Abstract

The present paper seeks to illustrate approximation theorems to the fixed point for generalized -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 refine many of the recently reported results in the literature.

1. Introduction

Consider a Banach space (BS) , together with its subset . Let us also consider the following notations , , and to represent the set of fixed points of , weak convergence, and strong convergence, correspondingly.

A self-mapping defined on a subset is referred to as (1)nonexpansive provided that for all (2)quasi-nonexpansive provided that , and for all and , the following assertion holds:

Notably, there is a relationship between a nonexpansive mapping and a quasi-nonexpansive mapping. That is, each nonexpansive mapping satisfying is quasi-nonexpansive; 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 [27]). Browder in 1965 [8] and Kirk [9] have shown that self-nonexpansive 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 -nonexpansive mapping in BS was proposed by Aoyama and Kohsaka [13] in 2011. This notion was further partially extended to a generalized (glz) -nonexpansive mapping by Pant and Shukla [14] in 2017 as follows: consider a BS with its subset , and the mapping is considered to be glz -nonexpansive provided that such that ,

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 and as sequences belonging to This iteration is known as the -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 is a sequence belonging to .

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 and denoting sequences lying in and satisfying some conditions.

Also, observe that the Mann iterative procedure is a particular case of the Ishikawa iteration by the choice of .

More recently, Piri et al. [17] in 2019 have shown some interesting examples of the glz -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 -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 -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].

2. Preliminaries

Recall that a BS satisfies the Opial property [22] for every sequence in such that ; then, with ,

For example, all Hilbert spaces, all finite dimensional BS, and have satisfied the Opial property, while has not satisfied the Opial property [23].

A BS is uniformly convex provided that for each such that for any , together with and ; then, is said to hold.

Let and be a bounded sequence and subset of a BS , respectively. Then, we define (i)the asymptotic radius of the bounded sequence at as(ii)the asymptotic radius of the bounded sequence relative to as(iii)the asymptotic center of the bounded sequence relative to as

We observe that Moreover, if is UCBS, then has exactly one point [23].

Let denote a dual space of BS . Recall that possesses the Fréchet differentiable norm provided that for each in the sphere (unit) of , there exists the following limit: which is attained uniformly for .

Thus, as rightly given in [23], , where and is a function (increasing) defined on of which .

Accordingly, we give an illustrative example for a glz -nonexpansive mapping in what follows.

Example 1 [14]. Consider of which a usual norm is endowed on. Let be defined by Therefore, is indeed a glz -nonexpansive mapping with .

Definition 2. Mapping which satisfies condition (I) [10]. “Let be a normed space and let . A map satisfies condition provided that there exists a nondecreasing function that satisfies and , for every such that , for each , where denotes the distance of from .

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 together with its subset . Let us also consider a glz -nonexpansive mapping given by with a FP . Then, is quasi-nonexpansive.

Lemma 4. Consider a BS together with its subset . Let us also consider a glz -nonexpansive mapping given by . Therefore, for every ,

Proposition 5. Demiclosedness principle [14]. “Consider a BS together with the Opial property, and let be a closed subset of . Let be a glz -nonexpansive mapping. If and , then . Meaning, is demiclosed at zero, with denoting the identity mapping on .

The lemma below gives the convexity and closedness of the set of FP for the glz -nonexpansive mapping.

Lemma 6 [14]. “Consider a glz -nonexpansive mapping , where is a subset of a BS . Then, is closed. In addition, if is convex and is strictly convex, then 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 and , . Moreover, consider the two sequences and such that , , and hold for some . Then,

3. Main Results

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

3.1. Main Results for glz -Nonexpansive with the Mann Iteration

Lemma 8. Consider a glz -nonexpansive self-mapping defined on a closed convex subset of a BS . Let the sequence be defined by the Mann iteration (1), and assume to be a FP of ; then, exists.

Proof. By referring to the definition of the Mann iteration (1) and Proposition 3, we get Therefore, the sequence is bounded and nonincreasing. Thus, we conclude that exists.

Theorem 9. Consider a glz -nonexpansive self-mapping defined on a closed convex subset of a UCBS . Let the sequence with be defined by the Mann iteration (1). Then, iff the sequence is bounded and

Proof. Consider a bounded sequence , and . As is UCBS, then and it contains exactly one point.
Let , and we want to demonstrate that .
Using the asymptotic radius definition as given above, we obtain Also, using Lemma 4, we get Hence, . However, with regard to the uniqueness of the asymptotic center of , we obtain . That means , and thus, .
Conversely, let and ; then, from Lemma 8, 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 -nonexpansive mapping, the following lemma is needed.

Lemma 10 [14]. “Suppose that the conditions of Theorem 9 are fulfilled. Then, exists for any ; in particular, , , where represents the set of all weak limit points of

Theorem 11. Weak convergent theorem. Consider a glz -nonexpansive self-mapping with defined on a closed convex subset of a UCBS which satisfies the Opial property or which has a Fréchet differentiable norm such that is demiclosed at zero. Let the sequence be defined by the Mann iteration (1) with such that a sequence in and . Then, the sequence converges weakly to a FP of .

Proof. Consider to be the set of all weak limit points of . Then, from the fact that , is a bounded sequence and from Theorem 9. Therefore, without loss of generality, let , which means Now, we want to show that From (24), (25), and Proposition 5, we have then, Thus, , and we deduce that is a subset of .
Now, to prove that the sequence converges weakly to a FP of , it is sufficient to prove that is a singleton set.
First, we assume to fulfil the Opial property and suppose and such that ; then, by the reflexiveness of , we have for some .
By Lemma 8, exists, since .
Using the Opial property on , we get that arriving at a contradiction.
Consequently, . Hence, is a singleton. This proves our result for which satisfies the Opial property.
Secondly, we assume to have a Fréchert differentiable norm given that is demiclosed at zero.
Substituting and for and , respectively, in where and , we obtain By referring to Lemma 10, the limit exists.
In particular, this implies that for all and .
By replacing in (32) by , respectively, we obtain , since .
Thus, . This shows that must be a singleton.

Theorem 12. Strong convergent theorem. Consider a glz -nonexpansive self-mapping with defined on a closed convex subset of a UCBS . Then, for arbitrary , the sequence defined by the Mann iteration (1) converges strongly to a member of provided that satisfies condition .

Proof. Since from Proposition 3, each glz -nonexpansive mapping that possesses at least one FP is a quasi-nonexpansive mapping, then our conclusion follows from Theorem 2 in [10].

Theorem 13. Strong convergent theorem. Consider a glz-nonexpansive self-mapping with defined on a closed convex subset of a BS Let the sequence be defined by the Mann iteration (1). Then, the sequence converges strongly to a FP of provided that

Proof. Assume that the , then a subsequence of of which By (34), suppose again to be a subsequence of of which such that is a sequence in . Then, by Lemma 8, we have Now, we want to show that is a Cauchy sequence in . By the triangular inequality and (36), we conclude that A standard argument refers to the fact that is a Cauchy sequence in . By Lemma 6, is a closed subset of the BS . Thus, converges to a FP . Then, we have Assume ; this means that converges strongly to . Accordingly, exists for by Lemma 8. Therefore, the sequence converges strongly to .

3.2. Main Results for glz -Nonexpansive with the Ishikawa Iteration

Now, let us state and prove some lemmas that will be utilized to prove the results as follows.

Lemma 14. Consider a glz -nonexpansive self-mapping defined on a closed convex subset of a BS . Let the sequence with be defined by the Ishikawa iteration (2). Suppose that ; then, the statements given below are true: (1).(2)

Proof. (1) By definition of the Ishikawa iteration (2) and Proposition 3, we have hence Again, using the definition of the Ishikawa iteration (2) and Proposition 3, one gets thus, Now, from (40) and (42), we get that (2) Using (42), the sequence is deduced to be nonincreasing and bounded. Thus, exists.

Theorem 15. Consider a glz -nonexpansive self-mapping defined on a closed convex subset of a UCBS Let the sequence with be defined by the Ishikawa iteration (2). Then, iff the sequence is bounded and also

Proof. Consider a bounded sequence and . Therefore, we get by following the same steps of the analogous part in the proof of Theorem 9.
Conversely, assume and , so from Lemma 14, exists. Suppose From equation (45) and Lemma 14, we have Hence, From equation (47) and Proposition 3, we get Thus, Now, by the definition of the Ishikawa iteration (2), one gets In view of equations (45), (49), and (50) and Lemma 7, one obtains Again, by the definition of the Ishikawa iteration (2), we have Now, letting , we get Hence, Then, we conclude By equations (52) and (55), we deduce We observe Now, taking the liminf of the last inequality, we get that Thus, From equations (47) and (60), we get Finally, from equations (45), (47), and (60) and via Lemma 7, one concludes that Hence,

Theorem 16. Weak convergent theorem. Consider a glz -nonexpansive self-mapping with defined on a closed convex subset of a UCBS which satisfies the Opial property or which has a Fréchet differentiable norm such that is demiclosed at zero. So for any initial value , the sequence defined by the Ishikawa iteration (2) together with restricting and converges weakly to a FP of .

Proof. The methodology of the proof is identical to that of Theorem 11.

Theorem 17. Strong convergent theorem. Consider a glz -nonexpansive self-mapping with defined on a closed convex subset of a UCBS , and suppose in addition that satisfies condition . So for any initial value , the sequence defined by the Ishikawa iteration given in (2) converges strongly to a FP of .

Proof. Since , Theorem 15 guarantees that the sequence is bounded and .
Also, from condition , we guarantee that Thus, as follows from equation (64). A standard argument happens when there exists of which as .

Theorem 18. Strong convergent theorem. Consider a glz -nonexpansive self-mapping with defined on a closed convex subset of a BS So for any initial value , the sequence defined by the Ishikawa iteration (2) converges strongly to a FP of provided that

Proof. Suppose that , so a subsequence of the sequence exists of which . In view of the previous step, consider a subsequence of such that , where is a sequence in .
So, with the help of Lemma 14, we guarantee that Now, we want to show that is a Cauchy sequence in . From (66), we conclude that A standard argument proves that is a Cauchy sequence in .
By referring to Lemma 6, we get that is a closed subset of BS ; then, converges to a FP .
Now, we have Assume ; this means that converges strongly to .
By Lemma 14, the limit exists for .
Thus, converges strongly to , where .

Theorem 19. Strong convergent theorem. Consider a glz -nonexpansive self-mapping with defined on a closed convex subset of a UCBS . Suppose also that the range of under is included in a subset of X that is compact (i.e., , where is compact). Then, for any initial value , the sequence defined by the Ishikawa iteration (2) converges strongly to a FP of .

Proof. Given that , Theorem 15 guarantees that the sequence is bounded and .
Now, from precompactness of , we conclude that , where is compact.
Hence, affirming as .
Then, has a convergent subsequence , so as .
Again, from Theorem 15, one gets which implies which further implies . Since by Lemma 14, for , exists, this means that the sequence converges strongly to .

4. Conclusion

In conclusion, the class of generalized -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 -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 -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.