On Monotone Asymptotic Pointwise Nonexpansive Mappings in Modular Function Spaces

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].


Introduction
In the light of the three main fixed point theorems [1][2][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].

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) Σ is a nontrivial -algebra of subsets of ; (ii) P ⊂ Σ a -ring such that  ∩  ∈ P for any  ∈ P and  ∈ Σ; (iii)  = ⋃   , where {  } ⊂ P is an increasing sequence.
Denote by E  , the vector space of simple functions whose support is in P. Next we consider M ∞ the space of all real valued functions  :  → [−∞, ∞] such that there exists a sequence of simple functions {  } which satisfy sup ∈N |  | ≤ ||, and lim →∞   () = (), for all  ∈ .
Definition (see [11,14]).A regular modular function  : M ∞ → [0, ∞] is an even function which satisfies the following conditions: We will assume throughout that function modulars are convex and regular.A subset  ∈ Σ is said to be -null if ( l  ) = 0, for any  ∈ E  , where 1  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 2 Journal of Function Spaces set where it does not hold is -null.Consider the set M = { ∈ M ∞ ; |()| < ∞  − .}.The MFS   is given by In the next theorem, we will review the most fundamental properties of the MFS needed in our work.
The following definition will represent the modular versions of the classical metric concepts.
ℎ will stand for the -limit of {ℎ  }. ( (3)  ⊂   is -closed if and only if the -limit of any -convergent sequence {ℎ  } ⊂  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  (⋅) ([0, +∞)), where the function  is defined by The function modular  is defined by If we take then for any  ∈ N. It is easy to see that lim In other words, {  } is -convergent to 0 and it is not -Cauchy.
The Next we present the definition of the modular uniform convexity which is an essential tool in metric fixed point theory.
Remark .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 Modular functions which are (UUC) have a similar property to the weak-compactness in Banach spaces.
en ⋂ ∈N   ̸ = 0 for any sequence {  } of nonempty bounded, -closed, and convex subsets of   such that  +1 ⊂   , for any  ∈ N. is intersection property is known as the property ().This property will be of huge help throughout our work.In particular, we have the following result.
Next we give the definition of the -type functions which will help us prove some interesting fixed point results.
Definition (see [19]).Let  be a function modular and  ⊂   be nonempty.A function  :  → [0, ∞] is said to be a -type if for any  ∈ , we have for some sequence is 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  :  → [0, ∞] be a -type.en 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 (see [21]).Let  ⊂   be nonempty.A mapping  :  →  is said to be (1) monotone if and only if we have  ≤   − ..  () ≤  ()  − .., (11) for any ,  ∈ ; (2) monotone asymptotically pointwise Lipschitzian if and only if  is monotone and there exists a sequence of mappings   :  → [0, ∞) such that for any  ∈ N, whenever  and  are comparable elements in .If lim sup →∞   () = 1, 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  ∈ .

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 Lemma 5.1 from the book [14], we introduce the following definition.
Definition .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 Lemma 5.1 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  0 ∈  such that  0 and ( 0 ) are comparable.en  has a fixed point comparable to  0 .
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.
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  1 of ℓ 2 , i.e.