Journal of Function Spaces
Volume 2020 (2020), Article ID 9851063, 7 pages
https://doi.org/10.1155/2020/9851063

Research Article

Approximating Fixed Points of Operators Satisfying (RCSC) Condition in Banach Spaces

Thabet Abdeljawad,1,2,3 Kifayat Ullah,⁴ Junaid Ahmad,⁴ Manuel de la Sen,⁵ and Junaid Khan⁴

¹Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia
²Department of Medical Research, China Medical University, Taichung 40402, Taiwan
³Department of Computer Sciences and Information Engineering, Asia University, Taichung, Taiwan
⁴Department of Mathematics, University of Science and Technology, Bannu, 28100 Khyber Pakhtunkhwa, Pakistan
⁵Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644 Bilbao, Barrio Sarriena, 48940 Leioa, Spain

Correspondence should be addressed to Junaid Ahmad; ahmadjunaid436@gmail.com

Received 13 July 2020; Revised 25 August 2020; Accepted 2 September 2020; Published 22 September 2020

Academic Editor: Yuanfang Ru

Copyright © 2020 Thabet Abdeljawad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a nonempty subset of a Banach space . A mapping is said to satisfy (RCSC) condition if each , . In this paper, we study, under some appropriate conditions, weak and strong convergence for this class of maps through iterates in uniformly convex Banach space. We also present a new example of mappings with condition (RCSC). We connect iteration and other well-known processes with this example to show the numerical efficiency of our results. The presented results improve and extend the corresponding results of the literature.

1. Introduction

will denote the set of all natural numbers throughout. In 2008, Suzuki [1] introduced a new class of mappings as follow. A self-map on a subset of a Banach space is said to satisfy condition if for all , we have

Obviously, when is nonexpansive mapping, that is, holds for all , then satisfies the condition. However, an example in [1] shows that there exists mappings, which satisfy the condition but not nonexpansive. A mapping with condition is often called Suzuki-type nonexpansive mapping. The class of Suzuki-type nonexpansive mappings is extensively studied by many authors (cf. [2–12] and others).

In 2012, motivated by Suzuki condition, Karapinar [13] suggested a new condition on mappings, the so-called (RCSC) condition (or Reich-Chatterjea-Suzuki condition). A self-map on a subset of a Banach space is said to satisfy the (RCSC) condition if for all , we have

The purpose of this work is to prove some weak and strong convergence results for this class of mappings through the iteration process [12] in the context of Banach spaces. We also give a numerical example to show the usefulness of our results. In this way, we extend and improve many well-known corresponding results of the current literature.

Approximating fixed points of nonlinear mappings played an important role and solved many problems [14–20]. It is now well known that if is nonexpansive, then the sequence of Picard iterates may not converge to a fixed point of . To overcome such problems and to get better a rate of convergence, many iterative processes are available in the literature. The well-known iterative processes are the Mann [21], Ishikawa [22], Noor [23], Agarwal et al. [24], Abbas and Nazir [25], Thakur et al. [7], Ullah and Arshad [12], and so on. Let , , and be a self-map on a nonempty convex subset of a Banach space.

The Mann iteration process [21] is a sequence defined as follows:

The Ishikawa iteration process [22] is a sequence defined as follows:

The Noor iteration process [23] is a sequence defined as follows:

The iteration process [24] is a sequence defined as follows:

The Abbas and Nazir iteration process [25] is a sequence defined as follows:

The Thakur et al. iteration process [7] is a sequence defined as follows:

The iteration process [12] is a sequence defined as follows:

In this paper, we will present some weak and strong convergence results using the iteration process (9) for mappings with (RCSC) condition. Similar results for the processes (3)–(8) can be proved on the same line of proofs.

2. Preliminaries

is called a fixed point of a self-map on if . We will denote by throughout the set of all fixed points of . A Banach space is said to satisfy Opial condition [26] if and only if for each weakly convergent sequence with a weak limit , we have the following property:

A self-map on a subset of a Banach space is said to satisfy the condition [27] if there is nondecreasing function with the properties , for every , and for all .

Let be a nonempty subset of a Banach space and a bounded sequence in . For each , define

(i)asymptotic radius of at by (ii)asymptotic radius of relative to by (iii)asymptotic center of relative to by

When the space is uniformly convex [28], then the set is always singleton. Notice also that the set is convex as well as nonempty provided that is weakly compact convex (see, e.g., [29, 30]).

Lemma 1. [13].
Let be a self-map on a subset of a Banach space. If satisfies the (RCSC) condition, then for all , the following holds:

The following facts are also needed.

Lemma 2. [13].
Let be a Banach space having Opial’s property, and . If satisfies the condition (RCSC), then the following condition holds:

The following lemma gives the structure of the fixed point set associated with a mapping satisfying (RCSC) condition.

Lemma 3. [13].
Let be a self-map on a subset of a Banach space. If satisfies the (RCSC) condition, then is closed. Moreover, if is strictly convex and is convex, then is also convex.

Lemma 4. [13].
Let be a self-map on a subset of a Banach space. If satisfies (RCSC) condition, then for all and , holds.

Lemma 5. [31].
Let for each and and be any two sequences in a uniformly convex Banach space such that , , and for some ; then, .

3. Main Results

We begin this section by proving a crucial lemma.

Lemma 6. Let be a self-map on a subset of a Banach space. Assume that satisfies the (RCSC) condition and let be a sequence generated by (9). If , then exists for each .

Proof. Let and . By Lemma 4, we have

which implies that

Hence, for all and . Thus, is bounded and nonincreasing, which implies that exists for each .

Now we give the necessary and sufficient condition for the existence of a fixed point for mapping with (RCSC) condition defined on a nonempty closed convex subset of a complete uniformly convex Banach space.

Theorem 7. Let be a self-map on a closed convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition and let be a sequence generated by (9). Then, if and only if is bounded and .

Proof. Let be bounded and . Let . By Lemma 1, we have

Hence, we conclude that . Since is uniformly convex, consists of a unique element. Thus, we have .
Conversely, suppose that and . By Lemma 6, exists and is bounded. Put

From (13), we have

By Lemma 4, we have

From (14), we have

From (17) and (19), we have

From (20), we have

Hence,

By Lemma 5, we have

Now we can prove the following weak convergence theorem.

Theorem 8. Let be a self-map on a closed convex subset of a uniformly convex Banach space having Opial’s property. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges weakly to a fixed point of .

Proof. Since is uniformly convex, is reflexive. By Theorem 7, is bounded and for all . By the reflexivity, one can find a weakly convergent subsequence of with a weak limit say . By Lemma 2, we have . It is suffice to show that is the weak limit of . is not the weak limit of . Then, one can find another weakly convergent subsequence of with a weak limit such that . Again by Lemma 2, . By Lemma 6 together with Opial’s property, we have

This is a contradiction. So, we have . Hence, is the weak limit of

Now we prove a strong convergence theorem as follows.

Theorem 9. Let be a self-map on a compact convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges strongly to a fixed point of .

Proof. By Theorem 7, for all . Since is compact and convex, we can find a strongly convergent subsequence of with a strong limit say . By Lemma 1, we have

By the uniqueness of limits in Banach spaces, . By Lemma 6, exists and hence is the strong limit of .

Now we state the following theorem. Since the proof is elementary, we will not include the details.

Theorem 10. Let be a self-map on a closed convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges strongly to a fixed point of provided that .

The following convergence theorem is based on condition .

Theorem 11. Let be a self-map on a closed convex subset of a uniformly convex Banach space. Assume that satisfies the (RCSC) condition with and let be a sequence generated by (9). Then, converges strongly to a fixed point of provided that satisifes the condition .

Proof. By Theorem 7, it follows that . By the condition , we have . The conclusion follows from Theorem 10.

4. Example

In this section, we compare the rate of convergence of the iteration process with other iterations in the setting of mappings with (RCSC) condition.

Example 1. Let be endowed with the usual norm. Set if and if . We shall prove that satisifes the (RCSC) condition. The case when is trivial. We consider only the following three nontrivial cases.
When , then and . Using triangle inequality, we have

When and , then and . Now

Finally, when and , then and . Now

Next, for and , but . Hence, does not satisfy the condition. Let , , and . The strong convergence of the (9), Thakur et al. (8), Abbas and Nazir (7), (6), Noor (5), Ishikawa (4), and Mann (3) iterates to a fixed point is given in Table 1.

Remark 12. From Table 1 and Figure 1, we see that the iteration process converges faster to than the others.

Table 1: Computation table obtained from the , Thakur et al., Abbas and Nazir, , Noor, Ishikawa, and Mann iterates for mapping defined in Example 1.

Thakur et al.
Abbas and Nazir

Noor
Ishikawa
Mann

3
3
3
3
3
3
3

2.1625
2.1931
2.2456
2.3863
2.4851
2.5363
2.6500

2.0264
2.0373
2.0603
2.1492
2.2353
2.2876
2.4225

2.0043
2.0072
2.0148
2.0576
2.1141
2.1542
2.2746

2.0007
2.0014
2.0036
2.0223
2.0554
2.0827
2.1785

2.0001
2.0003
2.0009
2.0086
2.0269
2.0443
2.1160

2
2.0001
2.0002
2.0033
2.0130
2.0238
2.0754

2
2
2.0001
2.0013
2.0063
2.0123
2.0490

2
2
2
2.0005
2.0031
2.0068
2.0318

2
2
2
2.0002
2.0015
2.0037
2.0207

2
2
2
2.0001
2.0007
2.0020
2.0134

2
2
2
2
2.0003
2.0011
2.0088

2
2
2
2
2.0002
2.0006
2.0057

2
2
2
2
2.0001
2.0003
2.0037

2
2
2
2
2
2.0002
2.0024

2
2
2
2
2
2.0001
2.0016

2
2
2
2
2
2
2.0010

2
2
2
2
2
2
2.0007

2
2
2
2
2
2
2.0004

2
2
2
2
2
2
2.0003

2
2
2
2
2
2
2.0002

2
2
2
2
2
2
2.0001

2
2
2
2
2
2
2

Figure 1: Convergence behavior of the (red line), Thakur et al. (green line), Abbas and Nazir (yellow line), Agarwal et al. (blue line), Noor (cyan line), Ishikawa (magenta line), and Mann (black line) iterates for mapping defined in Example 1 where .

Now using the above example, we make different choices of parameters and and intial points and also we get as our stopping criterion where is a fixed point of . The number of iterations for (9) to reach is compared with the leading three steps of Thakur et al. (8) and leading two steps of (6) iterations. The numbers in italic in Tables 2–4 show that iteration is better than the others. The "-" represents that the number of iterations exceeds 50.

Table 2: and . Number of iterations required to obtain the fixed point.

Initial points
(6)
Thakur et al. (8)
(9)

8

8

8

8

8

8

Table 3: and . Number of iterations required to obtain the fixed point.

Initial points
(6)
Thakur et al. (8)
(9)

13

14

14

14

14

14

Table 4: and . Number of iterations required to obtain the fixed point.

Initial points
(6)
Thakur et al. (8)
(9)

–

10

–

11

–

11

–

11

–

12

–

12

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19.

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