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

Authors: Maliha Rashid; Amna Kalsoom; Shao-Wen Yao; Abdul Ghaffar; Mustafa Inc
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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


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## Body

## 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 S-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 ofδ-ring and σ-algebra with examples.Definition 1.
SupposeΣ≠∅, and R be a nonempty family of subsets of Σ, then R is called ring of sets if A,B∈R, satisfies(i)
A∪B∈R(ii)
A\B∈R
A ring of setsR is called δ-ring of sets if for any sequence of sets An∈R implies ∪n=1∞An∈R.Example 2.
LetR be the collection of all finite subsets of ℕ, and then R is a ring, but not δ-ring.Definition 3.
Assume thatΣ≠∅, a collection A of subsets of Σ is called algebra of sets if A,B∈A, satisfies(i)
A∪B∈A(ii)
A′∈A, whenever A is in A
An algebra of setsA is called σ-algebra of sets if for every sequence of sets An∈A implies ∪n=1∞An∈A.Example 4.
For any setΣ, PΣ and ϕ,Σ are σ-algebras.In the following, we list some basic concepts of the MF space presented by Kozlowski [12].Definition 5.
Suppose thatΣ be a vector space,(a)
A functionalμ:Σ⟶0,∞ is known as modular if for ù,τ∈Σ, μ, satisfies(i)
μù=0 if and only if ù=0(ii)
μγù=μù with γ=1(iii)
μγù+ντ≤μù+μτ, if γ+ν=1 and γ≥0,ν≥0(b)
If condition (iii) is replaced by(i)
μγù+ντ≤γμù+νμτ, if γ+ν=1 and γ≥0,ν≥0, then μ is known as convex modular(c)
A modularμ defines a respective MF space, that is, the vector space Σμ given by(1)Σμ=ù∈Σ;μλù⟶0asλ⟶0Definition 6.
A subsetA∈A is said to be μ-null if μν1A=0 (the notation 1A represents the characteristic of A), for any ν∈Σ, and a property μw holds μ-almost everywhere μ−a.e. if the set w∈Σ;μwdoesnothold is μ-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.
LetM∞ stands for the class of all extended functions which are also measurable. A convex and even function μ:M∞⟶0,∞ is said to be regular modular if(1)
μù=0⇒ù=0μ-almost everywhere(2)
ùt′τt for all t∈Ω⇒μù′μτ, where ù,τ∈M∞μ is monotone(3)
ùnt↑ùt for all t∈Ω⇒μùn↑μù, where ù,∈M∞μ has Fatou property
Consider(2)M=ù∈M∞;ùt<∞μ−almosteverywhere.
TheMF spaceLμ is defined as(3)Lμ=ù∈M;μλù⟶0asλ⟶0.Following few useful definitions are taken from [17, 18]. From onwards, we assume μ as a convex regular modular.Definition 8.
(i)
τn is termed as μ-convergent to τ if(4)limn⟶∞μτn−τ=0.(ii)
A sequenceτn is termed as μ-Cauchy if(5)limm,n⟶∞μτn−τm=0.(iii)
LetK⊂Lμ be μ-closed if for any sequence τn∈K, μ-converge to τ⇒τ∈K(iv)
LetK⊂Lμ be μ-bounded if its μ-diameter supμτ−h;h∈K<∞Definition 9.
Suppose thatΣ be a vector space, μ is said to satisfy the Δ2-condition, if supn≥1μ2ùn,Dk⟶0 as k⟶∞ whenever Dk decreases to ϕ and supn≥1μùn,Dk⟶0ask⟶∞.Remark 10.
Considerμ-convergence implies μ-Cauchy if and only if it satisfies the Δ2-condition.Definition 11.
Letr>0 and ε>0. Define(6)δμr,ε=inf1−1rμù+τ2;ù,τ∈Dr,ε,where(7)Dr,ε=ù,τ;ù,τ∈Lμ,μù≤r,μτ≤r,μù−τ≥rε.(a)
μ is said to satisfy condition UC if whenever R>0 and ε>0, we have δpR,ε>0(b)
μ is considered to satisfy condition UUC if whenever s>0 and ε>0, ηs,ε>0 exists such that(8)δμR,ε>ηs,ε>0,forR>s.(c)
μ is considered to satisfy condition SC if whenever any τ,h∈Lμ with μτ=μh and(9)μγτ+1−γh=γμτ+1−γμh,forsomeγ∈0,1,where ù=τ.Following definition ofμ-type function will be used in the main result taken from [18].Definition 12.
LetK⊂Lμ, and a mapping τ:K⟶0,∞ is said to be μ-type if a sequence τm∈Lμ exists such that(10)τù=limsupn⟶∞μτm−ù,for any ù∈K. Any sequence ùn in K is said to be a minimizing sequence of τ if(11)limn⟶∞τùn=infτù;ù∈K.Following are the definitions of monotone and monotone asymptotically nonexpansive mapping in modular space, and useful remark about propertyR, given in [13].Definition 13.
A mappingΓ:K⟶K, where K be a nonempty subset of Lμ, is said to be(i)
Monotone if(12)Γù′Γτμ−a.e.wheneverù′τμ−a.e.,forù,τ∈K.(ii)
Monotone asymptotically nonexpansive ifΓ is monotone, and there exists Ln⊂1,+∞ such that limn⟶∞Ln=1, and(13)μΓnτ−Γnh≤Lnμτ−h,forù,τ∈K,such that τ′hμ-a.e. and n≥1. Also τ is said to be fixed point if Γτ=τ.Remark 14.
LetK≠ϕ be a μ-bounded, convex, and μ-closed subset of Lμ where μ is a convex regular modular. Let ùn be a monotonically increasing sequence in K (due to the convexity and μ-closedness of order intervals in Lμ), then property R will imply that(14)∩n≥1ù∈K;ùn′ùμ−a.e≠ϕ.The following Lemmas taken from [19] will be used in main result.Lemma 15.
LetK≠ϕ be a μ-bounded, convex, and μ-closed subset of Lμ where μ is a convex regular modular satisfying condition UUC. Then, every μ-type minimizing sequence defined on K will be μ-convergent, and the limit will not depend upon the minimizing sequence.Lemma 16.
Letμ be a convex regular modular satisfying condition UUC. If there exists R>0 and γ∈0,1 with(15)limsupn⟶∞μùn≤R,limsupn⟶∞μτn≤Randlimn⟶∞μγùn+1−γτn=R,
then we have(16)limn⟶∞μùn−τn=0.
Theμ-distance from ù∈Lμ to K⊂Lμ is given as(17)distμù,K=infμù−h:h∈K.Following Lemma taken from [9] will be used in the existence result.Lemma 17.
Supposeln, mnandδn be sequences of nonnegative satisfying(18)ln+1≤1+δnln+mn∀n≥1.
IfΣδn<∞ and Σmn<∞, then limn⟶∞ln+1 exists.Following is the definition of conditionI taken from [20].Definition 18.
LetK≠ϕ be a subset of Lμ, and a mapping Γ:K⟶K is assumed to fulfill the condition I if a nondecreasing function(19)l:0,∞⟶0,∞withl0=0andlr>0,exists for all r∈0,∞, such that(20)μù−Γù≥ldistμù,FμΓ,for all ù∈K.


## 3. Fixed Point Results for Monotone Total Asymptotically Nonexpansive Mapping

Now, we will define monotone total asymptotically nonexpansive mapping in modular space.Definition 19.
LetK≠ϕ be a subset of Lμ where μ is a convex regular modular. A self map Γ of K is said to be monotone total asymptotically nonexpansive mapping if there exists nonnegative sequences ζn and ξn with ζn⟶0, ξn⟶0, as n⟶∞, and a strictly increasing continuous function(21)ϕ:0,∞⟶0,∞withϕ0=0,such that(22)μΓnτ−Γnh≤μτ−h+ζnϕμτ−h+ξnforalln≥1.
There exists a constantM∗>0 such that ϕλ≤M∗λ for λ>0, then(23)μΓnτ−Γnh≤1+M∗ζnμτ−h+ξn,for every τ,h∈K such that τ and h are comparable μ-a.e.Theorem 20.
LetK≠ϕ be a μ-bounded and μ-closed subset of Lμ where μ is a convex regular modular satisfying condition UUC. Let a self map Γ of K be a μ-continuous monotone total asymptotically nonexpansive mapping. Assume that there exists ù0∈K, such that ù0′Γù0 or Γù0′ù0μ-a.e. Then, Γ has a fixed point ù such that ù0′ù or ù′ù0μ-a.e.Proof.
Assume thatù0′Γù0μ-a.e. Since Γ is monotone, then we have(24)Γnù0′Γn+1ù0,for every n∈ℕ, and the sequence Γnù0 is monotone increasing. From the above Remark,(25)K∞=∩n≥1ù∈K;ùn′ùμ−a.e≠ϕ.
Consider theμ-type function τ:K∞⟶0,+∞ define by(26)τh=limsupn⟶∞μΓnù0−h,foranyh∈K∞,(27)τ0=infτh;h∈K∞.
Letτn be a minimizing sequence of τ, from the Lemma τnμ-converges to τ∈K∞. We have to show that τ is the fixed point of Γ. Since h∈K∞, we have Γmh∈K∞, for every m∈ℕ, which implies(28)τΓmh=limsupn⟶∞μΓnù0−Γmh≤limsupn⟶∞μΓnù0−h+μmϕμΓnù0−h+ξm=τh+μmlimsupn⟶∞ϕμΓnù0−h+ξm.
In particular, we have(29)τΓmτn=limsupn⟶∞μΓnx0−Γmτn≤ττn+μmlimsupn⟶∞ϕμΓnx0−h+ξm,for n,m∈ℕ. As Γ is total asymptotically nonexpansive, so μm⟶0, ξm⟶0, when m⟶∞. Hence,(30)limm⟶∞τΓmτn=ττn.
The sequenceΓn+pτn is a minimizing sequence in K∞, for any p∈ℕ. By Lemma 15, Γn+pτn is μ-converge to τ, for any p∈ℕ. Since Γ is μ-continuous and Γnτn is μ-convergent to τ, then Γn+1τn is μ-convergent to Γτ and τ. Since μ-limit of any μ-convergent is unique, we have Γτ=τ; also, τ∈K∞, we have ù0′τ, hence proved.☐Example 21.
Letf be an extended real valued function defined on a measureable set D, such that fx=c for all x∈D. The function f is measureable if the set(31)x∈D:fx>α=Difα<cϕifα≥c∣is measureable. And the measureability of above set follows directly from the measureability of D and ϕ. So, a constant function is a measureable function. Now, we define a set of extended real valued functions as(32)M∞=f:f:D⟶ℝ¯withfx=c.
Define a functionμ:M∞⟶0,∞ by μf=fx for all f∈M∞, which clearly it is well defined.
Firstly, we need to show thatμ is a convex function. For this, we show M∞ that is a convex set. Consider(33)λf+1−λgx=λfx+1−λgx,Pointwiseaddition=λ.fx+1−λ.gx,Scalermultiplication=λc1+1−λc2,Asf,g∈M∞,λ∈0,1=c3,which implies λf+1−λg∈M∞. Hence, M∞ is a convex set. Now, for every f,g∈M∞, it is easy to prove that(34)μλf+1−λg=λ.μf+1−λ.μg,which further implies that μ is a convex function. Now, we check the properties of regular modular.(1)
Ifμf=0⇒fx=0⇒c=0, which further implies f=0(2)
If(35)ft≤gt,forallt∈Dc1≤c2,as μf=ft=c1 and μg=gt=c2. So, μf≤μg. Thus, μ is monotone.(3)
Clearly,μ is strongly convergent which implies weak convergence
Hence,μ is convex regular modular. Define(36)M=f∈M∞:∣fx∣<∞,andLμ=f∈M:μλfx=λfx=λ.fx=λ.c⟶0,asλ⟶0,and a subset(37)K=f∈M:fx=c∈0,2withμλf⟶0asλ⟶0of Lμ. Clearly, K is μ-bounded and μ-closed. Let a mapping Γ:K⟶K be defined by Γf=αf, where α∈0,1. Let ξnn∈ℕ=1/2n,ηnn∈ℕ=2/3n2 be any positive sequences and ξn,ηn⟶0 as n⟶∞. Define a strictly increasing function φ:0,∞⟶0,∞ by φx=x/2, with φ0=0. Consider(38)μΓnτ−Γnh=μαnτ−αnh=αnτ−αnhx=αnc1−c2,μτ−h+ηnφμτ−h+ξn=τ−hx+23n2μτ−h2+12n=1+13n2τx−hx+12n=1+13n2c1−c2+12n.
Clearly,(39)μΓnτ−Γnh≤μτ−h+ηnφμτ−h+ξn.
Also, there exists a constantM∗=1, φλ=λ/2<1.λ, and(40)μΓnτ−Γnh≤1+M∗ηnμτ−h+ξn.
So,Γ is monotone asymptotically nonexpansive mapping. Since all conditions of theorem are satisfied; thus, Γ has a fixed point, since Γf=αf implies 1−αf=0. Thus, f=0. Hence, the 0 function is a fixed point of Γ.


## 4. Convergence Analysis

LetK≠ϕ be a convex subset of Lμ where μ is a convex regular modular.We modify S-iteration in MF space is defined as(41)ù1∈Kyl=νlΓlùl+1−νlùl,ùl+1=γlΓlyl+1−γlΓlùl,for l∈ℕ, where γl and νl are sequences in 0,1.Theorem 22.
LetK≠ϕ be a μ-bounded subset of Lμ where μ is a convex regular modular satisfying condition UUC. Let a self map Γ of K be a monotone total asymptotically nonexpansive mapping with ùΓ≠ϕ. Assume that there exists ù0∈K, such that ù0′Γù0 or Γù0′ù0μ-a.e. If the sequence ùl is defined by (41) where 0<a′γl, νl′b<1, then Γ has a fixed point ù such that ù0′ù or ù′ù0μ-a.e. Then, the following holds(a)
liml⟶∞μùl−ù exist for ù∈ùΓ.(b)
liml⟶∞μΓlùl−ùl=0.Proof.
Letù∈ùΓ, and assume that ù0′Γù0μ-a.e. Using (41)(42)μyl−ù=μνlΓlùl+1−νlùl−ù≤νlμΓlùl−ù+1−νlμùl−ù,using (22), we have(43)μyl−ù≤νlμùl−ù+ζlϕμùl−ù+ξl+1−νlμùl−ù=νlμùl−ù+νlζlϕμùl−ù+νlξl+μùl−ù−νlμùl−ù=νlζlϕμùl−ù+νlξl+μùl−ù,upon using (23), and we get(44)μyl−ù≤1+νlζlM∗μùl−ù+νlξl.
Now,(45)μùl+1−ù=μγlΓlyl+1−γlΓlùl−ù≤γlμΓlyl−ù+1−γlμΓlùl−ù,using (22), and we have(46)μùl+1−ù≤γlμyl−ù+ζlϕμyl−ù+ξl+1−γlμùl−ù+ζlϕμùl−ù+ξl,upon using (23), and we get(47)μùl+1−ù≤1+δlμùl−ù+blξl.where(48)δl=γlνl+γlνlM∗ζl+1M∗ζl,bl=γlνl+γlνlM∗ζl+1.
Using Lemma 17,liml⟶∞μùl−ù exists for ù∈ùΓ. For part (b), we have to show that(49)liml⟶∞μΓlùl−ù=0.
Assume that(50)liml⟶∞μùl−ù=c≥0.
Case 1. Ifc=0, then the conclusion is trivial.
Case 2. Forc>0, we know that(51)μyl−ù=1+M∗νlζlμùl−ù+νlξl.
Takinglimsup on both sides of (50),(52)limsupl⟶∞μyl−ù≤c.
Also,(53)μΓlùl−ù=μΓlùl−Γlù≤μùl−ù+ζlϕμùl−ù+ξlapplies limsupl⟶∞ on both sides:(54)limsupl⟶∞μΓlùl−ù≤c.
Also,(55)μΓlyl−ù≤μyl−ù+ζlϕμyl−ù+ξl.
Takinglimsup on both sides,(56)limsupl⟶∞μΓlyl−ù≤c.
Now,(57)liml⟶∞μxl+1−ù=liml⟶∞μWΓlùl,Γlyl,γl−ù=c.
By using Lemma 16 and from (54) and (56), we have(58)liml⟶∞μΓlùl−Γlyl=0.
From (41) and (58),(59)μxl+1,Γlùl=μWΓlùl,Γlyl,γl−Γlùl≤1−γlμΓlùl−Γlùl+γlμΓlùl−Γlyl,taking liml⟶∞, and we have(60)liml⟶∞μxl+1−Γlùl=0.
Similarly,(61)liml⟶∞μxl+1−Γlyl=0.
Next,(62)μxl+1−ù≤μxl+1−Γlyl+μΓlyl−ù≤μxl+1−Γlyl+μyl−ù+ζlϕμyl−ù+ξl,taking liminfl⟶∞, and we get(63)c≤liminfl⟶∞μyl−ù.
From (52) and (63), we get(64)c=liml⟶∞μyl−ù=liml⟶∞μWxl,Γlxl,νl−ù.
By using Lemma 16, we have(65)liml⟶∞μΓlùl−ùl=0,
hence proved.☐



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