Generalized Enriched Nonexpansive Mappings and Their Fixed Point Theorems

Shukla, Rahul; Panicker, Rekha

doi:https://doi.org/10.1155/2023/5572893

Abstract and Applied Analysis

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5572893 | https://doi.org/10.1155/2023/5572893

Generalized Enriched Nonexpansive Mappings and Their Fixed Point Theorems

Rahul Shukla¹and Rekha Panicker¹

Academic Editor: Paul Eloe

Received05 Apr 2023

Revised03 Jun 2023

Accepted15 Nov 2023

Published02 Dec 2023

Abstract

This paper introduces a novel category of nonlinear mappings and provides several theorems on their existence and convergence in Banach spaces, subject to various assumptions. Moreover, we obtain convergence theorems concerning iterates of -Krasnosel’skiĭ mapping associated with the newly defined class of mappings. Further, we present that -Krasnosel’skiĭ mapping associated with -enriched quasinonexpansive mapping is asymptotically regular. Furthermore, some new convergence theorems concerning -enriched quasinonexpansive mappings have been proved.

1. Introduction

Consider a Banach space and a nonempty subset . A mapping is nonexpansive if for all . A point is a fixed point of if . While nonexpansive mappings may not have fixed points in a general Banach space. Browder [1], Göhde [2], and Kirk [3] independently proved fixed point theorems for nonexpansive mappings with certain geometric properties, such as uniform convexity or normal structure. Since then, numerous authors have obtained various extensions and generalizations of nonexpansive mappings and their results. Some of these notable extensions and generalizations are summarized in Pant et al.’s [4] study.

In 2008, Suzuki [5] introduced a novel category of nonexpansive mappings called mappings that fulfill condition (C) and derived significant fixed point results for this category. García-Falset et al. [6] further generalized condition (C) into the class of mappings satisfying condition (E).

Definition 1 [6]. Let be a nonempty subset of a Banach space A mapping is said to satisfy condition on if there exists such that for all We say that satisfies condition on whenever satisfies for some

This class of mappings properly contains many important classes of generalized nonexpansive mappings, see Pant et al’s [7] study.

A novel category of nonlinear mappings was introduced by Berinde [8] in a recent publication.

Definition 2 [8]. Let be a Banach space. A mapping is said to be -enriched nonexpansive mapping if such that:

In the recent years, a number of papers have appeared in the literature dealing with the fixed point theorems for enriched nonexpansive type mappings [9–11]. It was proven that any enriched contraction mapping defined on a Banach space has a unique fixed point, which can be approximated by means of the Krasnoselskij iterative scheme. In Berinde’s [12] study (also [13]), it was demonstrated that every nonexpansive mapping is a zero-enriched mapping. Building in this work, Shukla and Pant [14] recently extended the category of enriched nonexpansive mappings in the vein of Suzuki and introduced the subsequent class of mappings.

Definition 3. Let be a Banach space and a nonempty subset of A mapping is said to be Suzuki-enriched nonexpansive mapping if there exists such that for all :

It can be seen that every Suzuki-nonexpansive mapping is a Suzuki-enriched nonexpansive mapping with Motivated by García-Falset et al. [6], we generalize Suzuki-enriched nonexpansive mappings and consider a new class of mappings known as (E)-enriched nonexpansive mappings. In fact, we introduce a class of mapping, which contains both the Suzuki-enriched nonexpansive mappings and the class of mappings satisfying condition (E). Indeed the class of -enriched nonexpansive mappings and that of mappings satisfying condition (E) are independent in nature. A couple of examples below illustrate these facts.

Example 1 [4]. Consider the mapping defined by , where . The set of fixed point of is , and we can say that . Moreover, is a nonexpansive mapping that satisfies the -enriched condition. However, at and , fails to satisfy condition (E).

Example 2 [5]. Let with the usual norm. Define by the following equation:Then, satisfies condition (E). However at and is not -enriched nonexpansive mapping for any

This paper is organized as follows: Section 2 deals with some preliminary results which are utilized throughout this paper. In Section 3, we coined a new class of mappings, namely, (E)-enriched nonexpansive mapping. We show that (E)-enriched nonexpansive mapping properly contains some nonlinear mappings and present an illustrative example. Section 4 is devoted to some existence and convergence theorems concerning (E)-enriched nonexpansive mapping. In Section 5, we present some new developments of enriched quasinonexpansive mapping. Particularly, an (E)-enriched nonexpansive mapping with fixed point is an enriched quasinonexpansive mapping. We discuss convergence of the iterates of -Krasnosel’skiĭ mapping associated with enriched quasinonexpansive mapping.

2. Preliminaries

We denote the set of all fixed points of mapping , i.e., A Banach space is said to be uniformly convex if for each , such that for all with and A Banach space is strictly convex if:whenever with [15].

Theorem 1 [15]. Let be a uniformly convex Banach space. Then for any , , and with , there exists a such that:

Definition 4 [16]. A Banach space satisfies Opial property if for every weakly convergent sequence with weak limit , it holds:for all with

Definition 5 [17]. Suppose is a nonempty subset of a Banach space . Let be an element in such that there exists a point in satisfying the following condition: for any , . In this case, we refer to as a metric projection of onto and denote it by . If exists and is uniquely determined for all , then we call the mapping , the metric projection onto .

Definition 6 [17]. A mapping is said to be quasinonexpansive if:for all and

The fact that a nonexpansive mapping with a fixed point is quasinonexpansive is widely recognized. However, it should be noted that the converse may not hold.

Proposition 1 [6]. Let be a nonempty subset of a Banach space If is a mapping satisfying condition with then is quasinonexpansive.

Definition 7 [18]. Let be a nonempty convex subset of a Banach space and be a mapping. A mapping is said to be an -Krasnosel’skiĭ mapping associated with if there exists such that:for all

Definition 8 [19]. Let be a nonempty subset of a Banach space A mapping is called asymptotically regular if:

Lemma 1 (Browder [20]; demiclosedness principle). Suppose is a nonempty subset of a Banach space that satisfies the Opial property. Let be a mapping that satisfies condition (E). Consider a sequence in such that weakly converges to , and Then, we have , which implies that is demiclosed at zero.

Proof. The proof directly follows from [6, Theorem 1].

Lemma 2 (Berinde [8]). Let be a nonempty convex subset of a Banach space , and be a mapping. Define as follows:for all and Then,

3. (E)-Enriched Nonexpansive Mapping

This section presents a novel category of mappings, which we describe as follows.

Definition 9. Let be a Banach space and be a nonempty subset of We define a mapping an (E)-enriched nonexpansive mapping if and such that:

It can be seen that every mapping satisfying condition (E) is an (E)-enriched nonexpansive mapping with

Remark 1. If is a nonempty subset of and is (E)-enriched nonexpansive, and there exists a sequence in such that . Such a sequence is called almost fixed point sequence (a.f.p.s.) for .

Proposition 2. Let be a Suzuki-enriched nonexpansive mapping with any Then, is an (E)-enriched nonexpansive mapping for any and

Proof. We assume that is a Suzuki-enriched nonexpansive mapping. Then, implies the following equation:Now, we show that either:Arguing by contradiction, we suppose that:By the triangle inequality, we get the following equation:By Equation (13), we get the following equation:which is a contradiction. Consequently, Equation (14) holds. Therefore, from Equation (14), we have either:In the first case, we have the following equation:In the other case, we have the following equation:Therefore in both the cases, we get the following equation:

The following example shows that (E)-enriched nonexpansive mapping properly contains the class of Suzuki-enriched nonexpansive mappings.

Example 3. Let be the set of real numbers equipped with the standard norm and a subset of . Let be a mapping defined by the following equation:First, we show that is (E)-enriched nonexpansive mapping. For this, we consider the following nontrivial cases:Case (1).If and , then the following equation is obtained:Case (2).If and , then the following equation is obtained:Case (3).If and , then the following equation is obtained:Moreover, for and , we get the following equations:andThus, is not a Suzuki-enriched nonexpansive mapping with any

4. Some Fixed Point Theorems

In this section, we present some new fixed point theorems for (E)-enriched nonexpansive mappings.

Theorem 2. Let be a Banach space and a nonempty subset of Let be a mapping. If:(a) is an (E)-enriched nonexpansive mapping on (b)there exists an a.f.p.s. for in such that converges weakly to a point in , and(c) satisfies the Opial property.Then,

Proof. By the definition of mapping , we have the following equation:for all . Take, and put in Equation (28), then the above inequality is equivalent to the following equations:andDefine the mapping as follows:Thus, the following equation is obtained:Then, from Equation (30), we get the following equation:for all Thus, is a mapping satisfying condition (E). From Equation (32), it follows that if is an a.f.p.s. for then is an a.f.p.s. for Thus, all the assumptions of [6, Theorem 1] are satisfied and has a fixed point in , that is, From Lemma 2, This completes the proof.

Corollary 1. Let be a Banach space and a nonempty weakly compact subset of a Banach space Suppose that satisfies the Opial property. Let be an (E)-enriched nonexpansive mapping on Then, has a fixed point in if and only if admits an a.f.p.s.

Theorem 3. Let be a Banach space and a nonempty compact subset of a Banach space Let be an (E)-enriched nonexpansive mapping on Then, has a fixed point in if and only if admits an a.f.p.s.

Proof. By following the proof of Theorem 1, we can construct a mapping that satisfies condition (E). Therefore, all the conditions of [6, Theorem 2] are satisfied, and we can guarantee that has at least one fixed point. Using Lemma 2, we can conclude that . This completes the proof.

Theorem 4. Let be a uniformly convex in every direction (or UCED) Banach space and a nonempty weakly compact convex subset of Let be a mapping. If:(a) is an (E)-enriched nonexpansive mapping on and(b),then admits a fixed point.

Proof. In view of Theorem 1 and [6, Theorem 3], one can complete the proof.

In the next theorem, we present the convergence of iterates of -Krasnosel’skiĭ mapping associated with (E)-enriched nonexpansive mapping.

Theorem 5. Let be a Banach space and be an (E)-enriched nonexpansive mapping. For a given and , if the sequence of iterates converges strongly to , where is the -Krasnosel’skiĭ mapping associated with , then .

Proof. Using the same technique as in Theorem 1, one can define a mapping as follows:and is a mapping satisfying condition (E). For a given and , one can define a sequence as follows:Using the definition of , we have the following equationTake , then:Since strongly converges to a point in Thus, all assumptions of [21, Theorem 1] are satisfied, and But . This completes the proof.

Theorem 6. Let be a nonempty closed convex subset of a uniformly convex Banach space and an (E)-enriched nonexpansive mapping with Assume that the mapping is demiclosed at zero. Then for each , the sequence of iterates converges weakly to where is the -Krasnosel’skiĭ mapping associated with and

Proof. Using the same technique as in Theorem 1, one can define a mapping such that is a mapping satisfying condition (E). In fact, the demiclosedness of and at zeros are equivalent. Further, demiclosedness of and at zeros are equivalent. Keeping [21, Theorem 2.2] in mind, one can complete the proof.

Theorem 7. Let be a nonempty closed convex subset of a uniformly convex Banach space which has the Opial property. Let be an (E)-enriched nonexpansive mapping with Then for each and , the sequence of iterates converges weakly to a fixed point of

Proof. From [21, Theorem 2.2] and Theorem 6, one can complete the proof.

5. Enriched Quasinonexpansive Mapping

In this section, we present some new convergence results for -enriched quasinonexpansive mapping. Shukla and Pant [14] introduced the following new class of mappings.

Definition 10. Let be a Banach space and a nonempty subset of . A mapping is said to be -enriched quasinonexpansive mapping if there exists such that for all and :

It is noted that every quasinonexpansive mapping is a zero-enriched quasinonexpansive mapping and every -enriched nonexpansive mapping with a fixed point is -enriched quasinonexpansive mapping.

Proposition 3. Let be an (E)-enriched nonexpansive mapping with any , and Then, is a -enriched quasinonexpansive mapping for any

Proof. Let and we have the following equation:Thus, is a -enriched quasinonexpansive mapping.

The following example demonstrates that converse of the above proposition does not hold.

Example 4 [6]. Let be a subset of . Let be a mapping defined by the following equation:Clearly, It can be seen that:for all and is a -enriched quasinonexpansive mapping with
On the other hand, let and for all we get the following equation:as Hence, is not an (E)-enriched nonexpansive mapping for any

Theorem 8. Let be a nonempty convex subset of a uniformly convex Banach space If is a -enriched quasinonexpansive mapping with Then, the -Krasnosel’skiĭ mapping for is asymptotically regular.

Proof. By the definition of -enriched quasinonexpansive mapping, we have the following equation:for all and Take and put in Equation (43), then the above inequality is equivalent to the following equation:Define the mapping as follows:From Lemma 2, . Then, from Equation (44), we get the following equation:for all and Thus, is a quasinonexpansive mapping. And, we obtain the following equation:Let For each define Thus, the following equations are obtained:andIn order to prove that is asymptotically regular, it suffices to prove that Since , let Since is a quasinonexpansive mapping and :for all From Equation (45), the following equations are obtained:andNow:From Equation (52), the following equation is obtained:Take , then From Equation (50), the following equation is obtained:Therefore, the sequence is bounded by If for any then from Equation (56), as If for all take the following conditions:If and from Equation (55), we have the following equation:Since is a uniformly convex Banach space, by using the definition of uniformly convex space with , and the following equation:Noting that modulus of convexity is a nondecreasing function of , we obtain the following equation:From Equations (60) and (62), the following equation is obtained:Using induction in the above inequality, it follows that:We shall prove that Arguing by contradiction, consider that does not converge to zero. Then, there exists a subsequence of such that converges to Since is nondecreasing and we have for all From Equation (63) and for sufficiently large we have the following equation:Making , it follows that By Equation (46), we get and as which is a contradiction.
If then because Now:and by the uniform convexity of we get the following equation:Using induction in the above inequality, we get the following equation:Using the similar argument as in the previous case, it can be easily shown that as Therefore, in both cases, as From Equation (47), as , is asymptotically regular and this completes the proof.

Remark 2. The above theorem is a generalization of [18, Theorem 1] for a more general class of mappings.

Theorem 9. Let be a nonempty closed subset of a Banach space and a quasinonexpansive mapping. Assume that is strictly convex and is a convex compact subset of If is continuous then for any the -Krasnosel’skiĭ process converges to some

Proof. Following the same line of proof of [7, Theorem 5], one can complete the proof.

Theorem 10. Let and be the same as in Theorem 6. Let be a -enriched quasinonexpansive mapping with If is continuous then for any the -Krasnosel’skiĭ process converges to some

Proof. Following the same proof technique as in Theorem 4, we can define a mapping as follows:and is a quasinonexpansive mapping. For given , , we can define a sequence as follows:From Theorem 6, the sequence converges to some Using the definition of , we have the following equation:Take , then, we obtain the following equation:Hence, strongly converges to a fixed point of . But . This completes the proof.

Theorem 11. Let be a nonempty closed convex subset of a uniformly convex Banach space Let be a quasinonexpansive mapping with and the metric projection from into Then for each , the sequence converges to some

Proof. Following the same line of proof of [7, Theorem 6], one can complete the proof.

Theorem 12. Let , and be same as in Theorem 8. Let be a -enriched quasinonexpansive mapping with Then for each , the -Krasnosel’skiĭ process converges to some

Proof. In view of Theorems 4 and 8, one can complete the proof.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Open access funding was enabled and organized by SANLiC Gold.

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Copyright © 2023 Rahul Shukla and Rekha Panicker. 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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