Abstract
In the recent progress, different iterative procedures have been constructed in order to find the fixed point for a given self-map in an effective way. Among the other things, an effective iterative procedure called the JK iterative scheme was recently constructed and its strong and weak convergence was established for the class of Suzuki mappings in the setting of Banach spaces. The first purpose of this research is to obtain the strong and weak convergence of this scheme in the wider setting of generalized -nonexpansive mappings. Secondly, by constructing an example of generalized -nonexpansive maps which is not a Suzuki map, we show that the JK iterative scheme converges faster as compared the other iterative schemes. The presented results of this paper properly extend and improve the corresponding results of the literature.
1. Introduction
A mapping on a subset of a Banach space is called contraction provided that for all follows thatwhere is fixed. A point is called a fixed point for if . Normally, we denote the set of all fixed points of by , that is, . The Banach–Caccioppoli fixed point theorem (BCFPT) [1, 2] provides the existence of a unique fixed point for every self-contraction of a complete metric space.
We say that a self-map is nonexpansive on the set provided that
We may observe that every contraction of a subset of a Banach space is nonexpansive but the converse may not hold in general. Unlike contractions, every self-nonexpansive mapping of a complete metric space does not admit a fixed point. After many years of BCFPT, Browder [3], Gohde [4] and Kirk [5] independently obtained that a self-nonexpansive mapping of a closed bounded convex subset of a uniformly convex Banach space (UCBS, for short) always has a fixed point.
In 2008, Suzuki [6] provided a new type of generalization of nonexpansive mappings and proved some related fixed point results for this class of mappings in Banach spaces. Notice that a self-map is mapping with the property (also called Suzuki mapping) if any follows that
In 2011, Aoyama and Kohsaka [7] provided the idea of -nonexpansive mappings. A self-map is called -nonexpansive if any follows thatwhere .
In 2017, Pant and Shukla [8] defined a very general class of nonexpansive mappings which properly contains the class of Suzuki mappings and partially extends the class of -nonexpansive mappings. A self-map is called generalized -nonexpansive if any follows thatwhere .
Fixed point approximation for nonexpansive mappings under a suitable iterative method is a very active field of research and provides many interesting and important applications in applied sciences (cf. [9–12] and others). Finding the fixed points for nonexpansive and generalized nonexpansive under Picard iteration is not possible in general. A simple situation of such a case which is the rotation of the unit disk about the origin in a plane is a best example of a nonexpansive mapping which has a unique fixed point but Picard iteration does not converge to this point. In order to find fixed points of nonexpansive and hence generalized nonexpansive mappings and secondly to obtain relatively high accuracy, some authors introduced different types of iterative procedures (cf. Mann [13], Ishikawa [14], Noor [15], Agarwal et al. [16], Abbas and Nazir [17], Thakur et al. [18] and references therein). Suppose that is a closed nonempty convex subset of a given Banach space, and assume further that , , and is a self-map of .
The Mann [13] iteration process is stated as follows:
The Ishikawa [14] iterative process may be viewed as a two-step Mann iteration, which is given by
In 2000, Noor [15] suggested a three-step iterative process which is more general than the Mann and Ishikawa iteration processes as follows:
In 2007, Agarwal et al. [16] suggested a new iteration process, which converges faster than the Mann iteration for contraction mappings in Banach spaces:
In 2014, Abbas and Nazir [17] proposed a new three-step iteration which converges faster than all of the Picard, Mann, Ishikawa, and Agarwal iterative processes for nonexpansive mappings, as follows:
In 2016, Thakur et al. [18] suggested the following iteration process, which converges faster than all of the above iterative processes for Suzuki mappings:
Very recently, Ahmad et al. [19] introduced a new iterative process named JK iteration, as follows:
They observed that JK iteration (12) can be used for fixed points of Suzuki mappings. Moreover, they proved by providing a novel example of Suzuki mappings that the JK iteration process converges faster than all of the above iterative processes including the leading Thakur iteration (11). In this paper, firstly, we improve and extend the main results of Ahmad et al. [19] from the context of Suzuki mappings to the more general framework of generalized -nonexpansive mappings. We then provide a novel example of generalized -nonexpansive mappings and show that its JK iterative process is better than the mentioned iterative processes. Our results can be used for finding the solutions of split feasibility problems, solutions of differential and integral equations provided that the operator is generalized -nonexpansive.
2. Preliminaries
We now provide some definitions.
Definition 1 [20]. A Banach space is said to be endowed with Opial’s property if every weakly convergence sequence having a weak limit follows that
Definition 2 [21]. A self-mapping of a subset of a Banach space is said to be endowed with condition if one has a nondecreasing function having and for every and for every ; here, stands for the distance of from .
Definition 3. Suppose that is any given Banach space and is bounded. Assume that is closed and convex. Then, the asymptotic radius of the sequence relative to the set is given by . Moreover, the asymptotic center of with respect to is given by .
Remark 4. 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 and nonempty also known in the case when is weakly compact and convex [23, 24].
Now, we combine some elementary properties of generalized -nonexpansive mappings, which can be found in [8].
Proposition 5. Suppose that is any nonempty subset of a Banach space and .(a)If is Suzuki mapping, then, is generalized -nonexpansive(b)If is generalized -nonexpansive having nonempty fixed point set, then, for any , for all (c)If is generalized -nonexpansive, then, the set is closed in . Also, is convex in the case when is strictly convex and is convex(d)If is a generalized -nonexpansive mapping, then, for every choice of , the following holds:(e)Suppose that is generalized -nonexpansive and is endowed with the Opial property. If is weakly convergent to and , it follows that
The following useful lemma can be found in [25].
Lemma 6. Suppose that for each and . If and are any sequences in a UCBS endowed with , , and , then .
3. Main Results
The aim of this section is at giving some important weak and strong convergence of JK (12) for the class of generalized -nonexpansive mappings. We start the section with a key lemma.
Lemma 7. Suppose that is UCBS, is closed convex, and is a generalized -nonexpansive having . If is a JK iteration sequence as provided in (12). Then, exists for each .
Proof. If we choose , then, using (12) along with Proposition 5 (b), we haveHence,Consequently, we conclude that is nonincreasing and bounded; accordingly, we must have that exists for every element of . ☐
Theorem 8. Suppose that is UCBS, is closed convex, and is a generalized -nonexpansive. If is a JK iteration sequence as provided in (12). Then, if and only if is bounded and .
Proof. Firstly, we may take . According to Lemma 7, one concludes that is bounded and exists for every element of . We now supposeWe need to obtain that . Then, using Lemma 7, we haveSince , so by Proposition 5 (b), we inferNow from (16), we getUsing this together with (17), we obtainFrom (18) and (21), we deduceUsing (22), we getThus,Using (17), (19), and (24) and keeping Lemma 6 in mind, one concludes thatConversely, we may suppose that is bounded in such that . The aim is to prove that . If we take any , then, using Proposition 5 (d), it follows thatIt follows that . Since is UCBS, contains only one element, that is, we must have . Hence, , that is, the fixed point is nonempty. ☐
Now, we are in the position to prove our weak convergence result.
Theorem 9. Suppose that is UCBS, is closed convex, and is a generalized -nonexpansive having . If is a JK iteration sequence as provided in (12) and has Opial’s property, then, converges weakly to a point of
Proof. By Theorem 8, is bounded. The uniform convexity of follows reflexivity of , that is, has a weakly convergent subsequence with a weak limit, namely, . According to Theorem 8, . Hence, using Proposition 5 (e), we get . We claim that is the weak limit of . We may suppose on the contrary that is not the weak limit of , that is, has another weakly convergent subsequence with a weak limit, namely, . According to Theorem 8, . Hence, using Proposition 5 (e), we get . Now using Lemma 7 and Opial’s property, we haveConsequently, we obtained , which suggests a contradiction. Therefore, we conclude that is the weak limit of the sequence . ☐
Now, we prove the following strong convergence result.
Theorem 10. Suppose that is UCBS, is compact convex, and is a generalized -nonexpansive having . If is a JK iteration sequence as provided in (12), then, converges strongly to a point of .
Proof. Since and is compact, so we can find a subsequence, namely, of such that for some element . Moreover, since , so according to the Theorem 8, . Applying Proposition 5 (d), we getConsequently, provided that . But is a Banach space, and so, the limit of a convergent sequence is always unique. Thus, . Lemma 7 provides us that exists. Hence, is the strong limit of . ☐
We now state and then prove another strong convergence theorem as follows.
Theorem 11. Suppose that is UCBS, is closed convex, and is a generalized -nonexpansive having . If is a JK iteration sequence as provided in (12), then, converges strongly to a point whenever
Proof. According to Lemma 7, exists, for every choice of fixed point of . It follows that exists. Accordingly, we haveThe above strong limit suggests the existence of two subsequences , in and , respectively, with the property for every natural constant . According to the proof of Lemma 7, the iterative sequence is nonincreasing. Accordingly, we haveUsing the above and triangle inequality, one hasAccordingly, we obtained , that is, form the Cauchy sequence in the closed set . It follows that for some . Cosequently, . By Lemma 7, exists, that is, is also the strong limit of . ☐
We finish this section with a strong convergence theorem under the condition .
Theorem 12. Suppose that is UCBS, is closed convex, and is a generalized -nonexpansive having . If is a JK iteration sequence as provided in (12). Then, converges strongly to a point of whenever is endowed with condition .
Proof. According to Theorem 8, one can conclude that . Applying the condition of , one obtain . It now follows from Theorem 11 that is strongly convergent in the set . ☐
4. Numerical Example
The aim of this section is to provide a new example of generalized -nonexpansive mappings that exceeds the class of Suzuki mappings. We connect the mentioned iterative schemes with this example to show the effectiveness of our obtained results.
Example 13. We take a set and set a self-map on by the following rule:We show that is generalized -nonexpansive having , but not Suzuki mapping. This example thus exceeds the class of Suzuki mappings.
Case 1. When , we have
Case 2. When , we have
Case 3. When and , we have
The above cases clearly suggest that is generalized nonexpansive mapping having . Choose and ; then, , , and . Thus, it is seen that, but . Thus, exceeds the class of Suzuki mappings.
Now, we by choosing , , and , we may observe in Table 1 and Figure 1 that JK (12) iterative process converges faster to as compared the other processes.
Now we show the further effectiveness of the JK iteration (12) in the class of generalized -nonexpansive mappings. Using defined in Example 13, we suggest some different values for the parameters and . To do this, we set the stopping criteria . The obtained results are provided in Table 2. The bold numbers show that JK iteration (12) requires less iteration numbers as compared to the leading three-step Thakur (11) and leading two-step Agarwal (9).
Remark 14. The main outcome of this paper extended the corresponding results of Ahmad et al. [19] from the class of Suzuki maps to the setting of generalized -nonexpansive maps. We have observed in Tables 1 and 2 as well as in Figure 1 that the JK iterative scheme (12) is still more effective than the other iterative schemes even in the general setting of generalized -nonexpansive maps.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
No competing interests are associated with this paper.
Authors’ Contributions
This research work is completed by the equal contribution of each of the author listed in this paper.