On Monotone Asymptotic Pointwise Nonexpansive Mappings in Modular Function Spaces

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.