# On Monotone Asymptotic Pointwise Nonexpansive Mappings in Modular Function Spaces

Authors: B. A. Bin Dehaish
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this work, we investigate the existence of the fixed points of a monotone asymptotic pointwise nonexpansive mapping defined in a modular function space. Our result extends the fixed point result of Khamsi and Kozlowski.


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## Body

## 1. Introduction

In the light of the three main fixed point theorems [1–3], Goebel and Kirk [4] came up with the concept of asymptotic nonexpansive mappings. Nonexpansive mappings are a particular case of asymptotic nonexpansive mappings. But the study of the existence of their fixed points appears to be extremely difficult. Kirk [5, 6] initiated the concept of pointwise Lipschitz mappings, which naturally extends the class of Lipschitz mappings. The monotone mapping fixed point theory is quite recent and attracted a lot of attention. It began with the study of Ran and Reurings [7], which extended the classical principle of Banach Contraction in partially ordered metric spaces. We suggest a recent survey for interested readers [8]. Carl and Heikkila’s book [9] offers a wonderful source of monotonous mappings applications. The theory of fixed points in modular function spaces (MFS) is rooted in Khamsi, Kozlowski, and Reich’s original work [10]. The Kozlowski book [11] and the recent Khamsi and Kozlowski book [12] are very important references to this subarea.In this work, we investigate the existence of fixed points of a monotone asymptotic pointwise mappings defined in MFS. In particular, we generalize the classical fixed point result of Kirk and Xu [13].


## 2. Preliminaries

Extensively, details of MFS appeared in the literature; therefore, for additional information, we refer the readers to the books [11, 14].LetA be a nonempty set such that(i)
Σ is a nontrivial σ-algebra of subsets of A;(ii)
P⊂Σ a δ-ring such that P∩S∈P for any P∈P and S∈Σ;(iii)
A=⋃An, where {An}⊂P is an increasing sequence.Denote byEs, the vector space of simple functions whose support is in P. Next we consider M∞ the space of all real valued functions f:A→[-∞,∞] such that there exists a sequence of simple functions {fn} which satisfy supn∈N|fn|≤|f|, and limn→∞fn(t)=f(t), for all t∈A.Definition 1 (see [11, 14]).
A regular modular functionϱ:M∞→[0,∞] is an even function which satisfies the following conditions:(i)
ϱ(f)=0 implies f=0;(ii)
ϱ is monotone; i.e., |h(t)|≤|H(t)| for all t∈A implies ϱ(h)≤ϱ(H);(iii)
|Hn(t)|↑|H(t)| for all t∈A implies ϱ(Hn)↑ϱ(H).We will assume throughout that function modulars are convex and regular. A subsetB∈Σ is said to be ϱ-null if ϱ(glB)=0, for any g∈Es, where 1B is the characteristic function of the subset B. This will allow us to say that a property holds ϱ-almost everywhere, and write ϱ-a.e., if the set where it does not hold is ϱ-null. Consider the set M={f∈M∞;|f(t)|<∞ϱ-a.e}. The MFS Lϱ is given by(1)Lϱ=f∈M;limλ→0⁡ϱλf=0.In the next theorem, we will review the most fundamental properties of the MFS needed in our work.Theorem 2 (see [11, 14]).
Letϱ be a function modular. (1)
Ifϱ(αhn)→0, for some α>0, then hψ(n)→0ϱ-a.e. holds for some subsequence {hψ(n)}.(2)
We haveϱ(g)≤liminfn→∞ϱ(gn), for any sequence {gn} such that gn→gϱ-a.e.The following definition will represent the modular versions of the classical metric concepts.Definition 3 (see [11, 14]).
Letϱ be a function modular.(1)
{hn} is said to ϱ-converge to h if limn→∞ϱ(hn-h)=0. h will stand for the ϱ-limit of {hn}.(2)
A sequence{hn} is called ϱ-Cauchy if limn,m→∞ϱ(hn-hm)=0.(3)
C⊂Lϱ is ϱ-closed if and only if the ϱ-limit of any ϱ-convergent sequence {hn}⊂C belongs to C.(4)
For a nonempty subsetC, we define its ϱ-diameter as(2)δϱC=supϱf-h;f,h∈C.C is ϱ-bounded if and only if δϱ(C)<+∞.Regardless the fact that the modular may not satisfy the triangle inequality, theϱ-limit is unique. But ϱ-convergent sequences may not be ϱ-Cauchy. Indeed, a simple example may be found in the variable exponent space Lp(·)([0,+∞)), where the function p is defined by(3)px=n,x∈n,n+1,n∈N.The function modular ϱ is defined by(4)ϱf=∑n=0∞∫n,n+1fxndx.If we take(5)fnt=-1n12ln,+∞,n∈N,then ϱ(fn)=∑i=n∞1/2i, and(6)ϱfn-fn+1=ϱ-1n12ln,+∞--1n+112ln+1,+∞=ϱ12ln,+∞+12ln+1,+∞=ϱ12ln,n+1+ln+1,+∞,for any n∈N. It is easy to see that(7)limn→∞⁡ϱfn=0,ϱfn-fn+1=+∞,n∈N.In other words, {fn} is ϱ-convergent to 0 and it is not ϱ-Cauchy.Note thatϱ-balls Bϱ(f,r)={h∈Lϱ;ϱ(f-h)≤r} are ϱ-closed. It is interesting to notice that ϱ-Cauchy sequences in Lϱ are ϱ-convergent; i.e., Lϱ is ϱ-complete [11, 14].The next result follows easily from Theorem2.Theorem 4.
Letϱ be a function modular. Let {hn} be a sequence which ϱ-converges to h in Lϱ. If {hn} is monotone increasing (resp., decreasing), i.e., hn≤hn+1ϱ-a.e. (resp. hn+1≤hnϱ-a.e.), for any n≥1, then hn≤hϱ-a.e. (resp., h≤hnϱ-a.e.), for any n≥1.Next we present the definition of the modular uniform convexity which is an essential tool in metric fixed point theory.Definition 5 (see [14]).
Letϱ be a function modular. Then we will say that(i)
ϱ is uniformly convex (in short (UC)) if for every R>0 and ε>0, we have(8)δϱR,ε=inf⁡1-1Rϱf+g2;f,g∈D>0,where D is the set of all f,g∈Lϱ such that ϱ(f)≤R, ϱ(g)≤R and ϱ(f-g)≥εR;(ii)
ϱ is (UUC) if there exists η(s,ε)>0 for every s≥0, ε>0 such that δϱ(R,ε)>η(s,ε)>0, for R>s.Remark 6.
The modular uniform convexity in Orlicz function spaces was initiated in the work of Khamsi et al. [15]. In particular, we know that the (UC) property of the modular in Orlicz spaces is satisfied if and only if the Orlicz function is (UC) [15, 16]. An example of an Orlicz function which is (UC) is φ(t)=et2-1 [17, 18].Modular functions which are (UUC) have a similar property to the weak-compactness in Banach spaces.Theorem 7 (see [14, 15]).
Letϱ be a (UUC) function modular. Then ⋂n∈NCn≠∅ for any sequence {Cn} of nonempty ϱ-bounded, ϱ-closed, and convex subsets of Lϱ such that Cn+1⊂Cn, for any n∈N. This intersection property is known as the property (R).This property will be of huge help throughout our work. In particular, we have the following result.Theorem 8 (see [19]).
Assume thatϱ is (UUC). Let C⊂Lϱ be ϱ-bounded convex ϱ-closed nonempty subset. Let {gn}⊂C be a monotone increasing sequence (resp., decreasing). Then ⋂n∈N{g∈C;gn≤gϱ-a.e.}≠∅ (resp., ⋂n∈N{g∈C;g≤gnϱ-a.e.}≠∅).This conclusion holds because order intervals inLϱ are convex and ϱ-closed combined with the property (R).Next we give the definition of theϱ-type functions which will help us prove some interesting fixed point results.Definition 9 (see [19]).
Letϱ be a function modular and C⊂Lϱ be nonempty. A function φ:C→[0,∞] is said to be a ϱ-type if for any f∈C, we have(9)φf=limsupm→∞⁡ϱgm-f,for some sequence {gm} in Lϱ. Any sequence {hn}⊂C such that(10)limn→∞φhn=infφf:f∈Cis called a minimizing sequence of φ.The following result played a major role in the study of fixed point problems in MFS.Lemma 10 (see [20]).
Letϱ be a function modular. Assume that ϱ is (UUC). Let C⊂Lϱ be a ϱ-bounded ϱ-closed convex nonempty subset. Let φ:C→[0,∞] be a ϱ-type. Then any minimizing sequence of φ is ϱ-convergent and its ϱ-limit is independent of the minimizing sequence.Next we give the modular definitions of monotone Lipschitzian mappings which mimic their metric equivalents. First, recall thatf and g are said to be comparable if f≤gϱ-a.e. or g≤fϱ-a.e., for any f,g∈Lϱ.Definition 11 (see [21]).
LetC⊂Lϱ be nonempty. A mapping T:C→C is said to be(1)
monotone if and only if we have(11)
f
≤
g
ϱ
-
a
.
e
.
i
m
p
l
i
e
s
T
f
≤
T
g
ϱ
-
a
.
e
.
,for any f,g∈C;(2)
monotone asymptotically pointwise Lipschitzian if and only ifT is monotone and there exists a sequence of mappings kn:C→[0,∞) such that(12)ϱTnf-Tng≤knfϱf-g,for any n∈N, whenever f and g are comparable elements in C. If limsupn→∞kn(f)=1, for any f∈C, then T is monotone asymptotically pointwise nonexpansive mapping.
A pointf∈C is a fixed point of T if and only if T(f)=f.We can always assume that{kn(f)} is a decreasing sequence for any f∈K.


## 3. Main Results

In this section, we will extend the result of Khamsi and Kozlowski [20] to the monotone case. The first result is the pointwise formulation of the main result of [21]. A powerful tool used to prove the existence of fixed points of asymptotic pointwise ϱ-nonexpansive mappings will be the existence of minimum points of ϱ-type functions. Since ϱ may fail to satisfy the triangle inequality, ϱ-type functions may fail to have any good continuity properties that may guarantee the existence of a minimum point. Using the conclusion of Lemma5.1 from the book [14], we introduce the following definition.Definition 12.
Letϱ be a regular modular. We will say that ϱ is type-lsc if every ϱ-type function τ defined on a ϱ-bounded, ϱ-closed, and convex nonempty subset of Lϱ is ϱ-lower semicontinuous, i.e.,(13)τf≤liminfn→∞⁡τfn,for any {fn} which ϱ-converges to f.According toLemma5.1 from the book [14], any uniformly continuous modular ϱ is type-lsc. In [19], the authors investigated the existence of a fixed point for any monotone asymptotically nonexpansive mapping in MFS. Next we prove the pointwise version of their result.Theorem 13.
Assume thatϱ is (UUC) and type-lsc. Let C⊂Lϱ be ϱ-bounded ϱ-closed convex nonempty subset. Let T:C→C be ϱ-continuous monotone asymptotically pointwise ϱ-nonexpansive. Assume there exists f0∈C such that f0 and T(f0) are comparable. Then T has a fixed point comparable to f0.Proof.
Without any loss of generality, we assume thatf0≤T(f0)ϱ-a.e. From the monotonicity of T, we deduce that the sequence {Tn(f0)} is monotone increasing. Let(14)C∞=f∈C;Tnf0≤fϱ-a.eforanyn∈N.Remark 14.
Implies thatC∞≠∅. Let φ:C∞→[0,+∞) be the ϱ-type generated by {Tn(f0)}, i.e.,(15)φh=limsupn→∞⁡ϱTnf0-h.Note that φ(h)<∞, for any h∈C∞, since C is ϱ-bounded. Let φ0=inf{φ(h);h∈C∞}, and let {fn}⊂C∞ be a minimizing sequence of φ. Using Lemma 10, we conclude that {fn}ϱ-converges to some f∈C∞. Since ϱ is type-lsc, we deduce that φ is ϱ-lower semicontinuous. Hence we have(16)φf≤liminfn→∞⁡φfn=φ0.Therefore, we must have φ(f)=φ0. Next, we show that f is a fixed point of T. Fix h∈C∞. Since T is monotone, we have Tm(h)∈C∞ and(17)φTmh=limsupn→∞⁡ϱTnf0-Tmh≤limsupn→∞⁡kmhϱTn-mf0-h=kmhφh,for any m≥1. In other words, the inequality φ(Tm(h))≤km(h)φ(h) is satisfied for any h∈C∞ and m≥1. Therefore, we have(18)φ0≤φTmf≤kmfφf=kmfφ0,for any m≥1. Since T is asymptotically pointwise ϱ-nonexpansive, we have limm→∞km(f)=1, which implies that {Tm(f)} is a minimizing sequence of φ. Using Lemma 10, we conclude that Tm(f)ϱ-converges to f. Since T is ϱ-continuous Tm+1(f) will ϱ-converge to T(f). The uniqueness of the ϱ-limit implies T(f)=f; i.e., f is a fixed point of T. Since f∈C∞, we get f0≤fϱ-a.e.In the proof of Theorem13, the assumption type-lsc is crucial to secure the existence of the minimum point of a type which happens to be the desired fixed point of the map. Therefore, if we relax the type-lsc, one expects the proof to get more complicated. In this case, we will follow the ideas developed by Khamsi and Kozlowski [20] which allowed them to prove the existence of a fixed point for asymptotic pointwise nonexpansive mapping defined in modular function spaces by using the existence of a minimizing sequence for a ρ-type function which is ρ-convergent.Theorem 15.
Assume thatϱ is (UUC). Let K⊂Lϱ be ϱ-bounded ϱ-closed convex nonempty subset. Let T:K→K be ϱ-continuous monotone asymptotically pointwise ϱ-nonexpansive. Assume there exists f0∈K such that f0 and T(f0) are comparable. Then T has a fixed point comparable to f0.Proof.
Without any loss of generality, we assume thatf0≤T(f0)ϱ-a.e. As we did in the proof of Theorem 13, let K∞={f∈K;Tn(f0)≤fϱ-a.e.andn∈N} and define the ϱ-type function φ:K∞→[0,+∞) generated by {Tn(f0)}, i.e.,(19)φh=limsupn→∞⁡ϱTnf0-h.Set φ0=inf{φ(h);h∈K∞}. Let {fn}⊂K∞ be a minimizing sequence of φ. As we did before, we know that {fn}ϱ-converges to some f∈K∞. Since we do not know that φ is ϱ-lower semicontinuous, we may not be able to show that f is a minimum point of φ. Recall that we have φ(Tm(h))≤km(h)φ(h), for any h∈K∞ and m≥1, which implies(20)φTmfn≤kmfnφfn,for any n,m≥1. Next, we build by induction an increasing sequence of integers {Mn}, such that(21)kmfMn≤1+1Mn,for any n≥1 and m≥Mn. Set M1=1. Since limm→∞km(fM1)=1, there exists M2>M1 such that(22)kmfM1≤1+1M1,for all m≥M2. Again since limm→∞km(fM2)=1, there exists M3>M2 such that(23)kmfM2≤1+1M2,for all m≥M3. By induction, we build the sequence {Mn} in N such that Mn<Mn+1 and(24)kmfMn≤1+1Mn,for all m≥Mn and n≥1. For any n≥1 and p∈N, take m=Mn+1+p. Hence(25)φTmfMn≤kmfMnφfMn≤1+1MnφfMn.Note that we have limn→∞Mn=∞. Therefore, if we let n→∞, we get(26)φ0≤limsupn→∞⁡φTMn+1+pfMn≤φ0.Therefore, {TMn+1+p(fMn)} is a ϱ-minimizing sequence of φ. Using Lemma 10, we conclude that {TMn+1+p(fMn)} is ϱ-convergent to f for any p∈N. Take p=0; we get the sequence {TMn+1(fMn)} is ϱ-convergent to f. Using the ϱ-continuity of T, we get {T(TMn+1(fMn))} is ϱ-convergent to T(f). Using the uniqueness of the ϱ-limit and T(TMn+1(fMn))=TMn+1+1(fMn), we conclude that T(f)=f; i.e., f is a fixed point of T. Since f∈K∞, we have f0≤fϱ-a.e as claimed.Remark. Examples of asymptotically nonexpansive mappings are not easily found. As it was pointed out by Kirk and Xu [13], the original example given by Goebel and Kirk may be modified to generate an example of a monotone asymptotically nonexpansive mapping. Indeed, let C be the positive part of the unite ball B1 of l2, i.e.(27)C=fn∈B1;fn≥0foranyn≥1.Define the mapping T:C→C by(28)Tfn=0,f12,C2f2,C3f3,⋯.If we assume Cn∈(0,1), for any n≥2, and ∏n=2∞Cn=1/2, then we can show that T is a monotone asymptotically nonexpansive mapping which is not nonexpansive.



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