Abstract
In this paper, an efficient new iterative method for approximating the fixed point of Suzuki mappings is proposed. Some important weak and strong convergence results of the proposed iterative method are established in the setting of Banach space. An example illustrates the theoretical outcome.
1. Introduction and Preliminaries
Throughout the present research, we shall write and to denote the set of natural numbers and real numbers set, respectively. We say that a self-map of a subset of a Banach space is called a contraction map whenever a real constant exists with the following property:
An element is said to be a fixed point of provided that . In this manuscript, the notation will throughout denote the fixed point set of . The Banach–Caccioppoli fixed point theorem (see, e.g., [1, 2] and others) states that any contraction mapping in the setting of complete metric spaces admits a unique fixed point , and this is, in fact, the limit of all the sequences obtained from the Picard iterates [3], that is, . However, one of the important classes of mappings in fixed point theory is the class of nonexpansive mappings. Notice that, is called a nonexpansive mapping whenever (1) holds true for . In 1965, Browder [4] and Gohde [5] differently proved the same result concerning the existence of fixed points for nonexpansive mappings. Indeed, they suggested that any self-nonexpansive map of always admits at least one fixed point whenever is assumed to be a bounded convex closed subset of some uniformly convex Banach space (UCBS). Nevertheless, the sequence defined by Picard iterates may not have a limit in the fixed point set associated with a nonexpansive map in general as shown in the next example. Let and set ; it is easy to see that is a self-nonexpansive mapping on having a unique fixed point . However, for any , we obtain the sequence of Picard iterates as follows: which is an oscillating sequence and, hence, diverges in . For providing comparatively better convergence speed and to overcome such situations, different iterative methods have been suggested by some authors (cf. the works of Mann [6], Ishikawa [7], Noor [8], Agarwal et al. [9], Abbas and Nazir [10], Thakur et al. [11], and references therein).
In 2008, Suzuki [12] gained a big break through introducing an interesting extension of nonexpansive mappings as follows. We recall that a self-map mapping with property (also called Suzuki mapping) if the following fact is valid:
One can easily notice that the Suzuki mappings satisfy the nonexpansive requirement for some elements of the domain. Hence, nonexpansive mappings obviously satisfy property of Suzuki [12]. Interestingly, an example in [12] (see also an example below) nicely shows that there exist many mappings in the class of Suzuki mappings, which do not belong to the class of nonexpansive mappings. Suzuki also extended the celebrating result of Browder [4] and Gohde [5] from the setting of nonexpansive mappings to the framework of Suzuki mappings.
New iterative methods for the investigation of fixed points and solution of functional equations is the busy research topic and has fruitful applications such as in image recovery and signal processing (see, e.g., [13–19] and others). Therefore, it is our purpose to construct a new iterative method for the larger class of nonexpansive mappings called Suzuki mappings. We also show by an example that this new iterative process gives better approximations as compared to other methods. Suppose is a closed nonempty convex subset of a given Banach space, and assume further that , , and is a self-map of .
The Mann iterative method [6] is defined as follows:
The Ishikawa [7] iterative method is the extension of the Mann method from one step to two steps as follows:
The Noor [8] iterative method is the extension of both of the Mann and Ishikawa iterative methods as follows:
Agarwal et al.’s [9] method is the slightly modification of the Ishikawa method as follows:
Abbas and Nazir’s [10] iterative method is a three-step method read as follows:
Thakur et al. [11] proposed a new iterative method as follows:
Thakur et al. [11] showed that method (8) is better than all of the methods, namely, Mann (3), Ishikawa (4), Noor (5), Agarwal (6), and Abbas and Nazir (7). Here, in the current research, we first suggest an efficient new iterative method and prove that it can be used for computations of fixed points of the larger class of nonexpansive maps called Suzuki maps. Furthermore, we shall provide a novel example of the so-called Suzuki mappings and prove that it exceeds the corresponding class of nonexpansive mappings.
2. Preliminaries
Here, first we present some earlier important definitions, which are needed for our theoretical outcome.
Let be a given Banach space, and suppose is weakly convergent to and satisfies the following:
Whenever any weakly convergent sequence in has the abovementioned property, is called a Banach space endowed with Opial’s property (for details, see [20]). We now recall a property introduced by Sentor and Dotson [21] for (where is a nonempty subset of a Banach space). We recall that has condition [21] if one can find a nondecreasing function, namely, , with the properties , for every , and for all .
Let be any nonempty subset of a general Banach space , and suppose is any given bounded sequence in . We fix and denote(a)by , the asymptotic radius of at given by (b)by , the asymptotic radius associated with of given by (c)by , the asymptotic center associated with of given by
The most well-known fact about the set is that it is always singleton whenever is UCBS [22]. The fact that the set is convex nonempty whenever is weakly compact and convex is also well known (see, e.g., [23, 24]).
Lemma 1 (See [12]). Assume that is any nonempty subset of a Banach space, and suppose . If is a Suzuki mapping, then for every element and for every element , the fact holds.
Lemma 2 (See [12]). Assume that is any nonempty subset of a Banach space, and suppose . If is a Suzuki mapping, then for every two elements , we have the following property:
The following result is known as the demiclosed principle.
Lemma 3 (See [12]). Assume that is any nonempty subset of a Banach space having the Opial property, and suppose . If is a Suzuki mapping, then the following condition holds:
The fixed-point set endowed with a Suzuki mapping enjoys the following properties.
Lemma 4 (See [12]). Assume that is any nonempty subset of a Banach space, and suppose . If is a Suzuki mapping, then the set is closed. Furthermore, if is a strictly convex Banach space and is convex, then is convex too.
The following useful lemma can be found in [25].
Lemma 5. Let for each and . If and are any sequences in a UCBS, with , , and . Then, .
3. Main Results
Strongly motivated by those mentioned above, we introduce a new iterative process, namely, JK iteration, as follows:where .
In the present research section, we establish very interesting and important results for the larger class of the so-called Suzuki maps under the newly suggested method (12). We will present a numerical example to show that the JK iterative process is better than the iterative process by Thakur et al. (8). Furthermore, in the last section, a novel example of the so-called Suzuki maps which is not nonexpansive shows that Suzuki maps properly include nonexpansive maps. The numerical observations suggest that JK iterative method is for better than the leading method of Thakur and, hence, many others.
We now state and prove a much needed lemma for our main outcome, which will play a significant role in each result of the sequel.
Lemma 6. Assume that is any nonempty closed convex subset of a Banach space , and suppose is a Suzuki mapping with . Suppose is a sequence given in (12). Then, exists for every fixed point of .
Proof. Take . By Lemma 1, we haveThey imply thatFrom the equations mentioned above, we conclude that is a bounded and nonincreasing sequence of reals, and hence, exists for every fixed point of .
Theorem 1. Assume that is any nonempty closed convex subset of a UCBC, and suppose is a Suzuki mapping. Assume further that is a sequence given in (12). Then, if and only if is bounded, and .
Proof. First, we assume that is bounded and . We shall prove that . For this, let . By Lemma 2, we haveIt follows that . Since in UCBS, asymptotic centers are singleton, we have . Hence, the fixed point is nonempty.
Conversely, we assume that . Conclusions of Lemma 6 provide that is bounded and exists for every fixed point of . Now, ifthen by observing the proof of Lemma 6 and keeping (16) in mind, we obtainAppling Lemma 1, we getand by observing the proof of Lemma 6, we seeIt gives, together with (16),From (17) and (20), we obtainFrom (21), we haveHence,Now, from (16), (18) and (23) together with Lemma 5, we obtainNow, we are in the position to prove our weak convergence result.
Theorem 2. Assume that is a UCBS with Opial’s property and is a nonempty convex closed subset of , and suppose be a Suzuki mapping with . Suppose is a sequence given in (12). Then, converges weakly to a fixed point of .
Proof. By Theorem 1, is bounded and . Since is uniformly convex, is reflexive. Hence, one can easily find a subsequence, namely, of such that for some . By Lemma 3, . We shall prove that is the weak limit of . Let not be the weak limit of . Then, one can find another subsequence, namely, of such that and . Again by Lemma 3, . Now, using Lemma 6 and Opial’s property, we haveHence, , clearly a contradiction, and so we must accept that is the only weak limit of .
Now, we prove the following strong convergence result.
Theorem 3. Assume that is any nonempty convex compact subset of a UCBC, and suppose be a Suzuki mapping. Assume further that is a sequence given in (12). Then, converges strongly to a fixed point of .
Proof. From Theorem in [12], we can write . By Theorem 1, . Since the domain is compact, one can easily find a strongly convergent subsequence, namely, of having a limit say . By using Lemma 2, the following holds:Hence, whenever , so the uniqueness of limits follows . By Lemma 6, exists. Hence, is the strong limit of .
The proof of the following theorem is elementary and, therefore, omitted.
Theorem 4. Assume that is any nonempty closed convex subset of a UCBS, and suppose be a Suzuki mapping. If and , where is a sequence given in (12), then converges strongly to a fixed point of .
We finish this section with a strong convergence theorem under the condition .
Theorem 5. Assume that is any nonempty convex closed subset of a UCBS, and suppose be a Suzuki mapping with . Assume further that is a sequence given in (12). If fulfils condition (I), then converges strongly to a fixed point of .
Proof. In view of Theorem 1, we can conclude that . Since fulfils condition (I), one has . The conclusions are now clear from Theorem 4.
4. Numerical Example
This section introduces a novel example of self-Suzuki maps on a closed convex bounded subset of a Banach space. We suggest with many different cases that the novel JK scheme is far better than the earlier iterative methods using this example. Since we are using Suzuki maps in our work, the provided outcome holds simultaneously for nonexpansive maps as well.
Example 1. Consider a closed convex of a Banach space . Set a self-map as follows:One can conclude that is a Suzuki mapping and, however, not nonexpansive by studying the computations given below. Select and , and observe thatwhich proves that is not a nonexpansive on .
Next, we suggest the proof of the Suzuki property of on . The proof can be divided as given below.
Case 1. Select ; then, . For , one has , i.e., ; therefore, . So, one hasHence, .
Case 2. Select ; then, . For , one has , and so the following possible situations occur:(a)Whenever , . So, Therefore, .(b)Whenever , . Since , . So, the case is and .When and are already included in , we assume and ; then,For convenience, first we assume and , then and . Hence, .
Next, we consider and , then and . Hence, . So, . Hence, is a Suzuki mapping on . Let and , . A strong convergence of JK (12), leading Thakur (8), and leading two-step Agarwal (6) iterates to can be seen in the Table 1 and Figure 1. Clearly, JK iterates converge faster to .
Now, to show further effectiveness of JK iteration, we set stopping criteria , where is a fixed point of . The observations are given below.
Remark 1. One can observe from Tables 1–4 that the new JK iterative method converges faster than the fastest Thakur and fastest two-step Agarwal methods for the larger class of Suzuki mappings.
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
J. A., K. U., M. A., and Z. M. provided equal contributions to this article.
Acknowledgments
This research was funded by Nature Science Foundation of Heibei Province under Grant A2019404009, Postdoctoral Foundation of Heibei Province under Grant B2019003016, Innovation and Improvement Project of Academic Team of Hebei University of Architecture (Mathematics and Applied Mathematics) NO. TD202006, and Major Project of Hebei Provincial Department of Education of 2021.