Kannan Nonexpansive Mappings on Variable Exponent Function Space of Complex Variables with Some Applications

Mohamed, Elsayed A. E.; Bakery, Awad A.

doi:https://doi.org/10.1155/2022/9575121

Journal of Function Spaces

On this page

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

Special Issue

Developments in Functions and Operators of Complex Variables

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 9575121 | https://doi.org/10.1155/2022/9575121

Kannan Nonexpansive Mappings on Variable Exponent Function Space of Complex Variables with Some Applications

Elsayed A. E. Mohamed^1,2and Awad A. Bakery³

Academic Editor: Sarfraz Nawaz Malik

Received24 Jun 2021

Accepted19 Jan 2022

Published15 Feb 2022

Abstract

This paper establishes the reality of a fixed point of Kannan’s prequasinorm contraction mapping on the variable exponent function space of complex variables, demonstrating that it satisfies the property (R) and possesses a prequasinormal structure. We have established the presence of a fixed point of Kannan prequasinorm nonexpansive mapping on it and Kannan prequasinorm contraction mapping in the prequasi-Banach operator ideal, created by this function space and -numbers. Finally, we provide some applications for solutions to summable equations and illustrate instances to corroborate our findings.

1. Introduction

Many mathematicians have worked on feasible extensions to the Banach fixed point theorem since the publication of book [1] on the Banach fixed point theorem. Kannan [2] approved an instance of a class of operators that perform the same fixed point operations as contractions but with a continuous flop. Ghoncheh [3] made the first attempt to characterize Kannan operators in modular vector spaces. The variable exponent Lebesgue spaces contain Nakano sequence spaces. Throughout the second half of the twentieth century, it was assumed that these variable exponent spaces provided an acceptable framework for the mathematical components of several problems for which the conventional Lebesgue spaces were insufficient. Due to the relevance of these spaces and their effects, they have become a well-known and efficient tool for solving a variety of problems; nowadays, the area of spaces is a burgeoning area of research, with ramifications extending into a wide variety of mathematical specialties [4]. The study of variable exponent Lebesgue spaces received additional impetus from the mathematical description of non-Newtonian fluid hydrodynamics [5, 6]. Non-Newtonian fluids, also known as electrorheological fluids, have various applications ranging from military science to civil engineering and orthopedics. Operator ideal theory has a variety of applications in Banach space geometry, fixed point theory, spectral theory, and other branches of mathematics, among other branches of knowledge; for more details, see [7–13]. Bakery and Mohamed [14] investigated the concept of a prequasinorm on Nakano sequence space with a variable exponent in the range . They discussed the adequate circumstances for it to generate prequasi-Banach and closed space when endowed with a definite prequasinorm and the Fatou property of various prequasinorms on it. Additionally, they established the existence of a fixed point for Kannan prequasinorm contraction mappings on it and the prequasi-Banach operator ideal generated from -numbers belonging to this sequence space. Also, in [15], they found some fixed points results of Kannan nonexpansive mappings on generalized Cesàro backward difference sequence space of nonabsolute type. For more recent developments in contractive mappings and the existence of fixed points of nonlinear operators in various Banach spaces, Nguyen and Tram [16] examined various fixed point results with applications to involution mappings. Dehici and Redjel [17] introduced some fixed point results for nonexpansive mappings in Banach spaces. Benavides and Ramírez [18] presented some fixed points for multivalued nonexpansive mappings.

We denote the set of complex numbers by and Assuming that , Bakery and El Dewaik [19] defined the following function space:where

They developed a multitude of topological and geometric characteristics for this variable exponent weighted formal power series space, as well as the prequasi-ideal construction utilizing -number and . Upper bounds for -numbers of infinite series of the weighted n-th power forward shift operator on were also introduced for some entire functions. Further, they evaluated Caristi’s fixed point theorem in . For extra information on formal power series spaces and their behaviors, see [20–23]. The purpose of this paper is to develop an insight into how to think about the existence of a fixed point of Kannan prequasinorm contraction mapping in the prequasi-Banach special space of formal power series, where satisfies the property (R) and possesses the -normal structure property. It has been established that a fixed point of the Kannan prequasinorm nonexpansive mapping exists in the prequasi-Banach special space of formal power series. Additionally, we discuss the Kannan prequasinorm contraction mapping in terms of the prequasioperator ideal. The existence of a fixed point of the Kannan prequasi norm contraction mapping in the prequasi Banach operator ideal is offered, where is the class of all bounded linear mappings between any two Banach spaces with the sequence -numbers. Finally, we discuss several applications of solutions to summable equations and illustrate our findings with some instances.

2. Definitions and Preliminaries

Definition 2.1 (see [19]). The linear space is called a special space of formal power series (or in short (ssfps), if it shows the following settings:(1) , for all , where .(2)If and , for every , then .(3)Suppose ; then , with and marks the integral part of .

Definition 2.2 (see [19]). A subspace of is said to be a premodular (ssfps), if there is a function that verifies the next conditions:(i)For , we have and , where is the zero function of .(ii)Suppose and ; then there is with .(iii)Let ; then there is such that .(iv)Suppose , for every ; then .(v)There is so that .(vi), where indicates the space of finite formal power series; that is, for , we have with .(vii)One has with , where .

It is worth noting that the continuity of at is due to condition (ii). Condition (1) in Definition 2.1 and condition (vi) in Definition 2.2 analyze the notion that is a Schauder basis for .

The (ssfps) is called a prequasinormed (ssfps) if shows conditions (i)–(iii) of Definition 2.2, and if the space is complete under , then is called a prequasi-Banach (ssfps). By , we denote the ideal of all bounded linear operators between any arbitrary Banach spaces. Also, marks the space of all bounded linear operators from a Banach space into a Banach space .

Definition 2.3 (see [24]). A function is said to be an -number, if the sequence , for all , shows the following settings:(a)If , then .(b), for every ,,.(c)The inequality holds, if ,, and , where and are arbitrary Banach spaces.(d)Suppose and ; then .(e)Let ; then , whenever .(f)Assume that denotes the identity mapping on the -dimensional Hilbert space ; then or .

Definition 2.4. (see [7]) A class is said to be an operator ideal if every vector shows the following settings:(i), where is the ideal of all finite rank operators between any arbitrary Banach spaces.(ii) is linear space on .(iii)If ,, and , then .

Definition 2.5. (see [10]) A function is called a prequasinorm on the ideal if it shows the following settings:(1)For each , and .(2)One has with , for all and .(3)One has with , for every .(4)There is so that if ,, and , then , where and are normed spaces.

Definition 2.6. (see [19])(a)The prequasinormed (ssfps) on is said to be -convex, if , 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 , where ; then .(e) is -bounded, if .(f)The -ball of radius and center , for all , is defined as(g)A prequasinormed (ssfps) on verifies the Fatou property, if for every sequence with and every ,

Take note that the Fatou property determined the -balls’ closedness. By and , we denote the space of real bounded sequences and the space of all monotonic increasing sequences of positive reals.

Lemma 2.7. (see [19]) The function , for all , verifies the Fatou property, when .

Lemma 2.8. (see [19]) If , then the following settings hold:(1)The function space is a prequasiclosed and Banach (ssfps), with(2)The class is a prequasi-Banach and closed operator ideal, where , where , for every .

Lemma 2.9. The following inequalities will be utilized in the continuation:(i)[25] If and for each , then(ii)[26] Let , and for every with ; then(iii)[27] Assume and , for each ; then where .

3. Some Topological and Geometric Properties

In this section, first, we will talk about the uniform convexity (UUC 2) defined in [28] of the prequasinormed (ssfps) .

Definition 3.1 (see [4, 29]). We define the prequasinorm ’s uniform convexity type behavior as follows:(1)[30] Suppose and . Let When , we put When , we put . The function holds the uniform convexity (UC) if for each and , we have . Observe that, for all , then , for very small .(2)[28] The function verifies (UUC) if for every and , there is with(3)[28] Suppose and . Let When , we put When , we place . The function satisfies (UC 2) if for every and , one has . Observe that, for each ,, for very small .(4)[28] The function verifies (UUC 2) if for all and , there is with(5)[30] The function is strictly convex (SC), if for all so that and , we get .

We will require the following comment here and in the next: , for every and . When , we put .

Theorem 3.2. The function , for all , is (UUC2), if with .

Proof. Let the condition be satisfied, , and . Suppose so thatFrom the definition of , we haveand this implies . Consequent, let and . For every , we get . From the setup, one has or . Assume first . By using Lemma 2.9, condition (i), we obtainThis explainsAsby adding inequalities 2 and 3, and from inequality 1, we haveThis givesNext, suppose . Set ,As and the power function is convex, Since , we getFor any , we haveBy Lemma 2.9, condition (ii), we have Hence,This investigatesSinceby adding inequalities (27) and (28), one hasSinceby adding inequalities (29) and (30) and from inequality 1, we obtainThis impliesIt is clear thatBy using inequalities 4 and 9 and Definition 3.1, we putTherefore, we have , and we conclude that is (UUC2).We will examine the property (R) of the prequasinormed (ssfps) in this second part.

Theorem 3.3. Let with ; then the next setups are satisfied:(1)Assume that , , -closed and -convex, where , for all . Suppose so that Hence, one has a unique with .(2) satisfies the property (R). This means that, for every decreasing sequence of -closed and -convex nonempty subsets of such that , for some , then .

Proof. Suppose the setups are satisfied. To show (1), let as is -closed. Then, one has . So, for every , we have with . Assume is not -Cauchy. Therefore, one obtains a subsequence and so that , for all . Furthermore, one has , for each . AsandUnder , we getTherefore,with . If we let , we getWe have a contradiction. Then is -Cauchy. As is -complete, converges to some . For all , we have the sequence that converges to . As is -closed and -convex, one gets . Surely converges to , so . For and using Theorem 2.7, since satisfies the Fatou property, we getTherefore, . As is (UUC2), so it is SC, which implies that is unique. To show (2), let , for some . is increasing. Let . Suppose . Else , for every . By using Part (1), we have one point with , for every . A consistent proof will show that converges to some . When are -convex, decreasing, and -closed, we get .
This third part discusses the prequasinormed structure’s -normal structure feature (ssfps) .

Definition 3.4. satisfies the -normal structure property if for all nonempty -bounded, -convex, and -closed subset of did not decrease to one point, we have with

Theorem 3.5. If with , then holds the -normal structure property, where , for every .

Proof. Assume the setups are satisfied. Theorem 3.2 explains that is (UUC2). Let be a -bounded, -convex, and -closed subset of not decreased to unique point. Hence, . Let . Suppose with . So . For all , one obtains and . Since is -convex, one has . Hence,For all ,

4. Kannan Contraction Mapping

In the prequasinormed space, we now develop Kannan -Lipschitzian mapping (ssfps). We study enough conditions on with a defined prequasinorm such that Kannan prequasinorm contraction mapping has a unique fixed point.

Definition 4.1. An operator is called a Kannan -Lipschitzian, if there is , so thatFor all , one has the following:(1)If , then the operator is said to be Kannan -contraction.(2)If , then the operator is said to be Kannan -nonexpansive.A vector is called a fixed point of , when .

Theorem 4.2. If and is Kannan -contraction mapping, where , for all , then has a unique fixed point.

Proof. Let the setups be satisfied. For every , then . Since is a Kannan -contraction mapping, we haveTherefore, for every with , then we getSo is a Cauchy sequence in . As the space is prequasi-Banach (ssfps). Therefore, there is such that . To prove that , by Theorem 2.7, holds the Fatou property, and we haveHence, . Then is a fixed point of . To show that the fixed point is unique, assume we have two different fixed points of . Then, one hasTherefore, .

Corollary 4.3. Let and be Kannan -contraction mapping, where , for all ; then has one and only one fixed point with .

Proof. It is obvious, so it is omitted.

Definition 4.4. Assume is a prequasinormed (ssfps) and . The operator is called sequentially continuous at , if and only if when , .

Theorem 4.5. Let with and , where , for all . The point is the unique fixed point of , if the following conditions are satisfied:(a) is Kannan -contraction mapping.(b) is sequentially continuous at .(c)One has with the sequence of iterates having a subsequence converging to .

Proof. Suppose the settings are verified. If is not a fixed point of , then . By conditions (b) and (c), we haveSince the mapping is Kannan -contraction, one can seeSince , one has a contradiction. Hence, is a fixed point of . To explain that the fixed point is unique, suppose we have two different fixed points of . Therefore, one getsSo, .

Theorem 4.6. Assume is an increasing, and , where , for all . The point is the only fixed point of , if the following conditions are satisfied:(a) is Kannan -contraction mapping.(b) is sequentially continuous at .(c)One has so that the sequence of iterates has a subsequence converging to .

Proof. Let the conditions be verified. If is not a fixed point of , then . By conditions (b) and (c), we haveAs the operator is Kannan -contraction, one can seeSince , one obtains a contradiction. Hence, is a fixed point of . To explain that the fixed point is unique, assume we have two different fixed points of . Then, one getsSo, .

Example 4.7. Pick up , where , for all andFor all with , we haveFor all with , we haveFor all with and , we haveHence, is Kannan -contraction mapping. By Theorem 2.7, the function satisfies the Fatou property. By Theorem 4.2, the map has a unique fixed point .
Let with , where and . Since the prequasinorm is continuous, one getsTherefore, is not sequentially continuous at . Then, the map is not continuous at .
If , for all . For all with , one obtainsFor all with , we haveFor all with and , we haveTherefore, the map is Kannan -contraction mapping and
Obviously, is sequentially continuous at and has a subsequence converging to . By Theorem 4.5, the point is the unique fixed point of .

Example 4.8. Assume , where , for all andFor all with , we haveFor all with , we haveFor all with and , we haveHence, is Kannan -contraction mapping. From Theorem 2.7, the function satisfies the Fatou property. By Theorem 4.2, the map has a unique fixed point .
Let with , where and . Since the prequasinorm is continuous, we haveTherefore, is not sequentially continuous at . So, the map is not continuous at .
Suppose , for all . For all with , we haveFor all with , we haveFor all with and , we haveSo, the map is Kannan -contraction mapping and
Obviously, is sequentially continuous at and has a subsequence converging to . By Theorem 4.5, the point is the unique fixed point of .

Example 4.9. Let , where , for all andFor all with , we haveFor all with and then for any , we haveFor all with and , we haveHence, is Kannan -contraction mapping. Clearly, is sequentially continuous at , and there is with such that the sequence of iterates has a subsequence converging to . By Theorem 4.5, the map has one fixed point . Note that is not continuous at .
If , . For all with , we haveFor all with and then for any , we haveFor all with and , we haveSo, is Kannan -contraction mapping. By Theorem 2.7, the function satisfies the Fatou property. By Theorem 4.2, the map has a unique fixed point .

Example 4.10. Let , where , for all andFor all with , we haveFor all with and then for any , we haveFor all with and , we haveHence, is Kannan -contraction mapping. Evidently, is sequentially continuous at and there is with such that the sequence of iterates has a subsequence converging to . By Theorem 4.5, the map has one fixed point . Note that is not continuous at .
If , . For all with , we haveFor all with and then for any , we haveFor all with and , we haveHence, is Kannan -contraction mapping. By Theorem 2.7, the function satisfies the Fatou property. By Theorem 4.2, the map has a unique fixed point .

5. Kannan Nonexpansive Mapping

We examine enough conditions on the prequasinormed (ssfps) for the Kannan prequasinorm nonexpansive mapping on it to have a fixed point in this section.

Lemma 5.1. Allow the prequasinormed (ssfps) to validate the (R) and -quasi-normal properties. If is a nonempty -bounded, -convex, and -closed subset of , suppose is a Kannan -nonexpansive mapping. For , let . Let

Then, is a nonempty, -convex, -closed subset of and

Proof. As , this gives . As the -balls are -convex and -closed, one has is a -closed and -convex subset of . To prove that , suppose . If , we get . Otherwise, suppose . LetFrom the definition of , . So, , and we have . Assume . Hence, there is so that . Then,Since is randomly positive, we obtain , so we have . As , one has ; this indicates that is -invariant, consequent to prove that , asFor every , let . Hence, . The definition of provides . Hence, . So, one has , for every , this means . This completes the proof.

Theorem 5.2. Let the prequasinormed (ssfps) verify the -quasi-normal property and the (R) property. Assume is a nonempty, -convex, -closed, and -bounded subset of . Suppose is a Kannan -nonexpansive mapping. Hence, has a fixed point.

Proof. Suppose and , for every . By using the definition of , we have , with . Assume is described as in Lemma 5.1. Obviously, is a decreasing sequence of nonempty -bounded, -closed, and -convex subsets of . The property (R) proves that . Assume ; one can see , for every . Suppose ; we have ; this gives . Therefore, . We have . Else, ; this gives that fails to have a fixed point. Let be defined as in Lemma 5.1. As misses to have a fixed point and is -invariant, so has more than one point, which implies, . By the -quasinormal property, there is so that

For all , by Lemma 5.1, we have . By definition of , then . Obviously, we have

which contradicts the definition of . So , which implies that any point in is a fixed point of ; that is, has a fixed point in .

In view of Theorems 3.3, 3.5, and 5.2, it is easy to conclude the following theorem.

Theorem 5.3. If with , is a nonempty, -convex, -closed, and -bounded subset of , where , for every , and is a Kannan -nonexpansive mapping. Then, has a fixed point.

Example 5.4. Let with where and , for all . In view of example 4.8, the map is Kannan -contraction mapping. This implies Kannan -nonexpansive mapping. Evidently, is a nonempty, -convex, -closed, and -bounded subset of . By Theorem 5.3, the map has one fixed point in .

6. Kannan Contraction Mappings on the Operator Ideal

We study in this section the presence of a fixed point for the Kannan prequasinorm contraction mapping in the prequasi-Banach operator ideal defined by the and -numbers.

Notations 6.1. [19]

Definition 6.2. If and are Banach spaces, a prequasinorm on the ideal , where and converge for any , satisfies the Fatou property if for every sequence with and any ,

Theorem 6.3. Suppose and are Banach spaces. The prequasinorm , for all does not satisfy the Fatou property, if .

Proof. Let the condition be satisfied and with . By Theorem 2.8, the space is a prequasiclosed ideal, and then . Hence, for all , we haveHence, does not satisfy the Fatou property.

Definition 6.4. Suppose and are Banach spaces. For the prequasinorm on the ideal , where , where converges for any , an operator is called a Kannan -Lipschitzian, if there is , so thatfor all . An operator is said to be(1)Kannan -contraction, when .(2)Kannan -nonexpansive, when .

Definition 6.5. Suppose and are Banach spaces. For the prequasinorm on the ideal , where , where converges for any , and . The operator is said to be sequentially continuous at , if and only if when , .

Theorem 6.6. Suppose and are Banach spaces. Let and , where , for all . The point is the unique fixed point of , if the following conditions are satisfied:(a) is Kannan -contraction mapping.(b) is sequentially continuous at a point .(c)There is so that the sequence of iterates has a subsequence converging to .

Proof. Let the conditions be verified. If is not a fixed point of , then . From conditions (b) and (c), we haveSince is Kannan -contraction mapping, one can seeAs , we have a contradiction. Therefore, is a fixed point of . To show that the fixed point is unique. Let us have two different fixed points of . Therefore, one hasSo, .

Example 6.7. Suppose and are Banach spaces; , where , for every andFor all with , we haveFor all with , we haveFor all with and , we haveHence, is Kannan -contraction mapping and
Evidently, is sequentially continuous at the zero operator and has a subsequence converging to . By Theorem 6.6, the zero operator is the only fixed point of . Assume with , where and . Since the prequasinorm is continuous, one obtainsTherefore, is not sequentially continuous at . Then, the map is not continuous at .

Example 6.8. If and are Banach spaces, , where , for every andFor all with , we haveFor all with , we haveFor all with and , we haveHence, is Kannan -contraction mapping and
Obviously, is sequentially continuous at the zero operator and has a subsequence converging to . By Theorem 6.6, the zero operator is the only fixed point of . Suppose with , where and . Since the prequasinorm is continuous, one getsTherefore, is not sequentially continuous at . Then, the map is not continuous at .

7. Application to Nonlinear Summable Equations

Numerous authors have examined nonlinear summable equations such as (10); see [31–33]. This section is dedicated to locating a solution to (10) in , where and , for every . Take a look at the equations that are summable:and assume defined by

Theorem 7.1. The summable equation (10) has one solution in , if , , , , and for every , we have , with

Proof. Let the setups be verified. Consider the mapping defined by (11). We haveAccording to Theorem 4.2, one obtains a unique solution of equation (10) in .

Example 7.2. Assume the function space , where , for all . Consider the summable equationwhere and and let defined byIt is easy to see thatBy Theorem 7.1, the summable equation (114) has one solution in .

Example 7.3. Given the function space , where , for all , consider the summable equation (12). It is easy to see thatBy Theorem 7.1, the summable equation (114) has one solution in .

Example 7.4. Given the function space , where , for all , consider the summable equation (114) with and let , where , defined byClearly, is a nonempty, -convex, -closed, and -bounded subset of . It is easy to see thatBy Theorem 7.1 and Theorem 5.3, the summable equation (114) with has a solution in .

In this part, we search for a solution to nonlinear matrix (120) at , where and are Banach spaces, , and , for all . Consider the summable equationand suppose defined by

Theorem 7.5. The summable equation (120) has one solution in , if the following conditions are satisfied:(a),,,, and for every , one has , 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. Consider the mapping defined by (16). We haveIn view of Theorem 6.6, one obtains a unique solution of (120) at .

Example 7.6. Assume the function space , where , for all .
Consider the nonlinear difference equationwhere ,, and let defined byIt is easy to see thatBy Theorem 7.1, the nonlinear difference equation (124) has one solution in .

Example 7.7. Given the function space , where , for all , consider the nonlinear difference equation (17). It is easy to see thatBy Theorem 7.1, the nonlinear difference equation (124) has one solution in .

Example 7.8. Given the function space , where , for all , consider the nonlinear difference equation (124) with and let , where , defined byClearly, is a nonempty, -convex, -closed, and -bounded subset of . It is easy to see thatBy Theorem 7.1 and Theorem 5.3, the nonlinear difference equation (124) with has a solution in .

8. Conclusion

This paper studies the existence of a fixed point for Kannan’s prequasinorm contractive mappings in function spaces of complex variables. We have studied the existence of fixed points of Kannan prequasinorm nonexpansive mapping and the existence of Kannan’s prequasinorm contractive mapping in the prequasi-Banach operator ideal created by this function space and -numbers. We have also presented some applications of summable equations. Several numerical experiments were introduced to illustrate our results. Moreover, some successful applications to the existence of solutions of nonlinear difference equations are discussed. This paper has several advantages for researchers, such as studying the fixed points of any contraction mappings on this prequasinormed function space, which is a generalization of the quasinormed function space, examining the eigenvalue problem in these new settings and noting that the closed operator ideals are certain to play an important function in the principle of Banach lattices.

Data Availability

No data were used to support 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 authors read and approved the final version of the paper.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under Grant no. (UJ-20-084-DR). The authors, therefore, acknowledge with thanks the University's technical and financial support.

References

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
R. Kannan, “Some results on fixed points-II,” The American Mathematical Monthly, vol. 76, no. 4, pp. 405–408, 1969.
View at: Publisher Site | Google Scholar
S. J. H. Ghoncheh, “Some fixed point theorems for Kannan mapping in the modular spaces,” Ciencia e Natura, vol. 37, pp. 462–466, 2015.
View at: Publisher Site | Google Scholar
L. Diening, P. Harjulehto, P. Hästö, and M. Ruẑiĉka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin, Germany, 2011.
K. R. Rajagopal and M. Růzǐčka, “On the modeling of electrorheological materials,” Mechanics Research Communications, vol. 23, no. 4, pp. 401–407, 1996.
View at: Publisher Site | Google Scholar
M. Ruẑiĉka, “Electrorheological fluids. Modeling and mathematical theory,” in Lecture Notes in Mathematics, p. 1748, Springer, Berlin, Germany, 2000.
View at: Google Scholar
A. Pietsch, Operator Ideals, ” North-Holland Publishing Company, Amsterdam-New York-Oxford, 1980.
A. Pietsch, “Small ideals of operators,” Studia Mathematica, vol. 51, no. 3, pp. 265–267, 1974.
View at: Publisher Site | Google Scholar
B. M. Makarov and N. Faried, “Some properties of operator ideals constructed by numbers (In Russian),” Academy of Science, Siberian section, Novosibirsk, Russia, 1977.
View at: Google Scholar
N. Faried and A. A. Bakery, “Small operator ideals formed by s numbers on generalized Cesáro and Orlicz sequence spaces,” Journal of Inequalities and Applications, vol. 2018, no. 1, p. 357, 2018.
View at: Publisher Site | Google Scholar
T. Yaying, B. Hazarika, and M. Mursaleen, “On sequence space derived by the domain of -Cesàro matrix in space and the associated operator ideal,” Journal of Mathematical Analysis and Applications, vol. 493, Article ID 124453, 2021.
View at: Publisher Site | Google Scholar
M. Mursaleen and A. K. Noman, “Compactness by the Hausdorff measure of noncompactness,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 8, pp. 2541–2557, 2010.
View at: Publisher Site | Google Scholar
M. Mursaleen and A. K. Noman, “Compactness of matrix operators on some new difference sequence spaces,” Linear Algebra and Its Applications, vol. 436, no. 1, pp. 41–52, 2012.
View at: Publisher Site | Google Scholar
A. A. Bakery and O. S. K. Mohamed, “Kannan prequasi contraction maps on Nakano sequence spaces,” Journal of Functional Spaces, vol. 2020, Article ID 8871563, 10 pages, 2020.
View at: Publisher Site | Google Scholar
A. A. Bakery and O. S. K. Mohamed, “Kannan nonexpansive maps on generalized Cesàro backward difference sequence space of non-absolute type with applications to summable equations,” Journal of Inequalities and Applications, vol. 103, 2021.
View at: Publisher Site | Google Scholar
L. V. Nguyen and N. T. N. Tram, “Fixed point results with applications to involution mappings,” Journal of Nonlinear Variable Analysis, vol. 4, pp. 415–426, 2020.
View at: Google Scholar
A. Dehici and N. Redjel, “Some fixed point results for nonexpansive mappings in Banach spaces,” Journal of Nonlinear Variable Analysis, 2020.
View at: Google Scholar
T. D. Benavides and P. L. Ramírez, “Fixed points for multivalued non-expansive mappings,” Applied Set-Valued Analyse Optimation, vol. 2, pp. 143–157, 2020.
View at: Google Scholar
A. A. Bakery and M. H. El Dewaik, “A generalization of Caristi’s fixed point theorem in the variable exponent weighted formal power series space,” Journal of Function Spaces, vol. 2021, Article ID 9919420, 18 pages, 2021.
View at: Publisher Site | Google Scholar
A. L. Shields, “Weighted shift operators and analytic function theory,” in Topics of Operator Theory, vol. 13, Amer. Math., Providence, RI, USA, 1974.
View at: Publisher Site | Google Scholar
K. Hedayatian, “On cyclicity in the space ,” Taiwanese Journal of Mathematics, vol. 8, no. 3, pp. 429–442, 2004.
View at: Publisher Site | Google Scholar
H. Emamirad and G. S. Heshmati, “Chaotic weighted shifts in Bargmann space,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 36–46, 2005.
View at: Publisher Site | Google Scholar
N. Faried, A. Morsy, and Z. A. Hassanain, “s-numbers of shift operators of formal entire functions,” Journal of Approximation Theory, vol. 176, pp. 15–22, 2013.
View at: Publisher Site | Google Scholar
A. Pietsch, Eigenvalues and -numbers, Cambridge University Press, New York, NY, USA, 1986.
J. A. Clarkson, “Uniformly convex spaces,” Transactions of the American Mathematical Society, vol. 40, no. 3, pp. 396–414, 1936.
View at: Publisher Site | Google Scholar
K. Sundaresan, “Uniform convexity of Banach spaces l({p_{i}}),” Studia Mathematica, vol. 39, no. 3, pp. 227–231, 1971.
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: Publisher Site | Google Scholar
M. A. Khamsi and W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, Birkhauser, New York, NY, USA, 2015.
J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin, Germany, 1983.
H. Nakano, Topology of Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, Japan, 1951.
P. Salimi, Abdul Latif, and N. Hussain, “Modified --contractive mappings with applications,” Fixed Point Theory and Applications, vol. 151, 2013.
View at: Google Scholar
R. P. Agarwal, N. Hussain, and M.-A. Taoudi, “Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations,” Abstract and Applied Analysis, vol. 2012, Article ID 245872, 15 pages, 2012.
View at: Publisher Site | Google Scholar
N. Hussain, A. R. Khan, and R. P. Agarwal, “Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces,” Journal of Nonlinear Convex Analysis, vol. 11, no. 3, pp. 475–489, 2010.
View at: Google Scholar

Copyright

Copyright © 2022 Elsayed A. E. Mohamed 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.

PDF Download Citation

Download other formats

Order printed copies

Views

289

Downloads

338

Citations