Some Fixed Point Results of Kannan Maps on the Nakano Sequence Space

Bakery, Awad A.; Mohamed, O. M. Kalthum S. K.

doi:https://doi.org/10.1155/2021/2578960

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Unique and Non-Unique Fixed Points and their Applications

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Research Article | Open Access

Volume 2021 | Article ID 2578960 | https://doi.org/10.1155/2021/2578960

Some Fixed Point Results of Kannan Maps on the Nakano Sequence Space

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

Academic Editor: Santosh Kumar

Received01 May 2021

Revised27 May 2021

Accepted04 Jun 2021

Published22 Jun 2021

Abstract

In the recent past, some researchers studied some fixed point results on the modular variable exponent sequence space , where and . They depended on their proof that the modular has the Fatou property. But we have explained that this result is incorrect. Hence, in this paper, the concept of the premodular, which generalizes the modular, on the Nakano sequence space such as its variable exponent in and the operator ideal constructed by this sequence space and -numbers is introduced. We construct the existence of a fixed point of Kannan contraction mapping and Kannan nonexpansive mapping acting on this space. It is interesting that several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of summable equations are introduced. The novelty lies in the fact that our main results have improved some well-known theorems before, which concerned the variable exponent in the aforementioned space.

1. Introduction

Ideal operators and summability theorems are awfully invaluable in mathematical models and have large executions, for example, the fixed point theory, geometry of Banach spaces, normal series theory, approximation theory, and ideal transformations. For added evidence, see [1–4]. By , , , and , we denote the spaces of all, bounded, -absolutely summable and null sequences of real numbers. We indicate the space of all bounded linear operators from a Banach space into a Banach space by , and if , we inscribe and , while 1 displays at the th place, for all .

Definition 1 [5]. An -number function is a map detailed on which sorts every map a nonnegative scaler sequence overbearing that the next setting encompasses (a), for all (b) for every , and , (c)Ideal property: , for every , and , where and are discretionary Banach spaces(d)For and , one has (e)Rank property: assume rank , then , for each (f)Norming property: or , where mirrors the unit map on the -dimensional Hilbert space

The th approximation number, established by , is defined as

Notation 2. The sets , , and (cf. [6]) indicate where Also, where .

Suppose that , the Nakano sequence space defined and studied in [7–9] is denoted by when

The space , where and , for all , is a Banach space. If , then,

The vector spaces are contained in the variable exponent spaces . In the second half of the twentieth century, it was assumed that these variable exponent spaces provided the proper framework for the mathematical components of numerous issues for which the classical Lebesgue spaces were insufficient. Because of the importance of these spaces and their surroundings, they have become a well-known and environmentally friendly tool in the treatment of a variety of conditions; currently, the region of spaces is a prolific subject of research, with ramifications extending into a wide range of mathematical specialties (see [10]). The mathematical description of the hydrodynamics of non-Newtonian fluids provides an impetus for learning about variable exponent Lebesgue spaces, (see [11, 12]). Applications of non-Newtonian fluids, known as electrorheological, vary from their use in army science to civil engineering and orthopedics. Faried and Bakery provided the theory of the pre-quasioperator ideal, which is more general than the quasioperator ideal, in [6]. In [7], Bakery and Abou Elmatty explained the sufficient (not necessary) setting on so that generated a simple Banach pre-quasioperator ideal. The pre-quasioperator ideal is strictly restricted to different powers. It was a small pre-quasioperator ideal. Because of the booklet of the Banach fixed point theorem [13], many mathematicians have worked on many developments. Kannan [14] 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 [15]. Bakery and Mohamed [16] explored the concept of the pre-quasinorm on the Nakano sequence space such that its variable exponent in . They explained the sufficient conditions on it, equipped with the definite pre-quasinorm to generate pre-quasi-Banach and closed space, and examined the Fatou property of different pre-quasinorms on it. Moreover, they showed the existence of a fixed point of Kannan pre-quasinorm contraction maps on it and on the pre-quasi-Banach operator ideal constructed by -numbers which belong to this sequence space. For more details on Kannan’s fixed point theorems, see [17–24]. The aim of this paper is to examine the concept of the pre-quasinorm on with a variable exponent in . We study the sufficient conditions on equipped with the definite pre-quasinorm to form pre-quasi-Banach and closed (), the existence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach (), satisfies the property (), and has the -normal structure property. The existence of a fixed point of Kannan pre-quasinorm nonexpansive mapping in the pre-quasi-Banach () has been given. Finally, we examine the idea of Kannan pre-quasinorm contraction mapping in the pre-quasioperator ideal. As well, the existence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach operator ideal has been introduced. Finally, some illustrative examples and applications to the existence of solutions of summable equations are given.

2. Definitions and Preliminaries

By , we denote the space of all functions . Nakano [25] introduced the concept of modular vector spaces.

Definition 3. Suppose that is a vector space. A function is called modular if the next conditions hold (a)For , with , where is the zero vector of (b) holds, for all and (c)The inequality satisfies, for all and

The concept of premodular vector spaces is more general than modular vector spaces.

Definition 4 [2]. The linear space of sequences is called a special space of sequences (), if (a)(b) is solid, i.e., assume that , , and , for each , and then (c), where indicates the integral part of , in case

Definition 5 [6]. A subclass of is named a premodular (), if there is , it satisfies the next setting: (i)For , with (ii)For some , the inequality holds, for all and (iii)For some , the inequality satisfies, for all (iv)For and , we have (v)The inequality, includes, for some (vi)Let be the space of finite sequences, then (vii)we have such that for all This is an example of a premodular vector space but not a modular vector space.

Example 6. The function is a premodular (not a modular) on the vector space . Since for all , we have

Definition 7 [26]. Let be a (). The function is named a pre-quasinorm on , if it provides the following setting: (i)For , with (ii)For some , the inequality holds, for all and (iii)For some , the inequality satisfies, for all

Theorem 8 [26]. Let be a premodular (), and then, it is pre-quasinormed ().

Theorem 9 [26]. is a pre-quasinormed (), if it is quasinormed ().

Definition 10 [3]. Suppose that is the class of all bounded linear operators within any two arbitrary Banach spaces. A subclass of is called an operator ideal, if every element satisfies the next conditions: (i), where describes the Banach space of one dimension(ii)The space is linear over (iii)If , , and , then, , where and are normed spaces (see [27, 28])This is the concept of the pre-quasioperator ideal which is added in general to the quasioperator ideal.

Definition 11 [6]. A function is called a pre-quasinorm on the ideal if the next conditions hold: (1)Let , , and , if and only if, (2)We have so as to , for every and (3)We have so that , for each (4)We have if , , and , and then,

Theorem 12 [29]. Assuming that is a pre-modular (sss), then, is a pre-quasinorm on .

Theorem 13 [7]. Let and be Banach spaces and be a premodular (), and then, is a pre-quasi-Banach operator ideal, such that .

Theorem 14 [6]. is a pre-quasinorm on the ideal , if is a quasinorm on the ideal .

Lemma 15. The given inequalities will be used in the sequel: (i)Let , and for every [30], then(ii)Assume that , and for all so that [31], then(iii)Suppose that and , for every , then, where [32]

3. Pre-Quasinormed ()

We explain the sufficient setting of equipped with a pre-quasinorm to generate pre-quasi-Banach and closed (). The Fatou property of a pre-quasinorm on has been given.

Definition 16. (a)The function on is named convex, if , for all and (b) is convergent to , if and only if, If the limit exists, then it is unique(c) is Cauchy, when (d) is closed, if for every -converges to , then (e) is bounded, when (f)The ball of radius and center , for all , is detailed as(g)A pre-quasinorm on provides the Fatou property, if for all sequence with and any , then Note that the Fatou property implies the closedness of the balls.

Theorem 17. , where , for each , is a premodular (), if is increasing with .

Proof. To begin with, we have to show that is a (): (1)Assume . As is bounded, we getHence, .
And suppose that and . Since is bounded, we obtain So, . Therefore, by using equations (8) and (9), we have that is linear. Also, for every as (2)Suppose , for every and . We haveThen, . (3)Assuming that and is an increasing sequence, we haveThen, . As well, we prove that the functional on is a premodular: (i)Clearly, and (ii)We have such that , for every and (iii)We have so that , for every (iv)Evidently, from (101)(v)From (104), we have (vi)Evidently, (vii)We have , for or , for so that☐

Theorem 18. Let be an increase with , and then, is a pre-quasi-Banach (), where , for all .

Proof. Suppose that the setup is satisfied. From Theorem 17, the space is a premodular (). By Theorem 8, the space is a pre-quasinormed (). To explain that is a pre-quasi-Banach (), suppose that is a Cauchy sequence in . Therefore, for all , there is so that for every , we have Hence, for , and , we have Hence, is a Cauchy sequence in , for fixed , which gives , for constant . So, , for all . Conclusively, to prove that , one has Hence, . This gives that is a pre-quasi-Banach (). ☐

Theorem 19. Assuming that is increasing with , then, is a pre-quasiclosed (), where , for all .

Proof. Let the setup be satisfied. From Theorem 17, the space is a premodular (). By Theorem 8, the space is a pre-quasinormed (). To prove that is a pre-quasiclosed (), let and ; then, for each , there is such that for every , one can see Therefore, for and , we have Hence, is a convergent sequence in , for constant . So, , for fixed . Finally, to show that , one has Hence, . This implies that is a pre-quasiclosed (). ☐

Theorem 20. The function satisfies the Fatou property, if is increasing with , for every .

Proof. Assume that the setup is verified and with As the space is a pre-quasiclosed space, then, . Hence, for all , we have ☐

Theorem 21. The function does not verify the Fatou property, for every , if and , for each .

Proof. Assume that the setting is verified and with As the space is a pre-quasiclosed space, then, . Then, for all , one can see Therefore, does not verify the Fatou property. ☐

Similarly as Theorems 17 and 19 under the conditions is increasing with , it is easy to prove that the space , which is studied in [33], is a pre-quasiclosed (), where .

Theorem 22. The function satisfies the Fatou property, if is increasing with , for every .

Proof. Assume that the setup is verified and with As the space is a pre-quasiclosed space, then, . Hence, for all , we have

Theorem 23. The function does not verify the Fatou property, for every , if and , for each .

Proof. Assume that the setting is confirmed and with As the space is a pre-quasiclosed space, then, . Then, for all , one can see Therefore, does not verify the Fatou property. ☐

Example 24. The function is a pre-quasinorm (not a quasinorm) on .

Example 25. The function is a pre-quasinorm (not a norm) on .

Example 26. The function is a pre-quasinorm, quasi norm, and not a norm on , for .

Example 27. For , the function is a pre-quasinorm (a quasinorm and a norm) on .

4. Kannan Prequasi Contraction Mapping

In this section, we will define Kannan -Lipschitzian mapping in the pre-quasinormed (). We study the sufficient setting on constructed with definite pre-quasinorm so that there is one and only one fixed point of Kannan pre-quasinorm contraction mapping.

Definition 28. An operator is named a Kannan -Lipschitzian, if there is , such that for every . The operator is named (1)Kannan contraction, if (2)Kannan nonexpansive, if An element is called a fixed point of , when

In fact, the authors of reference [33] in Theorem 1 proved that the Kannan modular contraction mapping on a nonempty modular-closed subset of the modular space , where and , for all , has a unique fixed point. They depended on their proof that the modular has the Fatou property. But from Theorem 23, this result is incorrect. We have improved it in the next theorem.

Theorem 29. Let be an increase with and be Kannan contraction mapping, where , for every , so has a unique fixed point.

Proof. Assume that the conditions are verified. For all , then, . Since is a Kannan contraction mapping, we have Therefore, for every with , then, we have Hence, is a Cauchy sequence in . Since the space is a pre-quasi-Banach space, so, there is so that . To show that , as has the Fatou property, we get So, . Then, is a fixed point of . To prove that the fixed point is unique, assume that we have two different fixed points of . Therefore, one can see Hence, ☐

Corollary 30. Suppose that is increasing with and is Kannan contraction mapping, where , for every , then has unique fixed point with

Proof. Assume that the setup is verified. By Theorem 29, there is a unique fixed point of . Therefore, one can see ☐

Definition 31. Let be a pre-quasinormed (), and The operator is named sequentially continuous at , if and only if then .

Theorem 32. If is increasing with and , where , for every , the point is the only fixed point of , if the next settings are verified: (a) is Kannan contraction mapping(b) is sequentially continuous at (c)We have such that the sequence of iterates has a subsequence converges to

Proof. If the settings are satisfied, let be not a fixed point of , and then, . By the setups (b) and (c), one can see Since the operator is Kannan contraction, we have Since , we get a contradiction. Hence, is a fixed point of . To show that the fixed point is unique, suppose that we have two different fixed points of . Therefore, we have So, ☐

Example 33. Let , where , for all and

Since for all with , we have

For all with , we have

For all with and , we have

Therefore, the map is Kannan contraction mapping, since satisfies the Fatou property. By Theorem 29, the map has a unique fixed point

Let be such that where with . Since the pre-quasinorm is continuous, we have

Hence, is not sequentially continuous at . So, the map is not continuous at .

If , for all . Since for all with , we have

For all with , we have

For all with and , we have

Therefore, the map is Kannan contraction mapping and

It is clear that is sequentially continuous at and has a subsequence that converges to . By Theorem 32, the point is the only fixed point of .

Example 34. Let , where , for all and

Since for all with , we have

For all with , then, for any , we have

For all with and , we have

Therefore, the map is Kannan contraction mapping. It is clear that is sequentially continuous at and there is with such that the sequence of iterates has a subsequence converges to . By Theorem 32, the map has one fixed point . Note that is not continuous at .

If , for all . Since for all with , we have

For all with , then, for any , we have

For all with and , we have

Therefore, the map is Kannan contraction mapping. Since satisfies the Fatou property. By Theorem 29, the map has a unique fixed point .

5. Pre-Quasinormed Uniform Convexity

In this part, we investigate the uniform convexity (UUC 2) defined in [35] of the pre-quasinormed () .

Definition 35 [10, 34]. We define the following uniform convexity-type behavior of the pre-quasinorm : (1)Assume that and [35]. Indicate thatWhen , we put When , we put The function investigates the uniform convexity (UC) if for every and , we have Note that for all , then, , for very small (2)The function provides (UUC) if for every and , there is based on and so that [36](3)Suppose that and . Indicate [36]When , we put When , we put The function supports (UC 2) if for all and , we have Observe that for each , , for very small (4)The function satisfies (UUC 2) if for every and , there is based on and so that [36](5)The function is strictly convex (SC), if for each so that and we get [35]Here and after, we will need the following notation: for each and When we set

Theorem 36. The pre-quasinorm on is (UUC 2), where , for every , if is an increasing with .

Proof. Supposing that the setting is verified, and . Let such that From the definition of , one has This implies that So, set and For every we have By using the setting, we obtain or Suppose first that Using Lemma 15, one can see This gives Since one has This implies Next, assume that Put As the power function is convex and . Hence, As we have For all one can see By Lemma 15, we have So, this gives As we have As we get Obviously, If we set Hence, one has and we deduce that is (UUC 2). ☐

In fact, the authors of reference [37] in Proposition 3.5 proved that if , then, the modular space , where , has the property (). They depended on their proof that the modular has the Fatou property. But from Theorem 23, this result is incorrect. Consequently, all the related results in the two references [33, 37] are incorrect. In this part, we investigate the property () of the pre-quasinormed () .

Theorem 37. Let be an increase with , then (1)The space is a pre-quasi-Banach (), where , for every (2)Suppose that is a nonempty closed and convex subset of Assume that so thatThen we have a unique so that (3) verifies the property (), i.e., for each decreasing sequence of closed and convex nonempty subsets of such that for some so we have

Proof. Assume that the setting is verified. The proof of (100) comes from Theorem 18. To prove (101), suppose that as is closed. Therefore, we have . So, for every , there is such that . Assume that is not Cauchy. So there is a subsequence and such that for all More, we have for each As For all , one has Hence, for every . If we set we have This implies a contradiction. Therefore, is Cauchy. Since is complete, hence, converges to some . For every , we get the sequence converges to . As is closed and convex, one has Clearly, converges to ; this implies that By setting and using Theorem 20, as verifies the Fatou property, we have Hence, As the function is (UUC 2), then, it is (SC), which gives the uniqueness of . To prove (104), suppose , for some As is increasing, set , if . Otherwise , for every . From (2), there is one point such that , for all . A consistent proof will show that converges to some . Since are convex, decreasing, and closed, we have ☐

In this part, we investigate the normal structure property of the pre-quasinormed () .

Definition 38. verifies the normal structure property if for every nonempty bounded, convex, and closed subset of that are not decreased to one point, there is such that

Theorem 39. Let be an increase with ; then, has the normal structure property, where , for all .

Proof. Suppose that the setting is verified. Theorem 36 implies that is (UUC 2). Assume that is a bounded, convex, and closed subset of that are not decreased to one point. Hence, Set Suppose that such that Hence For every we have and Since is convex, we have . So, for every Hence,

6. Kannan Nonexpansive Mapping on

We examine here the sufficient conditions on the pre-quasi normed () so that the Kannan pre-quasinorm nonexpansive mapping on it has a fixed point.

Lemma 40. Let the pre-quasinormed () verify the () property and the quasinormal property. Assume that is a nonempty bounded, convex, and closed subset of . Suppose that is a Kannan nonexpansive mapping. For , assume that . Set Then, , convex, and closed subset of and

Proof. As , this implies that . Since the balls are convex and closed, so is a closed and convex subset of . To prove that , suppose that If we have Otherwise, let Set By using the definition of , then, Hence, ; this implies that Suppose that Hence, there is such that . Hence, Since is an arbitrary positive, we have so one has . As , we have ; this investigates that is invariant, consequent to prove that As for each Assume that So The definition of implies So, Hence, we have ; for every this gives This finishes the proof. ☐

Theorem 41. Picking up the pre-quasinormed () verifies the quasinormal property and the () property. Assume that is a nonempty, convex, closed, and bounded subset of . Suppose that is a Kannan nonexpansive mapping. Then, has a fixed point.

Proof. Set and for all From the definition of we have , for all Let be studied as in Lemma 40. Obviously, is a decreasing sequence of nonempty bounded, closed, and convex subsets of . The property () gives that Supposing that we get for each Assuming that one has ; this implies that Hence, We obtain . Otherwise, ; this implies that fails to have a fixed point. Suppose as defined in Lemma 40. Since fails to have a fixed point and is invariant, hence, has more than one point; this gives, . By the quasinormal property, there is such that for each By Lemma 40, one has By definition of then, Evidently, this implies that This contradicts the definition of . Hence, which gives that any point in is a fixed point of , i.e., has a fixed point in . ☐

Using Theorems 37, 39, and 41, we have the following corollary:

Corollary 42. Let be an increase with . Suppose that is a nonempty, convex, closed, and bounded subset of , where , for all . Assume that is a Kannan nonexpansive mapping. Then, has a fixed point.

Example 43. Let with where and , for all . From Example 33, the map is Kannan contraction mapping. So, it is Kannan nonexpansive mapping. Clearly, is a nonempty, convex, closed, and bounded subset of . By Corollary 42, the map has one fixed point in .

7. Kannan Pre-Quasicontraction on Prequasi Ideal

We study the presence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach operator ideal constructed by and -numbers.

Theorem 44 [9]. Pick up and to be Banach spaces, and is an increasing with , and then, , where is a pre-quasi-Banach operator ideal.

Theorem 45. If and are Banach spaces and is increasing with , then, , where is a pre-quasiclosed operator ideal.

Proof. By Theorem 17, the space is a premodular (). So, from Theorem 12, we have which is a pre-quasinorm on . Suppose that , for all and . Therefore, there is and as ; one has Then, is convergent in . i.e., , and since , for all and is a premodular (). Hence, one can see We obtain ; hence, . ☐

Definition 46. A pre-quasinorm on the ideal , where , verifies the Fatou property if for all sequence with and every , then

Theorem 47. The pre-quasinorm , for each does not verify the Fatou property, if is increasing with .

Proof. Suppose that the conditions are verified and with As the space is a pre-quasiclosed ideal; hence, . Then, for all , one has Therefore, does not verify the Fatou property. ☐

Now, we explain the definition of Kannan -Lipschitzian mapping in the pre-quasioperator ideal.

Definition 48. For the pre-quasinorm on the ideal , where . An operator is named a Kannan -Lipschitzian, if there is such that for every . An operator is named (1)Kannan contraction, if (2)Kannan nonexpansive, if

Definition 49. For the pre-quasi norm on the ideal , where , and The operator is named sequentially continuous at , if and only if, when then .

Theorem 50. Let be an increase with and , where , for all . The point is the unique fixed point of , if the next settings are verified: (a) is Kannan contraction mapping(b) is sequentially continuous at a point ,(c)We have such that the sequence of iterates has a subsequence which converges to

Proof. Suppose that the settings are satisfied. If is not a fixed point of , then, . From conditions (b) and (c), one has As is Kannan contraction mapping, we have Since , one has a contradiction. Hence, is a fixed point of . To prove that the fixed point is unique, assume that we have two different fixed points of . Therefore, we have So, ☐

Example 51. Let and be Banach spaces, , where , for every and

Since for all with , we have

For all with , we have

For all with and , we have

Therefore, the map is Kannan contraction mapping and

It is clear that is sequentially continuous at the zero operator and has a subsequence which converges to . By Theorem 50, the zero operator is the only fixed point of . Let be such that where with . Since the pre-quasinorm is continuous, we have

Hence, is not sequentially continuous at . So, the map is not continuous at .

8. Application to the Existence of Solutions of Summable Equations

Summable equations like (100) studied by Salimi et al. [38], Agarwal et al. [39], and Hussain et al. [40]. In this section, we search for a solution to (100) in where is increasing with and , for all . Consider the summable equations

and let defined by

Theorem 52. The summable equation (100) has a solution in if and for all , suppose that

Proof. Let the conditions be verified. Consider the mapping defined by (101). We have Then, from Theorem 41, we have a solution for equation (100) in ☐

Example 53. Given the sequence space , where , for all . Consider the summable equations with and and let where , defined by Clearly, is a nonempty, convex, closed, and bounded subset of . It is easy to see that By Theorem 52, the summable equation (104) has a solution in .

9. Conclusion

We have introduced the concept of the pre-quasinormed space, which is more general than the quasinormed space. We investigate the sufficient conditions on Nakano () such as its variable exponent in with the known pre-quasinorm to form pre-quasi-Banach and closed (), the concept of a fixed point of Kannan pre-quasi-norm contraction mapping in the pre-quasi-Banach (), which supports the property (), and the pre-quasinormal structure property. The existence of a fixed point of Kannan pre-quasinorm nonexpansive mapping in the pre-quasi-Banach () has been examined. Also, the existence of a fixed point of Kannan pre-quasinorm contraction mapping in the pre-quasi-Banach operator ideal formed by Nakano () and -numbers has been investigated. Finally, we have explained some examples to show that the results obtained can solve a problem. The strength of this approach is that the existence results are established under flexible conditions provided by controlling the power of the Nakano sequence space. The novelty lies in the fact that our main results have improved some well-known theorems before, which concerned the variable exponent in the aforementioned space.

Data Availability

No data were used.

Disclosure

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-20-078-DR. The authors, therefore, acknowledge with thanks the university technical and financial support.

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Copyright © 2021 Awad A. Bakery and O. M. 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.

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