Abstract

For different premodular, which is a generalization of modular, defined by weighted Orlicz sequence space and its prequasi operator ideal, we have examined the existence of a fixed point for both Kannan contraction and nonexpansive mappings acting on these spaces. Some numerous numerical experiments and practical applications are presented to support our results.

1. Introduction

The spaces of all, bounded, r-absolutely summable, and null sequences of real numbers will be denoted throughout the article by , , , and , respectively, where is the set of nonnegative integers.

Definition 1. [1, 2] An Orlicz function is a function, which is continuous and strictly increasing with,for, and, as.

Definition 2. An Orlicz functionis said to satisfy-condition for every values of, if there is, such that. The-condition is equivalent tofor every values ofand.

Lindentrauss and Tzafriri [3] utilized the idea of a convex Orlicz function to define Orlicz sequence space:

is a Banach space with the Luxemburg norm:

Every Orlicz sequence space contains a subspace that is isomorphic to or , for some ([4], Theorem 4.a.9). The space of all bounded linear operators from a Banach space into a Banach space will be denoted by and if , we write . , while 1 lies in the place, with .

Definition 3. [5] An-number function is a mapping fromintowhich transforms every maptosatisfying the next conditions:(i), for every,(ii), for every, and,,(iii)ideal property:, for every,and, whereandare any two Banach spaces,(iv)forand, we have,(v)rank property: If, then, for all,(vi)norming property:or, whereexplains the unit map on the-dimensional Hilbert space.

The th approximation number, , is defined as

Notations 1. The sets , , , and (cf. [6]) are defined as follows:

Fixed point theory, Banach space geometry, normal series theory, ideal transformations, and approximation theory are all examples of ideal operator theorems and summability. Faried and Bakery [6] established the concept of a prequasi operator ideal that encapsulates the quasi operator ideal. Bakery and Abou Elmatty investigated the sufficient (but not necessary) conditions on that allowed to build a simple Banach prequasi operator ideal in [7]. For varied weights and powers, the prequasi operator ideal was once rigorously contained and small prequasi operator ideal. Several mathematicians were able to investigate many extensions for contraction maps defined on the space or on the space itself thanks to the Banach fixed point theorem [8]. Kannan [9] investigated an example of a class of operators that perform the same fixed point actions as contractions but are not continuous. Kannan operators in modular vector spaces have only been described by Ghoncheh [10]. He demonstrated the existence of a Kannan mapping fixed point in complete modular spaces with Fatou property. For more details on Kannan’s fixed point theorems and modular vector spaces (see [1114]). Bakery and Mohamed [15] introduced the concept of the prequasi norm on with variable exponent in . They looked at the Fatou property of different prequasi norms on , as well as the sufficient requirements on with the definite prequasi norm to construct prequasi Banach and closed space. They also demonstrated the existence of a fixed point of Kannan prequasi norm contraction maps on and the prequasi Banach operator ideal constructed by and -numbers. Recently, Reich and Zaslavski [16] showed the existence of a unique fixed point for nonlinear contractive self-mappings of a nonbounded closed subset of a Banach space. They extended this conclusion to contractive mappings, which map into a Banach space a closed subset of the space. For nonexpansive mappings defined by an intersection of a finite number of closed bounded and convex nonempty subsets in Banach spaces, Dehici and Redjel [17] obtained certain fixed point results. According to Bendahmane and Bendoukha [18], a -metric space is a generalization of the metric and -metric spaces. They equipped them a Hausdorff topology and specified several fundamental features. Several well-known findings from fixed point theory are generalized to these new spaces. The paper is structured as follows: we present conditions on the weighted Orlicz sequence space , under definite prequasi norm of to construct prequasi Banach and closed sequence space in Section 3. The Fatou property of has been investigated for various prequasi norms. In Section 4, the existence of fixed point for Kannan -contraction mapping acting on equipped with different prequasi norms are presented. Several numerical experiments are shown to demonstrate our results. In Section 5, the conditions for which the space satisfies the property (R) and has the -normal structure property are presented. The existence of a fixed point of Kannan prequasi norm nonexpansive mapping on has been given. In Section 6, we explain the existence of a fixed point of Kannan prequasi norm contraction mapping in the prequasi Banach operator ideal . In Section 7, we give some applications to the existence of solutions of summable equations.

2. Definitions and Preliminaries

Here and after, the space of all functions is , is the zero vector of , is the integral part of , is the space of finite sequences, and is the class of each bounded linear mapping between any two Banach spaces. Nakano [19] introduced the concept of modular vector spaces.

Definition 4. Letbe a vector space. A functionis called modular if the following conditions hold:(i)If,and,(ii)ifand, then,(iii)assumeand, then.

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

Definition 5. [6] The linear space of sequences is said to be a special space of sequences (sss), if:(1),(2)is solid, i.e., for,and, for all, then,(3)If, then.

Definition 6. [6] A subclass of is called a premodular (sss), if we have that satisfies the following conditions:(i)When,,(ii)For everyand, then there iswith,(iii), for all, holds for some,(iv)Ifand, then,(v)For some, we have,(vi),(vii)There existssuch that, for all.

Example 1. The functionis a premodular (not a modular) on the vector space. As for every, one has

Definition 7. [15] Suppose is a (sss). The function is said to be prequasi norm on , if it holds the settings (i), (ii), and (iii) of Definition 6.

Theorem 1. [15] Let be a premodular (sss), then it is prequasi normed (sss).

Theorem 2. [15] is a prequasi normed (sss), when it is quasi-normed (sss).

Definition 8. [20](i)The prequasi normonis said to be-convex, when, for alland.(ii)is-convergent to, if and only if,. If the-limit exists, hence it is unique.(iii)is-Cauchy, if.(iv)is-closed, if for every-convergingto, then.(v)is-bounded, if.(vi)The-ball of radiusand center, for every, is defined as(vii)A prequasi normonsatisfies the Fatou property, if for every sequencewithand any, we have.

Recall that the -balls are -closed under the Fatou property.

Definition 9. [21] A subclass of is called an operator ideal, if every vector holds the following conditions:(i), whereindicates Banach space of one dimension.(ii)The spaceis linear over.(iii)If,, and, then, whereandare normed spaces.

Recall that the quasi operator ideals are a special case of the prequasi operator ideals.

Definition 10. [6] A function is said to be a prequasi norm on the ideal if the following conditions verify:(1)Suppose, and, if and only if,,(2)there existssuch that, for everyand,(3)we haveso that, for all,(4)we getso that if,, and, then.

Theorem 3. [15] The function is a prequasi norm on , when is a premodular (sss).

Theorem 4. [6] If is a quasi norm on the ideal , then is a prequasi norm on the ideal .

Lemma 1. [22, 23] Assumeis a continuous function and strictly increasing with, and if the functionsandare convex on, then, for all and .

3. Main Results

3.1. Properties of Different Prequasi Norms

In this section, we have studied some topological structures and the Fatou property of the weighted Orlicz sequence space, , for various prequasi norms.

Lemma 2. Ifis a concave Orlicz function, then, for all.

Proof. It is easy so omitted.

Theorem 5. , where, for each, is a premodular (sss), ifis a concave Orlicz function or convex Orlicz function satisfying-condition.

Proof. Suppose is a convex Orlicz function satisfying -condition. First, we must demonstrate that is a (sss):(1)(i)Let . As is a strictly increasing and convex function satisfying -condition, we getthis implies .(ii)Suppose and . Since satisfies -condition, we haveSo . Therefore, from conditions 1 (i) and (ii), one has is linear. We have , for every , as(2)Let , for every and . Since is a nondecreasing function, thenone has .(3)Assume , we getthen . Second, to prove that the functional on is a premodular:(i)Obviously, and .(ii)There are with , for every and .(iii)There exists with , for every .(iv)Follows the proof part (2).(v)Follows from the proof part (3) that .(vi)Obviously, .(vii)There exists , for or , for so that .If is a concave Orlicz function. By applying Lemma 2 and the parallel proof follows.

Theorem 6. Ifis a concave Orlicz function or convex Orlicz function satisfying-condition, thenis a prequasi Banach (sss), where, for each.

Proof. Suppose is a convex Orlicz function satisfying -condition. By using Theorem 5, the space is a premodular (sss). From Theorem 1, the space is a prequasi normed (sss). To prove that is a prequasi Banach (sss), let be a Cauchy sequence in . Therefore, for all , we have that for every , we getHence, for and , one has Then is a Cauchy sequence in , for fixed . This gives , for constant . Therefore, , for all . To investigate that , one has so . This implies that is a prequasi Banach (sss). If is a concave Orlicz function. By applying Lemma 2 and the parallel proof follows.

Theorem 7. If is a concave Orlicz function or convex Orlicz function satisfying -condition, then is a prequasi closed (sss), where , for every .

Proof. Let be a convex Orlicz function satisfying -condition. According to Theorem 5, the space is a premodular (sss). From Theorem 1, the space is a prequasi normed (sss). To prove that is a prequasi closed (sss), suppose and , hence for all , one has so that for every , we haveTherefore, for and , one has Hence, is a convergent sequence in , for constant . So, , for constant . Finally to show that , one hasHence, . This implies that is a prequasi closed (sss). If is a concave Orlicz function, by applying Lemma 2 and the parallel proof follows.

Theorem 8. If is a convex Orlicz function satisfying -condition and is convex, then the function verifies the Fatou property, for all .

Proof. Assume that such that . As the space is a prequasi closed space, one has . Hence, for every , from Lemma 1, we haveHence, satisfies the Fatou property.

Theorem 9. Ifis a concave Orlicz function, then the functionholds the Fatou property, for all.

Proof. Suppose so that . As the space is a prequasi closed space; hence, . As is continuous, concave and . Therefore, for every , one hasHence, satisfies the Fatou property.

Theorem 10. The functiondoes not satisfy the Fatou property, for all, ifis a strictly convex Orlicz function satisfying-condition.

Proof. Since is a strictly convex Orlicz function satisfying -condition, then there exists such that , for all . Let the conditions be fulfilled and with . As the space is a prequasi closed space; hence, . Since is continuous, then for any , we haveTherefore, does not hold the Fatou property.

Example 2. For every, the functionis a prequasi norm, not quasi, and not a norm.

Example 3. For all, the functionis a prequasi norm, quasi norm, and not a norm.

Example 4. The functionis a prequasi norm, a quasi norm, and a norm on.

4. Kannan -Contraction Operator

We now define Kannan -Lipschitzian mapping acting on . The sufficient conditions for a fixed point of Kannan contraction mapping on under various prequasi norms are investigated.

Definition 11. An operatoris called a Kannan-Lipschitzian, if there exists, so thatfor every .(1)The operator is said to be Kannan -contraction, when .(2)The operator is said to be Kannan -nonexpansive, whenever .A vector is called a fixed point of , when .

Theorem 11. Ifis a convex Orlicz function satisfying-condition andis convex, andis Kannan-contraction mapping, where, for all; hence,has a unique fixed point.

Proof. Assume that , one has . 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 space. Therefore, there is such that . To prove that . As holds the Fatou property, we obtainhence . Hence, is a fixed point of . To prove the uniqueness of the fixed point. For different fixed points of . We have thatTherefore, .

Corollary 1. Letbe a convex Orlicz function satisfying-condition andbe convex, andbe Kannan-contraction mapping, with, for every, thenhas a unique fixed pointsuch that.

Proof. From Theorem 11, there is a unique fixed point of . Hence, one has

Theorem 12. Supposeis a concave Orlicz function, andis Kannan-contraction mapping, where, for all; hence,has a unique fixed point.

Proof. It is easy so omitted.

Definition 13. Assumeis a pr-quasi normed (sss),and. The operatoris called-sequentially continuous at, if and only if, when, then.

Theorem 14. Letbe a strictly convex Orlicz function satisfying-condition, and, where, for every. The elementis the unique fixed point of, if the next conditions are satisfied:(i) is Kannan -contraction mapping,(ii) is -sequentially continuous at a point ,(iii)There exists such that the sequence of iterates has a subsequence converging to .

Proof. Since is a strictly convex Orlicz function satisfying -condition, then there exists such that , for all . Let the conditions be verified. If is not a fixed point of , then . By the conditions (ii) and (iii), we haveAs the operator is Kannan -contraction, one can seeSince , this gives a contradiction. Hence, is a fixed point of . To prove that the uniqueness of the fixed point . For different fixed points of . Therefore, one hasSo, .

Example 15. Assume, whereand, for allandAs for each with , one hasFor all with , one hasFor all with and , we obtainHence, the operator is Kannan -contraction. As verifies the Fatou property. From Theorem 11, the operator has a unique fixed point .
Assume is such that , where
with .
As the prequasi norm is continuous, one can seeTherefore, is not -sequentially continuous at . Hence, the operator is not continuous at .
Let , for all .
As for all with , one hasFor all with , one hasFor all with and , we getSo, the operator is Kannan -contraction and
Clearly, is -sequentially continuous at and contains a subsequence converging to . From Theorem 14, then is the unique fixed point of .

Example 5. Assume, whereand, for allandAs for each with , one hasFor all with , one hasFor all with and , we getHence, the operator is Kannan -contraction. As satisfies the Fatou property. From Theorem 11, the operator has one fixed point .
Suppose is so that , where.
with . As the prequasi norm is continuous, one can seeTherefore, is not -sequentially continuous at . Hence, the map is not continuous at .
Let , for every .
As for each with , one hasFor all with , one has