On Convergence Theorems for Generalized Alpha Nonexpansive Mappings in Banach Spaces

The present paper seeks to illustrate approximation theorems to the ﬁ xed point for generalized α -nonexpansive mapping with the Mann iteration process. Furthermore, the same results are established with the Ishikawa iteration process in the uniformly convex Banach space setting. The presented results expand and re ﬁ ne many of the recently reported results in the literature.


Introduction
Consider a Banach space (BS) X, together with its subset Dð ≠ ϕÞ. Let us also consider the following notations FixðTÞ, ⇀ , and ⟶ to represent the set of fixed points of T, weak convergence, and strong convergence, correspondingly.
A self-mapping T defined on a subset D is referred to as (1) nonexpansive provided that kTðuÞ − TðvÞk ≤ ku − v k, for all u, v ∈ D (2) quasi-nonexpansive provided that FixðTÞ ≠ ϕ, and for all u ∈ DðTÞ and v ∈ FixðTÞ, the following assertion holds: kTðuÞ − vk ≤ ku − vk Notably, there is a relationship between a nonexpansive mapping and a quasi-nonexpansive mapping. That is, each nonexpansive mapping satisfying FixðTÞ ≠ ϕ is quasinonexpansive; however, the opposite is not correct generally. Furthermore, the opposite is satisfied as shown in [1] when the linearity condition is added to the quasi-nonexpansive mapping. Thus, a linear quasi-nonexpansive mapping is nonexpansive. Yet, it can be straightforwardly verified that there exist nonlinear quasi-nonexpansive mappings which are continuous and are not nonexpansive; for example, Nonexpansive mapping and its generalization remain a central topic of interest in the fixed point (FP) theory among different mathematicians and mathematical theorists. Various considerations and a variety of in-depth investigations including generalizations to this mapping have been reported in the literature, in which we notice its development in different branches and under various conditions (see [2][3][4][5][6][7]). Browder in 1965 [8] and Kirk [9] have shown that selfnonexpansive mappings defined on a convex subset of a uniformly convex Banach space (UCBS) that is closed and bounded have fixed points. In 1974, Senter and Dotson [10] established a strong convergence fixed point theorem with regard to the Mann iteration of a nonexpansive mapping. Furthermore, in 1993, Xu and Tan [11] generalized the results of Reich [12] and Senter and Dotson [10] by using the Ishikawa iterative procedure instead of the Mann process.
Recently, the notion of α-nonexpansive mapping in BS was proposed by Aoyama and Kohsaka [13] in 2011. This notion was further partially extended to a generalized (glz) α-nonexpansive mapping by Pant and Shukla [14] in 2017 as follows: consider a BS X with its subset Dð≠ ϕÞ, and the mapping T : D ⟶ D is considered to be glz α-nonexpansive provided that ∃α ∈ ½0, 1Þ such that ∀u, v ∈ D, Also, in [14], they have obtained the existence of FP results and convergence theorems by using the iteration process defined by Agarwal et al. [15] that reads with ft n g and fs n g as sequences belonging to ð0, 1Þ: This iteration is known as the S-iteration, and it is independent of both the Ishikawa and Mann iteration processes as demonstrated in [15]. Over the most recent forty years, both the Ishikawa and Mann iteration processes have been effectively utilized by different mathematicians to approximate FP of different types of nonexpansive mappings in BS.
In 1953, Mann [11] devised a methodology that is termed as the Mann iterative process for approximating FP of continuous transformation in BS that reads where ft n g is a sequence belonging to ½0, 1. Moreover, Ishikawa [16] in 1974 generalized the Mann iterative process from one-to two-step iterations; he also obtained an iterative process to approximate FP of pseducontractive compact mapping in the Hilbert space given below: with ft n g and fs n g denoting sequences lying in ½0, 1 and satisfying some conditions. Also, observe that the Mann iterative procedure is a particular case of the Ishikawa iteration by the choice of s n = 0, ∀n ∈ ℕ.
More recently, Piri et al. [17] in 2019 have shown some interesting examples of the glz α-nonexpansive mapping and presented certain comparative convergence behaviors with regard to some powerful iteration procedures including the famous Mann and Ishikawa iterations among others.
As an application, fixed point theory of nonexpansive mapping and its generalization has many applications in different fields such as applications of nonexpansive mapping to solve an integral equation (see [18]) and to solve a variational inequality problem (see [19]). Also, there are applications of some classes of generalized nonexpansive mappings like quasi-nonexpansive mappings under contraction to find the minimum norm fixed point and generalized α-nonexpansive mappings to solve split feasibility problem (see [20,21]).
However, the present paper is aimed at establishing certain strong and weak convergence theorems of FP for the glz α-nonexpansive mapping via the application of the Mann iteration. Similar results are also set to be established by the application of the Ishikawa iteration process in the sense of UCBS. Remarkably, these results happen to be an extension of the results presented in [1,11].

Preliminaries
Recall that a BS X satisfies the Opial property [22] for every sequence fu n g in X such that fu n g ⇀ p; then, ∀q ∈ X with p ≠ q, For example, all Hilbert spaces, all finite dimensional BS, and ℓ p ð1 < p<∞Þ have satisfied the Opial property, while L p ½0, 2πðp ≠ 2Þ has not satisfied the Opial property [23].
Let fu n g and Dð≠ ϕÞ be a bounded sequence and subset of a BS X, respectively. Then, ∀u ∈ X, we define (i) the asymptotic radius of the bounded sequence fu n g at u as (ii) the asymptotic radius of the bounded sequence fu n g relative to D as (iii) the asymptotic center of the bounded sequence fu n g relative to D as We observe that AðD, fu n gÞ ≠ ϕ: Moreover, if X is UCBS, then AðD, fu n gÞ has exactly one point [23].
Let X * denote a dual space of BS X. Recall that X possesses the Fréchet differentiable norm provided that for each v in the sphere (unit) S of X, there exists the following limit: which is attained uniformly for v 0 ∈ S.

Journal of Function Spaces
Thus, as rightly given in [23], ∀u, w ∈ X, where JðuÞ = ∂ð1/2Þkuk 2 and g is a function (increasing) defined on ℝ + of which lim t↓0 ðgðtÞ/tÞ = 0. Accordingly, we give an illustrative example for a glz α -nonexpansive mapping in what follows.
Example 1 [14]. Consider D = ½0, 4 ⊂ ℝ of which a usual norm is endowed on. Let T : D ⟶ D be defined by Therefore, T is indeed a glz α-nonexpansive mapping with α ≥ 1/3. Definition 2. Mapping which satisfies condition (I) [10]. "Let X be a normed space and let D ⊆ X. A map T : D ⟶ D satisfies condition ðIÞ provided that there exists a nondecreasing function h : ½0,∞Þ ⟶ ½0,∞Þ that satisfies hð0Þ = 0 and hðtÞ > 0, for every t ∈ ð0,∞Þ such that ku − TðuÞk ≥ hðdðu, Fixð TÞÞÞ, for each u ∈ D, where dðu, FixðTÞÞ denotes the distance of u from FixðTÞ." Next, we state some important results that are essentially vital to the present work; these results were introduced in [14,24] together with their proofs. Proposition 3. Consider a BS X together with its subset Dð ≠ ϕÞ. Let us also consider a glz α-nonexpansive mapping given by T : D ⟶ D with a FP v ∈ D. Then, T is quasinonexpansive.

Lemma 4.
Consider a BS X together with its subset Dð≠ ϕÞ. Let us also consider a glz α-nonexpansive mapping given by T : D ⟶ D. Therefore, for every u, v ∈ D, Proposition 5. Demiclosedness principle [14]. "Consider a BS X together with the Opial property, and let Dð≠ ϕÞ be a closed subset of X. Let T : D ⟶ D be a glz α-nonexpansive mapping. If fu n g ⇀ z and lim n⟶∞ kTðu n Þ − u n k = 0, then TðzÞ = z. Meaning, ðI − TÞ is demiclosed at zero, with I denoting the identity mapping on X." The lemma below gives the convexity and closedness of the set of FP for the glz α-nonexpansive mapping.
Lemma 6 [14]. "Consider a glz α-nonexpansive mapping T : D ⟶ D, where Dð≠ ϕÞ is a subset of a BS X. Then, FixðTÞ is closed. In addition, if D is convex and X is strictly convex, then FixðTÞ is also convex." In the sequel, the next lemma will be used to navigate the main results of the paper.
Lemma 7 [24]. "Consider a UCBS X and 0 < a ≤ l n ≤ b < 1, ∀n ∈ ℕ. Moreover, consider the two sequences fu n g and fv n g such that limsup n⟶∞ ku n k ≤ r, limsup n⟶∞ kv n k ≤ r, and lim n⟶∞ kl n u n + ð1 − l n Þv n k = r hold for some r ≥ 0. Then, lim n⟶∞ ku n − v n k = 0:"

Main Results
This section starts off by investigating the weak and strong approximation FP for the glz α-nonexpansive mapping by using the Mann iteration process. Moreover, a similar examination will be looked at by the application of the Ishikawa iteration procedure.

Main Results for glz α-Nonexpansive with the Mann Iteration
Lemma 8. Consider a glz α-nonexpansive self-mapping T defined on a closed convex subset Dð≠ ϕÞ of a BS X. Let the sequence fu n g be defined by the Mann iteration (1), and assume ζ to be a FP of T; then, lim n⟶∞ ku n − ζk exists.
Proof. By referring to the definition of the Mann iteration (1) and Proposition 3, we get Therefore, the sequence fku n − ζkg is bounded and nonincreasing. Thus, we conclude that lim n⟶∞ ku n − ζk exists. Theorem 9. Consider a glz α-nonexpansive self-mapping T defined on a closed convex subset Dð≠ ϕÞ of a UCBS X. Let the sequence fu n g with u 1 ∈ D be defined by the Mann iteration (1). Then, FixðTÞ ≠ ϕ iff the sequence fu n g is bounded and Proof. Consider a bounded sequence fu n g, and lim n⟶∞ kTðu n Þ − u n k = 0. As X is UCBS, then AðD, fu n gÞ ≠ ϕ and it contains exactly one point. Let z ∈ AðD, fu n gÞ, and we want to demonstrate that FixðTÞ ≠ ϕ.

Journal of Function Spaces
Using the asymptotic radius definition as given above, we obtain Also, using Lemma 4, we get Hence, TðzÞ ∈ AðD, fu n gÞ. However, with regard to the uniqueness of the asymptotic center of fu n g, we obtain Tðz Þ = z. That means z ∈ FixðTÞ, and thus, FixðTÞ ≠ ϕ.
Conversely, let FixðTÞ ≠ ϕ and w ∈ FixðTÞ; then, from Lemma 8, lim n⟶∞ ku n − wk exists. Suppose Equation (18) and Proposition 3 yield Hence, From equations (18) and (20) and the definition of the Mann iteration (1), we get In view of equations (18), (20), and (21) and Lemma 7, we deduce that Consequently, In order to prove weak convergence of both the Mann and Ishikawa iterative processes to a FP for glz α-nonexpansive mapping, the following lemma is needed.
Lemma 10 [14]. "Suppose that the conditions of Theorem 9 are fulfilled. Then, lim n⟶∞ hu n , Jðp 1 − p 2 Þi exists for any p 1 , p 2 where η w ðu n Þ represents the set of all weak limit points of fu n g:" Theorem 11. Weak convergent theorem. Consider a glz α -nonexpansive self-mapping T with FixðTÞ ≠ ϕ defined on a closed convex subset Dð≠ ϕÞ of a UCBS X which satisfies the Opial property or which has a Fréchet differentiable norm such that ðI − TÞ is demiclosed at zero. Let the sequence fu n g be defined by the Mann iteration (1) with u 1 ∈ D such that a sequence ft n g in ½0, 1 and ∑ ∞ n=1 t n ð1 − t n Þ = ∞. Then, the sequence fu n g converges weakly to a FP of T.
Proof. Consider η w ðu n Þ to be the set of all weak limit points of fu n g. Then, from the fact that FixðTÞ ≠ ϕ, fu n g is a bounded sequence and from Theorem 9. Therefore, without loss of generality, let p ∈ η w ðu n Þ, which means Now, we want to show that η w ðu n Þ ⊂ FixðTÞ: From (24), (25), and Proposition 5, we have u n ⇀ p, then, Thus, p ∈ FixðTÞ, and we deduce that η w ðu n Þ is a subset of FixðTÞ.
Now, to prove that the sequence fu n g converges weakly to a FP of T, it is sufficient to prove that η w ðu n Þ is a singleton set.
First, we assume X to fulfil the Opial property and suppose p 1 and p 2 ∈ η w ðu n Þ such that p 1 ≠ p 2 ; then, by the reflexiveness of X, we have for some n k ↑∞,n j ↑∞. By Lemma 8, lim n⟶∞ ku n − p 1 k exists, since p 1 ∈ η w ðu n Þ ⊂ FixðTÞ.

Journal of Function Spaces
Using the Opial property on X, we get that arriving at a contradiction. Consequently, p 1 = p 2 . Hence, η w ðu n Þ is a singleton. This proves our result for which X satisfies the Opial property.
Secondly, we assume X to have a Fréchert differentiable norm given that ðI − TÞ is demiclosed at zero.
Thus, p 1 = p 2 . This shows that η w ðu n Þ must be a singleton.
Theorem 12. Strong convergent theorem. Consider a glz α -nonexpansive self-mapping T with FixðTÞ ≠ ϕ defined on a closed convex subset Dð≠ ϕÞ of a UCBS X. Then, for arbitrary u 1 ∈ D, the sequence fu n g defined by the Mann iteration (1) converges strongly to a member of FixðTÞ provided that T satisfies condition ðIÞ.
Proof. Since from Proposition 3, each glz α-nonexpansive mapping that possesses at least one FP is a quasinonexpansive mapping, then our conclusion follows from Theorem 2 in [10].
Theorem 13. Strong convergent theorem. Consider a glz α -nonexpansive self-mapping T with FixðTÞ ≠ ϕ defined on a closed convex subset Dð≠ ϕÞ of a BS X: Let the sequence fu n g be defined by the Mann iteration (1). Then, the sequence f u n g converges strongly to a FP of T provided that Proof. Assume that the lim inf n⟶∞ dðu n , FixðTÞÞ = 0, then ∃fw n g a subsequence of fu n g of which By (34), suppose fw n j g again to be a subsequence of fw n g of which such that fz j g is a sequence in FixðTÞ. Then, by Lemma 8, we have Now, we want to show that fz j g is a Cauchy sequence in FixðTÞ. By the triangular inequality and (36), we conclude that A standard argument refers to the fact that fz j g is a Cauchy sequence in FixðTÞ. By Lemma 6, FixðTÞ is a closed subset of the BS X. Thus, fz j g converges to a FP z. Then, we have Assume j ⟶ ∞; this means that fw n j g converges strongly to z. Accordingly, lim n⟶∞ ku n − zk exists for z ∈ Fixð TÞ by Lemma 8. Therefore, the sequence fu n g converges strongly to z.

Main
Results for glz α-Nonexpansive with the Ishikawa Iteration. Now, let us state and prove some lemmas that will be utilized to prove the results as follows.

Conclusion
In conclusion, the class of generalized α-nonexpansive mapping has been extensively examined in a uniformly convex Banach space setting. Results of the existence of the fixed point have been established and proven in Theorems 9 and 15 via the applications of the Mann and Ishikawa iterations, respectively. The established results corresponded to the results of Theorem 5.6 in [14].
Moreover, to approximate the fixed point of a generalized α-nonexpansive mapping, we made use of the Mann and Ishikawa iterations and proved strong convergence results. For instance, the established Theorems 12 and 13 via the Mann iteration came as a special state of Theorem 2 in [10] and corresponded to Theorem 5.9 in [14], correspondingly; while through the Ishikawa iteration, Theorems 17, 18, and 19 generalized Theorem 2.4 in [1] and corresponded to Theorem 5.9 in [14] and Theorem 2 in [11], respectively. Furthermore, with regard to weak convergence results, Theorems 11 and 16 for the Mann and Ishikawa iterations, respectively, generalized Theorem 2 in [12] and Theorem 2.3 in [1], respectively, by considering a generalized α-nonexpansive mapping instead of a nonexpansive mapping.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.