Some Fixed Point Results in Premodular Special Space of Sequences and Their Associated Pre-Quasi-Operator Ideal

Abstract

A weighted Nakano sequence space and the -numbers it contains are the subject of this article, which explains the concept of the pre-quasi-norm and its operator ideal. We show that both Kannan contraction and nonexpansive mappings acting on these spaces have a fixed point. A slew of numerical experiments back up our findings. The presence of summable equations’ solutions is shown to be useful in a number of ways. Weight and power of the weighted Nakano sequence space are used to define the parameters for this technique, resulting in customizable solutions.

1. Introduction

The spaces of all, bounded, -absolutely summable, and null sequences of real numbers will be denoted throughout the article by , , , and , respectively, where is the set of nonnegative integers. , while 1 lies in the ^th place, with .

Definition 1 (see [1]). A function , where is the space of all bounded linear operators from a Banach space into a Banach space and if , we write , which transforms every mapping to is said to be -number, if it is satisfying the following conditions: (a), for every (b), for every and , (c)Ideal property: , for every , , and , where and are any two Banach spaces(d)If and , we have (e)Rank property: if , then , for all (f)Norming property: or

The th approximation number, , is defined as

Notations 2 (see [2]). , where . Also, .

Fixed point theory, Banach space geometry, normal series theory, ideal transformations, and approximation theory are all examples of ideal operator theorems and summability. The concept of a pre-quasi-operator ideal is introduced and studied by Faried and Bakery [2]. Bakery and Abou Elmatty investigated some topological and geometric structures of in [3]. They proved that the space is a small pre-quasi-operator ideal and gave a strictly inclusion relation for different weights and powers. Several mathematicians were able to investigate many extensions for contraction mappings defined on the space or on the space itself thanks to the Banach fixed point theorem [4]. Kannan [5] investigated an example of a class of operators that perform the same fixed point actions as contractions but are not continuous. Ghoncheh [6] demonstrated the existence of a Kannan mapping fixed point in complete modular spaces with Fatou property (also see [7–10]). Bakery and Mohamed [11] examined the sufficient requirements on , variable exponent in under definite pre-quasi-norm so that there is a fixed point of Kannan pre-quasi-norm contraction mappings on this space. For the construction of pre-quasi-Banach and closed spaces, we use a weighted Nakano sequence space, , with various pre-quasi-norms in this study. Weighted Nakano sequence space’s pre-quasi-normal structural features, including the fixed point idea of Kannan pre-quasi-norm contraction and the Kannan pre-quasi-norm nonexpansive mapping in weighted Nakano sequence space, are improved. The existence of a fixed point for the Kannan pre-quasi-norm contraction mapping has been demonstrated using weighted Nakano sequence space and -numbers. Our talk concluded with various instances of how the information gathered could be put to good use in resolving a problem.

2. Preliminaries and Definitions

We indicate the space of all mappings , by .

Definition 3 (see [12]). If is a vector space and , a mapping is said to be modular: (a)If , with (b) holds, for each and (c)The inequality verifies, for every and .

Definition 4 (see [2]). If the following conditions hold: (1) is solid. This means if , , and , for every , then (2), where denotes the integral part of , when Then, is said to be a special space of sequences (sss).

Definition 5 (see [2]). If we have with the following: (i)if , (ii)suppose and , then there is for which (iii)the inequality, , for each , verifies for some (iv)if and , then (v)the inequality, , satisfies, for some (vi)assume is the space of finite sequences, one has (vii)we have so that for every Then, is said to be a premodular sss.

Example 1. Since for all , we have Hence, is a premodular (not a modular) on .

Definition 6 (see [11]). Assume is a sss. The function is called a pre-quasi-norm on , if it satisfies the conditions (i), (ii), and (iii) of Definition 5.

Theorem 7 (see [11]. Suppose is a premodular sss; then, is a pre-quasi-normed sss.

Theorem 8 (see [11]). Quasinormed sss is contained in pre-quasi-normed sss.

Definition 9 (see [13]). (a)The pre-quasi-norm on is called -convex, if , for each and (b) is -convergent to , if and only if If the -limit exists, then it is unique(c) is -Cauchy, if (d) is -closed, if for every -converging to , then (e) is -bounded, if (f)The -ball of radius and center , for every , is defined as(g)A pre-quasi-norm on satisfies the Fatou property, when for every sequence with and every then Recall that the Fatou property implies the -closed of the -balls.

Definition 10 (see [14]). Let be the class of each bounded linear operators between any two Banach spaces. A subclass of is known as an operator ideal, if all element fulfills the following conditions: (i), where indicates Banach space of one dimension(ii)The space is linear over (iii)If , , and , then , where and are normed spaces

Pre-quasi-operator ideals are more general than quasioperator ideals.

Definition 11 (see [2]). A mapping is called a pre-quasi-norm when (a)let , , and (b)we have with , when and (c)we have so that , for all (d)we have such that , and then

Theorem 12 (see [11]). Suppose is a premodular sss, then is a pre-quasi-norm on .

Theorem 13 (see [2]). Quasi-normed ideal is contained in pre-quasi-normed ideal.

Lemma 14 (see [15]). If and for all , then

Lemma 15 (see [16]). Let and with ; then,

Lemma 16 (see [17]). Suppose and , for all ; then, where .

3. Main Results

3.1. The Sequence Space

Assume and , where denotes the set of positive reals. In [3], the weighted Nakano sequence space was defined as while

Theorem 17. If , then

Proof. (1)If , with , then defined and considered in [18, 19](2)If , with , then examined by many authors [16, 20, 21]

Theorem 18. The space is a Banach space, where .

Proof. Since we have the following: (i), for each and , if and only if, (ii)suppose , without loss of generality, let then(iii)assume , then there are and be such that Let ; since is nondecreasing and convex, one hasAs the ’s are nonnegative, one can see Then, the space is a normed space. Next, let be a Cauchy sequence in . Therefore, for every , we have such that for all , we obtain So, for and , one can see Hence, is a Cauchy sequence in , for fixed . This implies , for fixed . Hence, , for every . Since , therefore, . This implies that is a Banach space.

4. Pre-Quasi-Normed Sequence Space

To create pre-quasi-Banach and closed sequence space, we study the conditions on , where , for each . The Fatou property of has been investigated for various .

Theorem 19. (a1)Let be an increase.(a2)Either is a monotonic decrease or monotonic increase so that there is , where .Then, is a premodular sss.

Proof. (i)Evidently, and .(1-i) and (iii). Let . As , one gets Hence, .
(1-ii) and (ii). Let and . Since , one has where . Therefore, . From conditions (1-i) and (1-ii), one has which is linear. And for all , as (2) and (iv). Let , for all and . Since , for all , then we get .
(3) and (v). Suppose and is increasing. There is so that and is increasing; one can see Then, .
(vi) Obviously,
(vii) We have , for or , for such that

Theorem 20. Let the conditions (a1) and (a2) of Theorem 19 be satisfied, then be a pre-quasi-Banach sss.

Proof. From Theorems 19 and 7, we have which is a pre-quasi-normed sss. Suppose is a Cauchy sequence in . Therefore, for all , we have such that for all , we get Hence, for and , one obtains This implies is a Cauchy sequence in , for fixed . This explains , with fixed . Therefore, , for all . Also, one has ; hence, .

Theorem 21. The space is a pre-quasi-closed sss, whenever the conditions (a1) and (a2) of Theorem 19 are satisfied.

Proof. Let and ; hence, for all , one has such that for every , we obtain This gives Therefore, is a convergent sequence in , for constant . Hence, , with constant . Also, one gets Hence, .

Theorem 22. The function , for every , has the Fatou property, if the conditions (a1) and (a2) of Theorem 19conditions (a1) and (a2) of Theorem 19 are satisfied.

Proof. Assume and As is a pre-quasi-closed space, this implies . Hence, for all , we have

Theorem 23. The function does not hold the Fatou property, if the setups (a1) and (a2) of Theorem 19 are satisfied with .

Proof. Assume and As is a pre-quasi-closed space, this implies . Hence, for all , we have

Example 2. The function is a pre-quasi-norm and not quasi-norm, for all .

Example 3. The function is a pre-quasi-norm, quasi-norm, and not a norm on , for .

Example 4. The function is a norm on .

5. Kannan Contraction’s Fixed Points

Here, -Lipschitzian mapping acting on as Kannan -Lipschitzian mapping has been defined. We investigate the adequate requirements for a fixed point of Kannan contraction mapping on equipped with various pre-quasi-norms.

Definition 24. A mapping is said to be a Kannan -Lipschitzian, if there exists , such that for every . (1)Let ; then, the operator is called Kannan -contraction(2)For , then the operator is said to be Kannan -non-expansiveA vector is said to be a fixed point of , if

Theorem 25. Assume the conditions (a1) and (a2) of Theorem 19 are satisfied, and is Kannan -contraction mapping, where , for all ; hence, has a unique fixed point.

Proof. Let the setups be satisfied. Assume ; hence, . Since is a Kannan -contraction mapping, we obtain Therefore, for with , one has Hence, is a Cauchy sequence in, since is pre-quasi-Banach space. We have with, to show that . As verifies the Fatou property, we get Hence, . So is a fixed point of . To show the uniqueness of , let us have two different fixed points of . So, one has This implies

Example 5. Assume , where, for every and As for each with , one has For all with , one has For all with and , we get

Hence, is Kannan -contraction and holds one element , so that , by Theorem 25.

Corollary 26. Let conditions (a1) and (a2) of Theorem 19 be satisfied, and is Kannan -contraction mapping, where , for all ; then, has a unique fixed point with .

Proof. In view of Theorem 25, we have a unique fixed point of . Therefore, one gets

Definition 27. Assume is a pre-quasi-normed sss, and The operator is said to be -sequentially continuous at , if and only if, when , then .

Example 6. Suppose , where , for every and

is clearly both -sequentially continuous and discontinuous at .

Example 7. Assume is defined as in Example 5. Suppose is such that , where with .
As the pre-quasi-norm is continuous, we have

Therefore, is not -sequentially continuous at .

Theorem 28. If the conditions (a1) and (a2) of Theorem 19 are satisfied with and , where , for all , (1)suppose is Kannan -contraction mapping(2)assume is -sequentially continuous at a point (3)we have such that has a subsequence converging to ; then, is the only fixed point of

Proof. Suppose the settings are verified. Assume is not a fixed point of , then . From parts (54) and (55), one gets Since is Kannan -contraction, we obtain We get a contradiction when. To show the uniqueness of , suppose we have two different fixed points of . Therefore, one obtains Hence,

Example 8. Assume is defined as in Example 5. Let , for every .
As for each with , one has For all