#### 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.

#### 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].

Let be a nonempty set such that(i) is a nontrivial -algebra of subsets of ;(ii) a -ring such that for any and ;(iii), where is an increasing sequence.

Denote by , the vector space of simple functions whose support is in . Next we consider the space of all real valued functions such that there exists a sequence of simple functions which satisfy , and , for all .

*Definition 1 (see [11, 14]). *A regular modular function is an even function which satisfies the following conditions:(i) implies ;(ii) is monotone; i.e., for all implies ;(iii) for all implies .

We will assume throughout that function modulars are convex and regular. A subset is said to be -null if , for any , where is the characteristic function of the subset . This will allow us to say that a property holds -almost everywhere, and write -a.e., if the set where it does not hold is -null. Consider the set . The MFS is given by

In the next theorem, we will review the most fundamental properties of the MFS needed in our work.

Theorem 2 (see [11, 14]). *Let be a function modular. *(1)*If , for some , then holds for some subsequence .*(2)*We have , for any sequence such that *

The following definition will represent the modular versions of the classical metric concepts.

*Definition 3 (see [11, 14]). *Let be a function modular.(1) is said to -converge to if . will stand for the -limit of .(2)A sequence is called -Cauchy if .(3) is -closed if and only if the -limit of any -convergent sequence belongs to .(4)For a nonempty subset , we define its -diameter as is -bounded if and only if .

Regardless the fact that the modular may not satisfy the triangle inequality, the -limit is unique. But -convergent sequences may not be -Cauchy. Indeed, a simple example may be found in the variable exponent space , where the function is defined byThe function modular is defined byIf we takethen , andfor any . It is easy to see thatIn other words, is -convergent to 0 and it is not -Cauchy.

Note that -balls are -closed. It is interesting to notice that -Cauchy sequences in are -convergent; i.e., is -complete [11, 14].

The next result follows easily from Theorem 2.

Theorem 4. *Let be a function modular. Let be a sequence which -converges to in . If is monotone increasing (resp., decreasing), i.e., -a.e. (resp. -a.e.), for any , then -a.e. (resp., -a.e.), for any .*

Next we present the definition of the modular uniform convexity which is an essential tool in metric fixed point theory.

*Definition 5 (see [14]). *Let be a function modular. Then we will say that(i) is uniformly convex (in short (UC)) if for every and , we havewhere is the set of all such that , and ;(ii) is (UUC) if there exists for every , such that , for .

*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 [17, 18].

Modular functions which are (UUC) have a similar property to the weak-compactness in Banach spaces.

Theorem 7 (see [14, 15]). *Let be a (UUC) function modular. Then for any sequence of nonempty -bounded, -closed, and convex subsets of such that , for any . This intersection property is known as the property .*

This property will be of huge help throughout our work. In particular, we have the following result.

Theorem 8 (see [19]). *Assume that is (UUC). Let be -bounded convex -closed nonempty subset. Let be a monotone increasing sequence (resp., decreasing). Then (resp., ).*

This conclusion holds because order intervals in are convex and -closed combined with the property .

Next we give the definition of the -type functions which will help us prove some interesting fixed point results.

*Definition 9 (see [19]). *Let be a function modular and be nonempty. A function is said to be a -type if for any , we havefor some sequence in . Any sequence such thatis called a minimizing sequence of .

The following result played a major role in the study of fixed point problems in MFS.

Lemma 10 (see [20]). *Let be a function modular. Assume that is (UUC). Let be a -bounded -closed convex nonempty subset. Let be a -type. Then any minimizing sequence of is -convergent and its -limit is independent of the minimizing sequence.*

Next we give the modular definitions of monotone Lipschitzian mappings which mimic their metric equivalents. First, recall that and are said to be comparable if -a.e. or -a.e., for any .

*Definition 11 (see [21]). *Let be nonempty. A mapping is said to be(1)monotone if and only if we havefor any ;(2)monotone asymptotically pointwise Lipschitzian if and only if is monotone and there exists a sequence of mappings such thatfor any , whenever and are comparable elements in . If , for any , then is monotone asymptotically pointwise nonexpansive mapping. A point is a fixed point of if and only if .

We can always assume that is a decreasing sequence for any .

#### 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 -nonexpansive mappings will be the existence of minimum points of -type functions. Since may fail to satisfy the triangle inequality, -type functions may fail to have any good continuity properties that may guarantee the existence of a minimum point. Using the conclusion of from the book [14], we introduce the following definition.

*Definition 12. *Let be a regular modular. We will say that is type-lsc if every -type function defined on a -bounded, -closed, and convex nonempty subset of is -lower semicontinuous, i.e.,for any which -converges to .

According to from the book [14], any uniformly continuous modular 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 that is (UUC) and type-lsc. Let be -bounded -closed convex nonempty subset. Let be -continuous monotone asymptotically pointwise -nonexpansive. Assume there exists such that and are comparable. Then has a fixed point comparable to .*

*Proof. *Without any loss of generality, we assume that -a.e. From the monotonicity of , we deduce that the sequence is monotone increasing. Let

*Remark 14. *Implies that . Let be the -type generated by , i.e.,Note that , for any , since is -bounded. Let , and let be a minimizing sequence of . Using Lemma 10, we conclude that -converges to some . Since is type-lsc, we deduce that is -lower semicontinuous. Hence we haveTherefore, we must have . Next, we show that is a fixed point of . Fix . Since is monotone, we have andfor any . In other words, the inequality is satisfied for any and . Therefore, we havefor any . Since is asymptotically pointwise -nonexpansive, we have , which implies that is a minimizing sequence of . Using Lemma 10, we conclude that -converges to . Since is -continuous will -converge to . The uniqueness of the -limit implies ; i.e., is a fixed point of . Since , we get .

In the proof of Theorem 13, 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 -type function which is -convergent.

Theorem 15. *Assume that is (UUC). Let be -bounded -closed convex nonempty subset. Let be -continuous monotone asymptotically pointwise -nonexpansive. Assume there exists such that and are comparable. Then has a fixed point comparable to .*

*Proof. *Without any loss of generality, we assume that -a.e. As we did in the proof of Theorem 13, let and define the -type function generated by , i.e.,Set . Let be a minimizing sequence of . As we did before, we know that -converges to some . Since we do not know that is -lower semicontinuous, we may not be able to show that is a minimum point of . Recall that we have , for any and , which impliesfor any . Next, we build by induction an increasing sequence of integers , such thatfor any and . Set . Since , there exists such thatfor all . Again since , there exists such thatfor all . By induction, we build the sequence in such that andfor all and . For any and , take . HenceNote that we have . Therefore, if we let , we getTherefore, is a -minimizing sequence of . Using Lemma 10, we conclude that is -convergent to for any . Take ; we get the sequence is -convergent to . Using the -continuity of , we get is -convergent to . Using the uniqueness of the -limit and , we conclude that ; i.e., is a fixed point of . Since , we have 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 be the positive part of the unite ball of , i.e.Define the mapping byIf we assume , for any , and , then we can show that is a monotone asymptotically nonexpansive mapping which is not nonexpansive.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that they have no conflicts of interest.