Unique and Non-Unique Fixed Points and their ApplicationsView this Special Issue
Iterative Approximation of Fixed Points by Using Iteration Process in Banach Spaces
We connect the iteration process with the class of generalized -nonexpansive mappings. Under some appropriate assumption, we establish some weak and strong convergence theorems in Banach spaces. To show the numerical efficiency of our established results, we provide a new example of generalized -nonexpansive mappings and show that its iteration process is more efficient than many other iterative schemes. Our results are new and extend the corresponding known results of the current literature.
1. Introduction and Preliminaries
Once an existence of a solution for an operator equation is established then in many cases, such solution cannot be obtained by using ordinary analytical methods. To overcome such cases, one needs the approximate value of this solution. To do this, we first rearrange the operator equation in the form of fixed-point equation. We apply the most suitable iterative algorithm on the fixed point equation, and the limit of the sequence generated by this most suitable algorithm is in fact the value of the desired fixed point for the fixed point equation and the solution for the operator equation. The Banach Fixed Point Theorem  (BFPT, for short) suggests the elementry Picard iteration in the case of contraction mappings. Since for the class of nonexpansive mappings, Picard iterates do not always converge to a fixed point of a certain nonexpansive mapping, we, therefore use some other iterative processes involving different steps and set of parameters. Among the other things, Mann , Ishikawa , Noor , iteration of Agarwal et al. , SP iteration of Phuengrattana and Suantai , iteration of Karahan and Ozdemir , Normal-S , Picard-Mann hybrid , Krasnoselskii-Mann , Abbas , Thakur , and Picard-S  are the most studied iterative processes. In 2018, Ullah and Arshad introduced  iteration process for Suzuki mappings and proved that it converges faster than all of these iteration processes.
Very recently, Ali and Ali  introduced the novel iteration process, namely, iterative scheme for generalized contractions as follows: where .
They showed that the iteration (1) is stable and has a better rate of convergence when compared with the other iterations in the setting of generalized contractions.
Definition 1. Let . Then is said to be (i)nonexpansive provided that , for every two (ii)endowed with condition (C) provided that implies , for every two (iii)generalized -nonexpansive provided that implies , for every two and (iv)endowed with condition  if one has a nondecreasing function such that and at and for all
In 1965, Browder  and Gohde  are in a uniformly convex Banach space (UCBS), while Kirk  in a reflexive Banach space (RBS) established an existence of fixed point for nonexpansive maps. In 2008, Suzuki  showed that the class of maps endowed with condition is weaker than the notion of nonexpansive maps and proved some related fixed point theorems in Banach spaces. In 2017, Pant and Shukla  proved that the notion of generalized -nonexpansive maps is weaker than the notion of maps endowed with condition . They proved some convergence theorems using Agarwal iteration  for these maps. Very recently, Ullah et al.  used iteration for finding fixed points of generalized -nonexpansive maps in Banach spaces. In this paper, we show under some conditions that iteration converges better to a fixed point of generalized -nonexpansive map as compared to the leading iteration and hence many other iterative schemes.
Definition 2. Select a Banach space such that is nonempty and is bounded. We set for fix the following.
asymptotic radius of the bounded sequence at the point by ;
asymptotic radius of the bounded sequence with the connection of by ;
asymptotic center of the bounded sequence with the connection of by .
Definition 3 (see ). A Banach space is called with Opial’s condition in the case when every sequence which is weakly convergent to , then one has the following Pant and Shukla  observed the following facts about generalized -nonexpansive operators.
Proposition 4. If is a Banach space such that is closed and nonempty, then for and , the following hold (i)If is endowed with condition , then is generalized -nonexpansive(ii)If is generalized -nonexpansive endowed with a nonempty fixed point, then for and is a fixed point of (iii)If is generalized -nonexpansive, then is closed. Furthermore, when the underlying space is strictly convex and the set is convex, then the set is also convex(iv)If is generalized -nonexpansive, then for every choice of (v)If the underlying space is with Opial condition, the operator is generalized -nonexpansive, is weakly convergent to and , then We now state an interesting property of a UCBS from .
Lemma 5. Suppose is any UCBS. Choose and such that , , and for some . Then, consequently, .
2. Main Results
We first provide a very basic lemma.
Lemma 6. Suppose is any UCBS and is convex nonempty and closed. If is generalized -nonexpansive operator satisfying with and is a sequence of iterates (1), then, consequently, one has always exists for every taken .
Proof. We may take any . Using Proposition 4(ii), we see that
This implies that
Consequently, , that is, is bounded as well as nonincreasing. This follows that exists for each .
We now provide the necessary and sufficient requirements for the existence of fixed points for any given generalized nonexpansive mappings in a Banach space.
Theorem 7. Suppose is any UCBS and is convex nonempty and closed. If is generalized -nonexpansive operator and is a sequence of iterates (1). Then, if and only if is bounded and .
Proof. Suppose that and . Take any , and so applying Lemma 6, we have exists and is bounded. Suppose that this limit is equal to some , that is, As we have established in the proof of Lemma 6 that This together with (6) gives that Since is in the set , so we may apply Proposition 4(ii) to obtain the following Now, if we look in the proof of Lemma 6, we can see the following From (8) and (10), we have By (11) and (1), one has If and only if One can now apply the Lemma 5, to obtain Conversely, we want to show that the set is nonempty under the assumptions that is bounded such that . We may choose a point . If we apply Proposition 4(iv), then one can observe the following We observed that . By using the facts that this set has only element in the case of UCBS , one concludes , accordingly the set is nonempty.
The weak convergence of iteration is established as follows.
Theorem 8. Suppose is any UCBS with Opial condition and is convex nonempty and closed. If is generalized -nonexpansive operator with and is a sequence of iterates (1). Then, consequently, converges weakly to a fixed point of .
Proof. By Theorem 7, the given sequence is bounded. Since is UCBS, is RBS. Therefore, some one construct a weakly convergent sequence of . We may assume that be this subsequence having weak limit . If we apply Theorem 7 on this subsequence, we obtain . Thus, by Proposition 4(v), one has . It is sufficient to show that converges weakly to . In fact, if does not converge weakly to . Then, there exists a subsequence of and such that converges weakly to and . Again by Proposition 4(v), . By Lemma 6 together with Opial property, we have This is a contradiction. So, we have . Thus, converges weakly to .
Now we provide some strong convergence results.
Theorem 9. Suppose is any UCBS and is convex nonempty and compact. If is generalized -nonexpansive operator with and is a sequence of F iterates (1). Then, consequently, converges strongly to a fixed point of .
Proof. Since the domain is a compact subset of and . It follows that a subsequence of exists such that for some . In the view of Theorem 7, . Applying Proposition 4(iv), one has Hence, if we let , then . The fact that is the strong limit of now follows from the existence of .
Theorem 10. Suppose is any UCBS and is convex nonempty and closed. If is generalized -nonexpansive operator with and is a sequence of F iterates (1) and . Then, consequently, converges strongly to a fixed point of .
Proof. By using Lemma 6, one has exists, for every fixed point of . It follows that exists. Accordingly The above limit provides us two subsequence and of and , respectively, in the following way By looking into the proof of Lemma 6, we see that is nonicreasing, therefore It follows that Consequently, we obtained that which show that is Cauchy sequence in and so it converges to an element . Applying Proposition 4(iii), is closed and so . By Lemma 6, exists and hence is the strong limit of .
Theorem 11. Suppose is any UCBS and is convex nonempty and closed. If is generalized -nonexpansive operator satisfying condition with and is a sequence of iterates (1). Then, consequently, converges strongly to a fixed point of .
Proof. Keeping Theorem 7 in mind, one can write From the definition of condition (I), we see that Applying (22) on (23), we have It follows that Now applying Theorem 10, is strongly convergent to a fixed point of .
To support the main results, we provide an example of generalized -nonexpansive mappings, which is not endowed with condition (). Using this example, we compare with other iterations in the setting of generalized -nonexpansive mappings.
Example 12. 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 I. When , we have
Case II. Choose , we have
Case III. When and , we have Consequently, for every two points . Now if one chooses and , we must have , and . It has been observed, and . Thus, exceeded the class of Suzuki mappings.
We now compare the effectiveness of the iterative scheme  with the leading  and Picard  and the elementry , Ishikawa  and Mann  approximation scheme. We may take and . For the strating , we can see some values in Table 1. Furthermore, Figure 1 provides information about the behavior of the leading schemes. Clearly, iterative scheme is more effective than the other schemes in the general context of generalized -nonexpansive maps.
Remark 13. In the view of the above discussion, we noted that the main theorems and outcome of this paper improved and extended the main results of Ullah and Arshad  from Suzuki mappings to generalized -nonexpansive mappings and from the setting of iteration to the more general setting of iteration process. Moreover, the main results of this paper improved the results of Ali and Ali  from the setting of contractions to the general context of generalized -nonexpansive mappings. We have also improved the results of Ullah et al.  in the sense of better rate of convergence.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
The authors are grateful to the Basque Government for its support through grant IT1207-19.
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