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

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