Convergence Results for Total Asymptotically Nonexpansive Monotone Mappings in Modular Function Spaces

Abstract

In this article, we consider an extensive class of monotone nonexpansive mappings. We use -iteration to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the settings of modular function space.

1. Introduction

In 1965, the existence results for nonexpansive mapping were initiated by Browder [1], Kirk [2], and Göhde [3] independently. The idea about asymptotically nonexpansive mappings was introduced by Goebel and Kirk [4] in 1972. Fixed point results of nonexpansive mapping were extended for monotone case by Bachar and Khamsi [5] in 2015. Alfuraidan and Khamsi [6] extended the concept of asymptotically nonexpansive for the case of monotone in 2018. Alber et al. [7] introduced the concept of total asymptotically nonexpansive mappings that generalizes family of mapping that are the extension of asymptotically nonexpansive mappings in 2006. Example 2 of [8] and Example 3.1 of [9] show that total asymptotically nonexpansive mappings properly contain the asymptotically nonexpansive mappings.

The notation for modular function (MF) space was initiated in 1950 by Nakano [10], which was further generalized by Musielak and Orlicz [11] in 1959. In 1990, Khamsi et al. [12] were the first who initiated fixed point theory in MF space. Alfuraidan, Bachar, and Khamsi [13] in 2017 extended results of Goebel and Kirk [4] for monotone asymptotically nonexpansive mappings in MF spaces using Mann iteration process.

In this article, we extend the notion of monotone total asymptotically nonexpansive mappings in MF space and generalize the results of Alfuraidan and Khamsi presented in [6, 13]. We use -iteration process to approximate the fixed point, which is fastly convergent than the classic Picard [14], Mann [15], and Ishikawa [16] iterative processes.

2. Preliminaries

Firstly, we have the definitions of -ring and -algebra with examples.

Definition 1. Suppose , and be a nonempty family of subsets of , then is called ring of sets if satisfies(i)(ii)A ring of sets is called -ring of sets if for any sequence of sets implies

Example 2. Let be the collection of all finite subsets of , and then is a ring, but not -ring.

Definition 3. Assume that , a collection of subsets of is called algebra of sets if , satisfies(i)(ii), whenever is in An algebra of sets is called -algebra of sets if for every sequence of sets implies

Example 4. For any set , and are -algebras.

In the following, we list some basic concepts of the MF space presented by Kozlowski [12].

Definition 5. Suppose that be a vector space,(a)A functional is known as modular if for , , satisfies(i) if and only if (ii) with (iii), if and (b)If condition (iii) is replaced by(i), if and then is known as convex modular(c)A modular defines a respective MF space, that is, the vector space given by

Definition 6. A subset is said to be -null if (the notation represents the characteristic of ), for any , and a property holds -almost everywhere if the set is -null.

A property is considered to hold almost everywhere (a.e) if there is a set of points where this property fails to hold has measure zero.

Definition 7. Let stands for the class of all extended functions which are also measurable. A convex and even function is said to be regular modular if(1) -almost everywhere(2) for all where is monotone(3) for all where has Fatou propertyConsiderThe MF space is defined as

Following few useful definitions are taken from [17, 18]. From onwards, we assume as a convex regular modular.

Definition 8. (i) is termed as -convergent to if(ii)A sequence is termed as -Cauchy if(iii)Let be -closed if for any sequence , -converge to (iv)Let be -bounded if its -diameter

Definition 9. Suppose that be a vector space, is said to satisfy the -condition, if as whenever decreases to and as

Remark 10. Consider -convergence implies -Cauchy if and only if it satisfies the -condition.

Definition 11. Let and . Definewhere(a)μ is said to satisfy condition if whenever and , we have (b) is considered to satisfy condition if whenever and , exists such that(c) is considered to satisfy condition if whenever any with andwhere .

Following definition of -type function will be used in the main result taken from [18].

Definition 12. Let , and a mapping is said to be -type if a sequence exists such thatfor any . Any sequence in is said to be a minimizing sequence of if

Following are the definitions of monotone and monotone asymptotically nonexpansive mapping in modular space, and useful remark about property , given in [13].

Definition 13. A mapping , where be a nonempty subset of , is said to be(i)Monotone if(ii)Monotone asymptotically nonexpansive if is monotone, and there exists such that , andsuch that -a.e. and . Also is said to be fixed point if .

Remark 14. Let be a -bounded, convex, and -closed subset of where is a convex regular modular. Let be a monotonically increasing sequence in (due to the convexity and -closedness of order intervals in ), then property will imply that

The following Lemmas taken from [19] will be used in main result.

Lemma 15. Let be a -bounded, convex, and -closed subset of where is a convex regular modular satisfying condition . Then, every -type minimizing sequence defined on will be -convergent, and the limit will not depend upon the minimizing sequence.

Lemma 16. Let be a convex regular modular satisfying condition . If there exists and withthen we haveThe -distance from to is given as

Following Lemma taken from [9] will be used in the existence result.

Lemma 17. Suppose , and be sequences of nonnegative satisfyingIf and , then exists.

Following is the definition of condition taken from [20].

Definition 18. Let be a subset of , and a mapping is assumed to fulfill the condition if a nondecreasing functionexists for all such thatfor all .

3. Fixed Point Results for Monotone Total Asymptotically Nonexpansive Mapping

Now, we will define monotone total asymptotically nonexpansive mapping in modular space.

Definition 19. Let be a subset of where is a convex regular modular. A self map of is said to be monotone total asymptotically nonexpansive mapping if there exists nonnegative sequences and with , , as , and a strictly increasing continuous functionsuch thatThere exists a constant such that for thenfor every such that and are comparable -a.e.

Theorem 20. Let be a -bounded and -closed subset of where is a convex regular modular satisfying condition . Let a self map of be a -continuous monotone total asymptotically nonexpansive mapping. Assume that there exists such that or -a.e. Then, has a fixed point such that or -a.e.

Proof. Assume that -a.e. Since is monotone, then we havefor every , and the sequence is monotone increasing. From the above Remark,Consider the -type function define byLet be a minimizing sequence of , from the Lemma -converges to . We have to show that is the fixed point of . Since we have , for every , which impliesIn particular, we havefor . As is total asymptotically nonexpansive, so , , when Hence,The sequence is a minimizing sequence in for any By Lemma 15, is -converge to for any Since is -continuous and is -convergent to , then is -convergent to and Since -limit of any -convergent is unique, we have ; also, we have hence proved.☐

Example 21. Let be an extended real valued function defined on a measureable set , such that for all . The function is measureable if the setis measureable. And the measureability of above set follows directly from the measureability of and So, a constant function is a measureable function. Now, we define a set of extended real valued functions asDefine a function by for all which clearly it is well defined.
Firstly, we need to show that is a convex function. For this, we show that is a convex set. Considerwhich implies Hence, is a convex set. Now, for every , it is easy to prove thatwhich further implies that is a convex function. Now, we check the properties of regular modular.(1)If which further implies (2)Ifas and So, Thus, is monotone.(3)Clearly, is strongly convergent which implies weak convergenceHence, is convex regular modular. Defineand a subsetof . Clearly, is -bounded and -closed. Let a mapping be defined by , where Let be any positive sequences and as . Define a strictly increasing function by , with . ConsiderClearly,Also, there exists a constant , andSo, is monotone asymptotically nonexpansive mapping. Since all conditions of theorem are satisfied; thus, has a fixed point, since implies . Thus, . Hence, the function is a fixed point of .

4. Convergence Analysis

Let be a convex subset of where is a convex regular modularWe modify -iteration in MF space is defined asfor , where and are sequences in

Theorem 22. Let be a -bounded subset of where is a convex regular modular satisfying condition . Let a self map of be a monotone total asymptotically nonexpansive mapping with . Assume that there exists , such that or -a.e. If the sequence is defined by (41) where , , then has a fixed point such that or -a.e. Then, the following holds(a) exist for (b)

Proof. Let and assume that -a.e. Using (41)using (22), we haveupon using (23), and we getNow,using (22), and we haveupon using (23), and we getwhereUsing Lemma 17, exists for For part (b), we have to show thatAssume thatCase 1. If , then the conclusion is trivial.
Case 2. For we know thatTaking on both sides of (50),Also,applies on both sides:Also,Taking on both sides,Now,By using Lemma 16 and from (54) and (56), we haveFrom (41) and (58),taking , and we haveSimilarly,Next,taking , and we getFrom (52) and (63), we getBy using Lemma 16, we havehence proved.☐

Data Availability

There is no any data available.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (No. 71601072), Key Scientific Research Project of Higher Education Institutions in Henan Province of China (No. 20B110006), and the Fundamental Research Funds for the Universities of Henan Province.