#### Abstract

In this work, we study the convergence of a new faster iteration in which two -nonexpansive mappings are involved in the setting of uniformly convex Banach spaces with a directed graph. Moreover, by constructing a numerical example, we show the fastness of our iteration procedure over other existing iteration procedures in the literature.

#### 1. Introduction

In 1922, Banach originated a great tool for solving the existence problems of nonlinear mappings, which is familiar as Banach’s contraction principle [1]. After it, this principle has been generalized in so many different directions.

Over the last 50 years, many authors introduced and studied various iteration schemes for different classes of contractive and nonexpansive mappings. In 2008, Jachymski [2] introduced the concept of joining fixed point theory and graph theory and established the Banach contraction principle in a complete metric space endowed with a directed graph. In 2012, Aleomraninejad et al. [3] presented some iterative scheme for -contraction and -nonexpansive mappings in a Banach space with a graph. Tiammee et al. [4] familiarized Browder’s convergence theorem for -nonexpansive mappings in a Hilbert space with a directed graph. In 2016, Tripak [5] showed the convergence of a sequence developed by the Ishikawa iteration to some common fixed points of two -nonexpansive mappings in a Banach space combined with a graph. In 2017, Suparatulatorn et al. [6] introduced and studied the modified S-iteration for two -nonexpansive mappings in a uniformly convex Banach space associated with a graph. Recently, Thianwan and Yambangwai [7] introduced a new two-step iteration process involving two -nonexpansive mappings and studied its convergence analysis in a uniformly convex Banach space endowed with a graph. There is vast literature in this direction for more details (see [8–19] and references therein).

Recently, Ullah et al. [20] introduced a new three-step iteration process known as the K-iteration process and proved its convergence for Suzuki’s generalized nonexpansive mapping.

Inspired by the above work, we proposed a modified iteration process containing two -nonexpansive mappings, by generating the sequence as follows:

Let be a nonempty convex subset of a Banach space , for any random , where and are appropriate real sequences in and are -nonexpansive mappings. Under some certain conditions, we demonstrate the convergence analysis of (1) for approximating common fixed points of two -nonexpansive mappings in a uniformly convex Banach space with graph. We also construct a numerical example, and by using MATLAB R2018a, we clarify that the proposed iteration procedure converges faster than modified Ishikawa iteration, modified S-iteration, and Thianwan’s new iteration (see [5–7]).

#### 2. Preliminaries

In this part, we gather some familiar concepts and applicable conclusions which will be used often.

Let be a nonempty subset of a Banach space . Let denote the diagonal of the cartesian product , i.e., .

denotes the set of vertices that coincides with in a directed graph , and the set of its edges contains all loops, i.e., . By assuming has no parallel edge to identify the graph with the pair . denotes the conversion of a graph . So we have

A set is said to be dominated by if for each and dominates if for each .

Let be a self map. An edge preserving mapping, i.e., , is said to be -nonexpansive if

A mapping is said to be -demiclosed at , if for any sequence in such that for all and then .

Let us recall that a Banach space is said to satisfy Opial’s property if and then

Lemma 1 [21]. *Let be a subset of a metric space . A mapping is semicompact if for a sequence in with there exists a subsequence of such that .*

Let be a subset of a normed space and let be a directed graph such that . Then, is said to have property WG (SG) if for each sequence in converging weakly (strongly) to and there is a subsequence of such that for all .

Lemma 2 [6]. *Suppose that is the Banach space having Opial’s condition, has property WG, and let be a -nonexpansive mapping. Then, is -demiclosed at 0, i.e., if and , then , where is the set of fixed points of .*

Lemma 3 [22]. *Let be uniformly convex Banach space and a sequence in for some . Suppose that the sequences and in are such that , , and for some . Then, .*

Lemma 4 [23]. *Let be the Banach space that satisfies Opial’s condition and let be a sequence in . Let be such that and exist. If and are subsequences of that converges weakly to and , respectively, then .*

Lemma 5 [24]. *Let be a bounded sequence in a reflexive Banach space . If for any weakly convergent subsequences of , both and converge weakly to the same point in , then the sequence is weakly convergent.*

Lemma 6 [7]. *Let be a nonempty closed convex subset of a uniformly convex Banach space and suppose that has property WG. Let be a -nonexpansive mapping on . Then, is -demiclosed at 0.*

#### 3. Main Results

We initiate this section by proving the following proposition.

Proposition 7. *Let and be two self -nonexpansive mappings on with nonempty, where is a nonempty closed convex subset of a uniformly convex Banach space endowed with a directed graph. Let , is convex and the graph is transitive. For random , define the sequence by (1). Let be such that , are in . Then, , , , , , , , , and are in .*

*Proof. *We go ahead by induction. By using the edge preserving property of and assumption , we get . By convexity of , we obtain . Again, by edge preservingness of and , we have . By convexity of and and applying the edge preservingness of again, we have . Again, by using the property of edge preserving of and , we get . Again, by applying edge preserving of , , we get as is convex. Thus, by edge preserving of , . Again, by using convexity of and edge preservingness of , we get . By edge preserving of , we get , and we get . Next, we assume that . By edge preserving of and convexity of , we get and . By applying edge preserving of on , we get . By using convexity of and and edge preserving property of , we have . As is edge preserving, we get . Owing to edge preserving of , we obtain and so , since is convex. By convexity of and and applying the edge preservingness of again, we have . Therefore, for all . Using a similar argument, we can show that under the assumption that . By using transitivity of , we get . This completes the proof.

Lemma 8. *Let , , , , , and be the same as in Proposition 7. Suppose that and are real sequences in and for arbitrary and . Then,
*(i)* exists*(ii)

*Proof. *(i)Let . By Proposition 7, we have . As and are -nonexpansive mappings and using the iterative sequence , we haveThis implies that sequence is decreasing and bounded below for all . Hence, exists.
(ii)Assume that . If ; then, by -nonexpansiveness of and , we getTherefore, the result follows.

Suppose that .

Taking the lim sup on both sides in the inequality (7) and (8), we obtain
On the other hand, using (1), we have
This implies that
So
This implies
By taking lim inf both sides, we have
From (11) and (17), we get
By using (11) and (19) and Lemma 3, we get
By taking lim inf both sides, we have
By using (12) and (22), we have
We also have
By taking limit infimum on both sides, we get
By using edge preserving property of and , we have
By taking limit sup on both sides in (26) and (27), we get
By using (28) and (29), we obtain
By taking limit sup on both sides in (30), we get
By using (25) and (31), we have
By (28), (29), and (32) and Lemma 3, we have
In addition,
By using (20), (33), and (34), we have
Therefore, we conclude .

The next proof is for the weak convergence of the sequence generated by (1) in a uniformly convex Banach space with directed graph satisfying Opial’s condition.

Theorem 9. *Let , , , , , and be the same as in Proposition 7 with satisfying Opial’s condition and has property WG. Suppose that and are real sequences in and for arbitrary and and then weakly converges to a common fixed point of and .*

*Proof. *Let be such that . From Lemm 8(i), exists, so is bounded. It follows from Lemma 8(ii) that . Since is uniformly convex and is bounded, we may assume that as , without loss of generality. By Lemma 2, we have . Suppose that subsequences and of converges weakly to and , respectively. By Lemma 8(ii), we obtain that and as . Using Lemma 2, we have . By Lemma 8(i), and exist. It follows from Lemma 4 that . Therefore, converges weakly to a common fixed point of and .

Next, we prove weak convergence of the sequence generated by (1) without assuming Opial’s condition in a uniformly convex Banach space with a directed graph.

Theorem 10. *Let , , , , , and be the same as in Proposition 7 with having property WG, and are real sequences in , is dominated by , and dominates . Then, weakly converges to a common fixed point of and .*

*Proof. *Let be such that . From Lemma 8(i), exists, so is bounded in . Since is nonempty closed convex subset of a uniformly convex Banach space , it is weakly compact and hence there exists a subsequence of the sequence which converges weakly to some point . By Lemma 8(ii), we obtain that
By using (20), (34), and (35), we have
In addition,
And by using (35) and (37),
Using Lemma 6, and are -demiclosed at so that . To complete the proof, it suffices to show that converges weakly to . To this end, we need to show that satisfies the hypothesis of Lemma 5. Let be a subsequence of which converges weakly to some . By similar argument as above . Now, for each , using (1), we have
By using (34), we get
By using (20), we have
We have
It follows from (38)
By using (41) and (44), we get
It follows from (39) and (45), we get
Therefore, the sequence satisfies the hypothesis of Lemma 5 which in turn implies that weakly converges to so that .

Next, we recall condition (B) for strong convergence.

Let be a nonempty closed convex subset of a uniformly convex Banach space . The mappings and on are said to satisfy condition (B) [21] if there exists a nondecreasing function with and for all such that for all , where

Theorem 11. *Let , , , , , and be the same as in Proposition 7. Suppose that and are real sequences in , satisfy condition (B), is dominated by , and dominates . Then, the sequence converges strongly to a common fixed point of and .*

*Proof. *From Lemma 8(i), exists and so exists for any . Also, from Lemma 8(ii), . Owing to condition (B),
We have . As is a nondecreasing function satisfying for all , we obtain that .

Hence, we can find a subsequence of and a sequence such that . Put for some . Then,
So is a Cauchy sequence. We assume that as . Since is closed, we get . So we have as . Since exists, we get .

We prove another strong convergence theorem as follows.

Theorem 12. *Let , , , , , and be the same as in Proposition 7 with having property SG, and are real sequences in , is dominated by , and dominates . If one of is semicompact, then converges strongly to a common fixed point of and .*

*Proof. *It follows from Lemma 8 that is bounded and . Since one of and is semicompact, then there exist subsequences of such that as . Since has property SG and transitivity of graph , we obtain . Notice that, for each , .

Then,
Hence, . Thus, exists by Theorem 11. We note that as . Hence, . It follows, as in the proof of Theorem 11, that converges strongly to a common fixed point of and .

#### 4. Numerical Examples

This section contains a numerical example which supports our main theorem. It is worth mentioning here that this example is motivated by [25].

*Example 1. *Let and . Let be a directed graph defined by and if and only if or . Define mappings for any .

It is easy to show that are -nonexpansive mappings but are not nonexpansive mappings because when , , , and . Choose , , for all and initial points . In Table 1 and Figure 1, we have shown the convergence rate of the modified Ishikawa iteration, modified S-iteration, Thianwan iteration, and proposed iteration (1).

Figure 1 shows the convergence of the modified Ishikawa iteration, modified S-iteration, Thianwan new iteration, and proposed iteration (1) to the common fixed point of and which is 1 in this numerical experiment, and it is clear that the proposed iteration process converges faster than others.

#### 5. Conclusion

The purpose of this paper was to study the convergence of a new faster iteration in which two -nonexpansive mappings were involved in the setting of uniformly convex Banach spaces with a directed graph. Also, we constructed a numerical example to show the fastness of our iteration procedure over other existing iteration procedures in the literature.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no competing interests.

#### Authors’ Contributions

All authors contributed equally and significantly in writing this paper. All authors have read and agreed to the published version of the manuscript.

#### Acknowledgments

This research is supported by Deanship of Scientific Research, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia.