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

Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China Department of Mathematics, Ghazi University, DG Khan 32200, Pakistan Department of Computer Engineering, Biruni University, Istanbul, Turkey Department of Mathematics, Science Faculty, Firat University, 23119 Elazig, Turkey Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan


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.

Preliminaries
Firstly, we have the definitions of δ-ring and σ-algebra with examples.
Definition 1. Suppose Σ ≠ ∅, and R be a nonempty family of subsets of Σ, then R is called ring of sets if A, B ∈ R, satisfies (ii) A \ B ∈ R A ring of sets R is called δ -ring of sets if for any sequence of sets fA n g ∈ R implies ∪ ∞ n=1 A n ∈ R: Example 2. Let R be the collection of all finite subsets of ℕ, and then R is a ring, but not δ-ring.
An algebra of sets A is called σ -algebra of sets if for every sequence of sets fA n g ∈ A implies ∪ ∞ n=1 A n ∈ A: Example 4. For any set Σ, PðΣÞ and fϕ, Σg are σ-algebras.
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. Let M ∞ stands for the class of all extended functions which are also measurable. A convex and even function μ : M ∞ ⟶ ½0,∞ is said to be regular modular if The MF space L μ is defined as Following few useful definitions are taken from [17,18]. From onwards, we assume μ as a convex regular modular.
Following definition of μ-type function will be used in the main result taken from [18].
for any u ∈ K. Any sequence fu n g in K is said to be a mini- Following are the definitions of monotone and monotone asymptotically nonexpansive mapping in modular space, and useful remark about property ðRÞ, given in [13].
(ii) Monotone asymptotically nonexpansive if Γ is monotone, and there exists fL n g ⊂ ½1,+∞Þ such that lim n⟶∞ L n = 1, and such that τ ′ hμ-a.e. and n ≥ 1. Also τ is said to be fixed Remark 14. Let K ≠ ϕ be a μ-bounded, convex, and μ-closed subset of L μ where μ is a convex regular modular. Let fu n g be a monotonically increasing sequence in K (due to the convexity and μ-closedness of order intervals in L μ ), then prop-erty ðRÞ will imply that The following Lemmas taken from [19] will be used in main result. Lemma 15. Let K ≠ ϕ be a μ-bounded, convex, and μ-closed subset of L μ where μ is a convex regular modular satisfying condition ðUUCÞ. 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. Let μ be a convex regular modular satisfying condition ðUUCÞ. If there exists R > 0 and γ ∈ ð0, 1Þ with then we have The μ-distance from u ∈ L μ to K ⊂ L μ is given as Following Lemma taken from [9] will be used in the existence result. Lemma 17. Suppose fl n g, fm n g and fδ n g be sequences of nonnegative satisfying If Σδ n < ∞ and Σm n < ∞, then lim n⟶∞ l n+1 exists.
Following is the definition of condition ðIÞ taken from [20]. Definition 18. Let K ≠ ϕ be a subset of L μ , and a mapping Γ : K ⟶ K is assumed to fulfill the condition ðIÞ if a nondecreasing function exists for all r ∈ ð0,∞Þ, such that for all u ∈ K.

Fixed Point Results for Monotone Total Asymptotically Nonexpansive Mapping
Now, we will define monotone total asymptotically nonexpansive mapping in modular space.
Definition 19. Let K ≠ ϕ be a subset of L μ 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 fζ n g and fξ n g with ζ n ⟶ 0, ξ n ⟶ 0, as n ⟶ ∞, and a strictly increasing continuous function such that There exists a constant M * > 0 such that ϕðλÞ ≤ M * λ for λ > 0, then for every τ, h ∈ K such that τ and h are comparable μ-a.e. Theorem 20. Let K ≠ ϕ be a μ-bounded and μ-closed subset of L μ where μ is a convex regular modular satisfying condition ðUUCÞ. Let a self map Γ of K be a μ-continuous monotone total asymptotically nonexpansive mapping. Assume that there exists u 0 ∈ K, such that u 0 ′ Γðu 0 Þ or ðΓðu 0 Þ ′ u 0 Þμ -a.e. Then, Γ has a fixed point u such that u 0 ′ u or ðu ′ u 0 Þμ-a.e.
Proof. Assume that u 0 ′Γðu 0 Þμ-a.e. Since Γ is monotone, then we have for every n ∈ ℕ, and the sequence fΓ n u 0 g is monotone increasing. From the above Remark, Consider the μ-type function τ : K ∞ ⟶ ½0,+∞Þ define by Let fτ n g be a minimizing sequence of τ, from the Lemma fτ n gμ-converges to τ ∈ K ∞ . We have to show that τ is the fixed point of Γ. Since h ∈ K ∞ , we have Γ m ðhÞ ∈ K ∞ , for every m ∈ ℕ, which implies In particular, we have for n, m ∈ ℕ. As Γ is total asymptotically nonexpansive, so μ m ⟶ 0, ξ m ⟶ 0, when m ⟶ ∞: Hence, The sequence fΓ n+p ðτ n Þg is a minimizing sequence in K ∞ , for any p ∈ ℕ: By Lemma 15, fΓ n+p ðτ n Þg is μ-converge to τ, for any p ∈ ℕ: Since Γ is μ-continuous and fΓ n ðτ n Þg is μ-convergent to τ, then fΓ n+1 ðτ n Þg 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 u 0 ′τ, hence proved. ☐ Example 21. Let f be an extended real valued function defined on a measureable set D, such that f ðxÞ = c for all x ∈ D. The function f is measureable if the set 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 Define a function μ : M ∞ ⟶ ½0,∞Þ by μðf Þ = f ðxÞ for all f ∈ M ∞ , which clearly it is well defined.
Firstly, we need to show that μ is a convex function. For this, we show M ∞ that is a convex set. Consider Journal of Function Spaces set. Now, for every f , g ∈ M ∞ , it is easy to prove that which further implies that μ is a convex function. Now, we check the properties of regular modular.

Convergence Analysis
Let K ≠ ϕ be a convex subset of L μ where μ is a convex regular modular:We modify S-iteration in MF space is defined as for l ∈ ℕ, where fγ l g and fν l g are sequences in ð0, 1Þ: Theorem 22. Let K ≠ ϕ be a μ-bounded subset of L μ where μ is a convex regular modular satisfying condition ðUUCÞ. Let a self map Γ of K be a monotone total asymptotically nonexpansive mapping with uðΓÞ ≠ ϕ. Assume that there exists u 0 ∈ K, such that u 0 ′ Γðu 0 Þ or ðΓðu 0 Þ ′ u 0 Þμ -a.e. If the sequence fu l g is defined by (41) where 0 < a ′ γ l , ν l ′ b < 1, then Γ has a fixed point u such that u 0 ′ u or ðu ′ u 0 Þμ -a.e. Then, the following holds Proof. Let u ∈ uðΓÞ, and assume that u 0 ′ Γðu 0 Þμ-a.e. Using (41) using (22), we have upon using (23), and we get Now, upon using (23), and we get where Using Lemma 17, lim l⟶∞ μðu l − uÞ exists for u ∈ uðΓÞ: For part (b), we have to show that Assume that Case 1. If c = 0, then the conclusion is trivial. Case 2. For c > 0, we know that Taking lim sup on both sides of (50), Also, From (41) and (58),

Data Availability
There is no any data available.

Conflicts of Interest
The authors declare that they have no conflicts of interest. 6 Journal of Function Spaces