Abstract
In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard–Mann hybrid iteration process. We also construct an example of mean nonexpansive mappings and show that it exceeds the class of nonexpansive mappings. To show the numerical accuracy of our main outcome, we show that Picard–Mann hybrid iteration process of this example is more effective than all of the Picard, Mann, and Ishikawa iterative processes.
1. Introduction and Preliminaries
Suppose is a Banach space and . Consider a selfmap . If an element exists such that , then we say that is a fixed point for . In this manuscript, we essentially represent the set by . The selfmap is called contraction [1] if
The selfmap is called nonexpansive if (1) holds for the value . In 1965, Browder [2] and Gohde [3] proved a fixed point theorem for a nonexpansive map under the restriction that is a uniformly convex Banach space (UCBS) and is bounded as well as closed and convex.
In [4], Zhang provided the following class of mappings.
Definition 1. Let be a subset of a Banach space. A selfmap is called mean nonexpansive if for all there are non-negative real numbers such that , we have
Zhang [4] provided an existence of fixed point result for mean nonexpansive mappings in Banach space setting under the normal structure assumption. After this, Wu and Zhang [5] and Zuo [6] investigated some other elementary properties and fixed point results for these mappings. In [7], Zhou and Cui used Ishikawa [8] iteration for approximating fixed points for these maps. The main aim here is to suggest some weak and strong convergence theorems for these mappings under the Picard-Mann hybrid [9] iteration and to show by a new example of mean nonexpansive maps that it converges better than the Ishikawa [8] and Mann [10] iteration processes.
Remark 2. It is easy to observe that each nonexpansive mapping is mean nonexpansive. Once again, in this research, we shall provide a new example to show that the converse is not true in general, that is, the class of mean nonexpansive maps properly includes the class of nonexpansive maps.
In the following example, is mean nonexpansive but not nonexpansive.
Example 3 (see [6]). Consider and set by
The Banach [1] celebrated fixed point theorem suggests the existence and uniqueness of a fixed point for a self contraction under the restriction that is complete metric space and is closed. Also this theorem essentially uses the Picard iteration [11] for finding this unique fixed point. Nevertheless, in the case of nonexpansive maps and hence for generalized nonexpansive maps, the Picard iteration fails to converge in the associated fixed point set. For some more literature on iterative schemes, please cite the work in [12–14]. Assume that is any nonempty subset of a Banach space and .
The Picard [11] iterative process is stated as:
Mann [10] iterative process is stated as: where .
Ishikawa [8] iteration process may be viewed as a two-step Mann iteration, stated as follows: where .
Khan [9] introduced the Picard–Mann hybrid iteration as follows: where .
Khan [9] provided the weak and strong convergence of the scheme (7) for the class of nonexpansive operators. Furthermore, he proved that the Picard–Mann hybrid iteration process is more effective than the Picard (4), Mann (5) and Ishikawa (6) iteration processes in the setting of nonexpansive maps. In this paper, we connect this scheme with the class of mean nonexpansive mappings, and in this way, we extend his results in the more general setting of mean nonexpansive mappings.
Now we provide some elementary definitions and results, which will be used in sequel.
Definition 4 [15]. A Banach space is called UCBS if and only if for every choice of , one has a such that for every in , whenever and
Definition 5 [16]. A Banach space is said to satisfy the Opial’s property if any weakly convergent sequence in which admits a weak limit , one has
Definition 6 [17]. Suppose is any subset of a Banach space . A selfmap is said to be endowed with the condition if and only if a function exists sucht that and for all and for each .
The following lemma holds, which suggests many examples of mean nonexpansive mappings.
Lemma 7. If is a selfmap and nonexpansive on a subset of a Banach space. Then is mean nonexpansive.
From the definition of mean nonexpansive maps, we have the following facts.
Lemma 8. If is a selfmap and mean nonexpansive on a subset of a Banach space . Then is closed. Moreover, if is strictly convex and is convex, then is also convex.
Theorem 9 [6]. Let be a subset of a reflexive Banach space (RBS) having Opial property. Let be a mean nonexpansive mapping. If be such that.
(a0) converges weakly to ,
(b0) ,
then .
Any UCBS can be characterized by the following way.
Lemma 10 [18]. If is a UCBS and If and are two sequences in such that , and for some and . Then, .
2. Main Results
The following results are the main outcome of this section. Notice that all these results improve and extend some main results of Khan [9] from the case of nonexpansive maps to case of mean nonexpansive maps.
Lemma 11. Let be a convex closed subset of a UCBS and be a mean nonexpansive mapping and . Assume that is a sequence of Picard–Mann hybrid iterative process (7). Consequently exists for each .
Proof. Let . Then using (7), we have This implies that We have showed that . It follows that is nonincreasing and bounded. Thus exists for each .☐
Theorem 12. Let be a convex closed subset of a UCBS and be a mean nonexpansive mapping and . Assume that is a sequence of Picard–Mann hybrid iterative process (7). Consequently, is bounded in with the property .
Proof. Since the set is nonempty so we may choose any . By Lemma 11, exists and is bounded. Suppose that By looking in the proof of Lemma 11, one see Now It follows that Again by looking in the proof of Lemma 11, one see It follows that From (12) and (16), we obtain From (17), we have Hence, Now from (11), (14) and (19) together with Lemma 10, we obtain ☐
We now provide a weak convergence theorem under the assumption of the Opial’s condition.
Theorem 13. Let be a convex closed subset of a UCBS and be a mean nonexpansive mapping and . Assume that is a sequence of Picard–Mann hybrid iterative process (7). If has the Opial property, then converges weakly to a point of of .
Proof. By Theorem 12, the sequence is bounded and . Since is UCBS, it follows that is RBS. Thus one has a weakly convergent subsequence of exists with some weak limit . By Theorem 9, . Next we show that is weakly convergent to . We may suppose that is not weakly convergent to , that is, one has a weakly convergent subsequence of with a weak limit and . Again applying Theorem 9, . By applying Opial’s condition and keeping Lemma 11 in mind, it follow that Hence we have seen a contradiction. Accordingly, we have . Thus, converges weakly to .☐.
The strong convergence theorem under the assumption of compactness is established as follows.
Theorem 14. Let be a convex compact subset of a UCBS and be a mean nonexpansive mapping and . Assume that is a sequence of Picard–Mann hybrid iterative process (7). If is compact, then converges strongly to an element of .
Proof. Since is compact, and . One can choose a strongly convergent subsequence of such that . Now we show that . For this ☐
Consequently, we obtained
According to Theorem 12, we have , so applying , we obtain . This shows that . By Lemma 11, exists. Consequently, is the strong limit of and element of .
The strong convergence theorem without the compactness assumption is established as follows.
Theorem 15. Let be a convex closed subset of a UCBS and be a mean nonexpansive mapping and . Assume that is a sequence of Picard–Mann hybrid iterative process (7). Then converges strongly to an element of if and only if .
Proof. The necessity is obvious.
Conversely, suppose that and . From the Lemma 11, exists. Therefore , by assumption. We prove that is a Cauchy sequence in . As , for a given , there exists such that for each ,
In particular . Therefore there exists such that
Now for ,
This shows that is a Cauchy sequence in . As is closed subset of a Banach space , so there exists a point such that . Now gives that . Since from Lemma 8, we have the set a closed set in . Hence .☐.
The below facts are essentially due to Sentor and Dotson [17].
Definition 16. Let be a subset of a Banach space . A selfmap is said to be endowed with the condition if and only if a function exists such that and for all and for each .
The strong convergence theorem under the assumption of condition is established as follows.
Theorem 17. Let be a convex closed subset of a UCBS and be a mean nonexpansive mapping and . Assume that is a sequence of Picard–Mann hybrid iterative process (7). If is endowed with condition , then converges strongly to an element of .
Proof. From Theorem 12, we have Condition (I) of provides Now all the requirements of the Theorem 15 are available, so we conclude that converges strongly to an element of .☐
Remark 18. In the view of Lemma 7, our results contains the case of nonexpansive mappings.
3. Example
Now we want to provide a new example of mean nonexpansive maps.
Example 19. Let be defined by if and for . Suppose . Then it is easy to show
Now we may consider the following cases.
(i)Suppose that . Then(ii)Suppose that . Then(iii)Suppose that and (iv)Suppose that and
Thus we conclude that is mean nonexpansive. Notice that, when and , then and so is not nonexpansive. Choose and , then the strong convergence and effectiveness of the Picard-Mann hybrid iteration can be seen in Table 1 and Figure 1.
4. Conclusions
We begun the finding of fixed points for mean nonexpansive operators in Banach spaces under the Picard–Mann hybrid iterative process. Some convergence results are established under different assumptions. It has been showen by an example that the notion of mean nonexpansive maps is properly more general than the notion of nonexpansive maps. Also, the Picard-Mann hybrid iterates of this example converge faster than the Picard, Ishikawa and Mann iterates. In particular, our results essentially improve and extended the results Khan [9] from the setting of nonexpansive operators to the larger frame work of mean nonexpansive operators.
Data Availability
No data were used to support this study.
Conflicts of Interest
We have no conflict of interest.
Authors’ Contributions
Every author listed in this manuscript has contributed equally to each part.
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.