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 getsSo, for all with , one getsThen, is a Cauchy sequence in . As the space is pre-quasi Banach space. One has with to prove that . Since verifies the Fatou property, one obtainsThen, . So, is a fixed point of to show the uniqueness. Let be two not equal fixed points of . One hasSo, .

Corollary 2. If the conditions of Theorem 7 are satisfied, and is Kannan -contraction mapping, where , for all , one has unique fixed point of so that .

Proof. In view of Theorem 26, one has a unique fixed point of . So

Example 8. assume , where , for every andAs for each with , one hasFor all with , one hasFor all with and , we getHence, is Kannan -contraction as satisfies the Fatou property. From Theorem 26, one has holds one fixed point .

Definition 14. pick up be a pre-quasi-normed (cssf), and . The operator is called -sequentially continuous at , if and only if, when , then .

Example 9. suppose that , where , for every and is clearly both -sequentially continuous and discontinuous at .

Example 10. assume that is defined as in Example 8. Suppose is such that , where with .
As the pre-quasi-norm is continuous, we haveTherefore, is not -sequentially continuous at .

Theorem 26. If the conditions of Theorem 7 are satisfied with , and , where , for all , then we suppose that(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. If we assume that is not a fixed point of , one has . From parts (2) and (3), we getAs is Kannan -contraction, one obtainsAs , one has a contradiction. Then, is a fixed point of to show that the uniqueness. Let be two not equal fixed points of . One obtainsHence, .

Example 11. assume that is defined as in Example 8. Let , for all . Since for all with , one getsFor all with , one getsFor all with and , one getsSo, is Kannan-contraction and
Obviously, is -sequentially continuous at and holds converges to . By Theorem 27, the point is the only fixed point of .

Definition 15. An operator is said to be a Kannan -contraction, if one has with for all .

Definition 16. An operator is said to be -sequentially continuous at , where , if and only if, .

Example 12. If,where , for every andEvidently, is -sequentially continuous at the zero operator . Let be such that , where with . Since the pre-quasi-norm is continuous, one getsTherefore, is not -sequentially continuous at .

Theorem 27. the conditions of Theorem 7 are satisfied and , then we assume that(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 getSince is Kannan -contraction, one obtainsAs , there is a contradiction. Hence, is a fixed point of to prove that the uniqueness of the fixed point . We suppose that one has two not equal fixed points of . So, one gets Then, .

Example 13. In view of Example 12. Since for all with , we haveFor all with , we haveFor all with and , we haveHence, is Kannan -contraction and
Obviously, is -sequentially continuous at and has a subsequence converges to . By Theorem 28, is the only fixed point of .

6. Applications

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

Theorem 28. consider the summable equationwhich presented by Salimi et al. [35], and assume , where the conditions of Theorem 7 are satisfied and , for every , defined byThe summable equation (4) has a unique solution in , if , , , , there is so that , and for all , we have

Proof. One hasBy Theorem 26, one gets a unique solution of equation (4) in .

Example 14. suppose, where , for all . We consider the summable equationwith . Let defined byObviously,By Theorem 29, the summable equation (75) has a unique solution in .

Example 15. suppose , where , for all . We consider the summable equationwith . Let defined byObviously,By Theorem 29, the summable equation (75) has a unique solution in .

Example 16. suppose , where , for every . We consider the nonlinear difference equations:with , , for all , and assume , defined byEvidently,By Theorem 29, the nonlinear difference (81) have a unique solution in .

Theorem 29. consider the summable equation (4) and assume is defined by (5), where the conditions of Theorem 7 are satisfied with and , for every . The summable equation (4) has a unique solution , if the following conditions are satisfied:(1)If , , , , there is so that , and for all , we have(2) is -sequentially continuous at .(3)There is with has converging to .

Proof. One hasBy Theorem 27, one gets a unique solution of (4).

Example 17. suppose, where , for all . We consider the summable equationwith . Let defined byWe assume is -sequentially continuous at , and there is with has converging to . Obviously,By Theorem 29, the summable equation (86) has a unique solution .

Example 18. suppose , where , for all . We consider the summable equationwith . Let defined byWe assume that is -sequentially continuous at , and there is with has converging to . Obviously,By Theorem 29, the summable equation (89) has a unique solution .

Example 19. suppose , where , for every . We consider the nonlinear difference equation:with , , for all , and assume , defined byWe suppose that is -sequentially continuous at , and there is with has converging to . Evidently,By Theorem 29, the nonlinear difference equation (18) has a unique solution .
In this part, we search for a solution to nonlinear matrix (81) at , where and are Banach spaces, the conditions of Theorem 7 are satisfied, and
, for all . We consider the summable equationAnd we suppose that is defined by

Theorem 30. The summable equation (18) has one solution , if the following conditions are satisfied:(a), , , , and for every , there is so that , with(b) is -sequentially continuous at a point .(c)There is so that the sequence of iterates has a subsequence converging to .

Proof. suppose the settings are verified. We consider the mapping defined by (19). We haveIn view of Theorem 27, one obtains a unique solution of equation (18) at .

Example 20. We assume the class , where , for all .
We consider the nonlinear difference equations:where and and let be defined asWe 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 thatBy Theorem 30, the nonlinear difference (99) has one solution .

Example 21. assume the class , where , for all . We consider the nonlinear difference equation (20) and let be defined as (21). 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 thatBy Theorem 30, the nonlinear difference (99) has one solution .

7. Conclusion

In this paper, we have explained sufficient settings of the space equipped with definite function to be pre-quasi Banach. The Fatou property of various pre-quasi-norms on has been investigated. The geometric and topological structures of the mappings’ ideal by this space and extended fuzzy functions have been explained. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated pre-quasi 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 fuzzy functions are introduced. As a future project, we can build the domain of second-order quantum backward difference in Nakano sequences of fuzzy functions space and look at its properties.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declared that they have no conflicts of interest.

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-92. The authors, therefore, acknowledge with thanks the University technical and financial support.

References

T. Yaying, B. Hazarika, B. Chandra Tripathy, and M. Mursaleen, “The spectrum of second order quantum difference operator,” Symmetry, vol. 14, no. 3, p. 557, 2022.
View at: Publisher Site | Google Scholar
M. Et, “On some difference sequence spaces,” Doğa-Tr. J. Math, vol. 17, pp. 18–24, 1993.
View at: Google Scholar
A. Pietsch, Operator Ideals, North-Holland Publishing Company, Amsterdam, Netherlands, 1980.
Gh. Constantin, “Operators of type,” Rend. Acc. Naz. Lincei, vol. 52, no. 8, pp. 875–878, 1972.
View at: Google Scholar
N. Tita, “On Stolz mappings,” Math. Japonica, vol. 26, no. 4, pp. 495-496, 1981.
View at: Google Scholar
N. Faried and A. A. Bakery, “Small operator ideals formed by numbers on generalized Cesàro and Orlicz sequence spaces,” Journal of Inequalities and Applications, vol. 1, 357 pages, 2018.
View at: Publisher Site | Google Scholar
A. A. Bakery and A. R. A. Elmatty, “A note on Nakano generalized difference sequence space,” Advances in Difference Equations, vol. 2020, no. 1, p. 620, 2020.
View at: Publisher Site | Google Scholar
S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
View at: Publisher Site | Google Scholar
M. Ruẑiĉka, Electrorheological fluids. Modeling and mathematical theory Lecture Notes in Mathematics, Springer, Berlin, Germany, vol. 1748, 2000.
W. Mao, Q. Zhu, and X. Mao, “Existence, uniqueness and almost surely asymptotic estimations of the solutions to neutral stochastic functional differential equations driven by pure jumps,” Applied Mathematics and Computation, vol. 254, pp. 252–265, 2015.
View at: Publisher Site | Google Scholar
M. Younis, D. Singh, and A. A. N. Abdou, “A fixed point approach for tuning circuit problem in dislocated -metric spaces,” Mathematical Methods in the Applied Sciences, vol. 45, no. 4, pp. 1801–2459, 2022.
View at: Google Scholar
M. Younis and D. Singh, “On the existence of the solution of Hammerstein integral equations and fractional differential equations,” J. Appl. Math. Comput, vol. 68, no. 2, pp. 1087–1105, 2022.
View at: Publisher Site | Google Scholar
M. Younis, D. Singh, I. Altun, and V. Chauhan, “Graphical structure of extended -metric spaces: an application to the transverse oscillations of a homogeneous bar,” International Journal of Nonlinear Sciences and Numerical Stimulation, Article ID 000010151520200126, 2021.
View at: Publisher Site | Google Scholar
M. Younis, A. Stretenović, and S. Radenović, “Some critical remarks on ’Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations,” Nonlinear Analysis Modelling and Control, vol. 27, no. 1, pp. 163–178, 2022.
View at: Publisher Site | Google Scholar
R. Kannan, “Some results on fixed points- II,” The American Mathematical Monthly, vol. 76, no. 4, p. 405, 1969.
View at: Publisher Site | Google Scholar
S. J. H. Ghoncheh, Some Fixed point theorems for Kannan mapping in the modular spaces, Ciĕncia eNatura, Santa Maria, CA, USA, vol. 37, pp. 462–466, 2015.
A. A. Bakery and O. S. K. Mohamed, “Kannan prequasi contraction maps on Nakano sequence spaces,” J. Funct. Spaces, vol. 2020, Article ID 8871563, 10 pages, 2020.
View at: Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
K. Javed, F. Uddin, H. Aydi, A. Mukheimer, and M. Arshad, “Ordered-theoretic fixed point results in fuzzy -metric spaces with an application,” Journal of Mathematics, vol. 2021, Article ID 6663707, 7 pages, 2021.
View at: Publisher Site | Google Scholar
S. U. Rehman and H. Aydi, “Rational fuzzy cone contractions on fuzzy cone metric spaces with an application to fredholm integral equations,” Journal of Function Spaces, vol. 2021, Article ID 5527864, 13 pages, 2021.
View at: Publisher Site | Google Scholar
H. A. H., M. S. Humaira, H. A. Hammad, M. Sarwar, and M. De L, “Existence theorem for a unique solution to a coupled system of impulsive fractional differential equations in complex-valued fuzzy metric spaces,” Advances in Difference Equations, vol. 2021, no. 1, 2021.
View at: Publisher Site | Google Scholar
B. Rome, M. Sarwar, and R. Rodríguez-López, “Fixed point results via -admissibility in extended fuzzy rectangular -metric spaces with applications to integral equations,” Mathematics, vol. 9, no. 16, 2021.
View at: Publisher Site | Google Scholar
S. U. Rehman, A. Bano, H. Aydi, and C. Park, “An approach of Banach algebra in fuzzy metric spaces with an application,” AIMS Mathematics, vol. 7, no. 5, pp. 9493–9507, 2022.
View at: Publisher Site | Google Scholar
H. Aydi, M. Aslam, D. Sagheer, S. Batul, R. Ali, and E. Ameer, “Kannan-type contractions on new extended -metric spaces,” Journal of Function Spaces, vol. 2021, Article ID 7613684, 12 pages, 2021.
View at: Publisher Site | Google Scholar
M. Samreen, S. Hussain, H. Aydi, and Y. U. Gaba, “-Contractive mappings in fuzzy metric spaces,” Journal of Mathematics, vol. 2022, Article ID 5920124, 10 pages, 2021.
View at: Publisher Site | Google Scholar
F. Nuray and E. Savaş, “Statistical convergence of sequences of fuzzy numbers,” Mathematica Slovaca, vol. 45, no. 3, pp. 269–273, 1995.
View at: Google Scholar
A. A. Bakery and E. A. E. Mohamed, “On the nonlinearity of extended -type weighted Nakano sequence spaces of fuzzy functions with some applications,” Journal of Function Spaces, vol. 2022, Article ID 2746942, 20 pages, 2022.
View at: Publisher Site | Google Scholar
M. Matloka, “Sequences of fuzzy numbers,” Fuzzy Sets and Systems, vol. 28, pp. 28–37, 1986.
View at: Google Scholar
S. Nanda, “On sequences of fuzzy numbers,” Fuzzy Sets and Systems, vol. 33, no. 1, pp. 123–126, 1989.
View at: Publisher Site | Google Scholar
V. Kumar, A. Sharma, K. Kumar, and N. Singh, “On I-limit points and I-cluster points of sequences of fuzzy numbers,” International Mathematical Forum, vol. 2, no. 57, pp. 2815–2822, 2007.
View at: Publisher Site | Google Scholar
A. Pietsch, Eigenvalues and -numbers, Cambridge University Press, New York. NY, USA, 1986.
A. A. Bakery and O. K. S. K. Mohamed, “Orlicz generalized difference sequence space and the linked pre-quasi operator ideal,” Journal of Mathematics, vol. 2020, Article ID 6664996, 9 pages, 2020.
View at: Publisher Site | Google Scholar
B. Altay and F. Başar, “Generalization of the sequence space derived by weighted means,” Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 147–185, 2007.
View at: Google Scholar
B. E. Rhoades, “Operators of type,” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur, vol. 59, no. 8, pp. 238–241, 1975.
View at: Google Scholar
P. Salimi, A. Latif, and N. Hussain, “Modified --contractive mappings with applications,” Fixed Point Theory and Applications, vol. 151, p. 2013, 2013.
View at: Google Scholar
H. Nakano, “Modulared sequence spaces,” in Proceedings of the Japan Academy Series A: Mathematical Sciences, vol. 27, no. 9, pp. 508–512, 1951.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Awad A. Bakery and OM Kalthum S. K. Mohamed. 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.

PDF Download Citation

Download other formats

Order printed copies