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.
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.
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.