Abstract
The purpose of this paper is to introduce a new four-step iteration scheme for approximation of fixed point of the nonexpansive mappings named as -iteration scheme which is faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Ullah iteration schemes. We show the stability of our proposed scheme. We present a numerical example to show that our iteration scheme is faster than the aforementioned schemes. Moreover, we present some weak and strong convergence theorems for Suzuki’s generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. Our results extend, improve, and unify many existing results in the literature.
1. Introduction
Most of the nonlinear equations can be transformed into a fixed point problem as follows:where is a self-map on a certain distance space and the solution of the aforementioned equation is considered as a fixed point of the mapping . Banach [1] proved that if a self-map on a complete metric space is such thatfor , then it possesses a unique fixed point . Moreover, the iterative processcalled the Picard iteration process, converges to . It is worth mentioning that Picard iteration process is useful for the approximation of the fixed point of the contraction mappings but the case when ones dealing with nonexpansive mappings it may fail to converge to the fixed point even if has a unique fixed point. Krasnosel’skii [2] showed that Mann [3] iteration process can approximate the fixed points of a nonexpansive mapping. In this iteration scheme, the sequence is generated by an arbitrary aswhere is in .
In 1974, Ishikawa [4] developed an iterative scheme to approximate the fixed point of nonexpansive mappings, where is defined iteratively starting from byfor all , where are in .
For the approximation of the fixed point of nonexpansive mappings, Mann and Ishikawa iterative methods have been studied by several authors (see e.g., [5–9]).
Another iteration scheme was proposed by Noor [10] in 2000, for , the sequence is defined byfor all , where , and are in .
Agarwal et al. [11], in 2007, proposed the following iterative scheme: for arbitrary , a sequence is generated byfor all , where are in . They proved that this procedure converges faster than Mann iteration for contraction mappings.
In 2014, Abbas and Nazir [12] developed an iterative scheme which is faster than Agarwal et al.’s [11] scheme, where a sequence is formulated from arbitrary byfor all , where , and are in .
Later in 2016, Thakur et al. [13] developed the following iterative procedure, where a sequence is generated iteratively by arbitrary andfor all , where , and are in .
Recently, in 2018, Ullah and Arshad developed a new iteration process which converges faster than all the aforementioned process, where the sequence is constructed by taking arbitrary andfor all , where , and are in .
Our aim is to introduce a new faster iteration process than those mentioned above and to prove the convergence results for Suzuki’s generalized nonexpansive mappings in the context of uniformly convex Banach spaces. We also show that our process is stable analytically. Numerically, we compare the rate of convergence of our iteration process with the existing iteration processes.
2. Preliminaries
Throughout this paper, is a nonempty closed convex subset of a uniformly convex Banach space , denotes the set of all positive integers and denotes the set of all fixed points of , that is,
Definition 1. (see [14]). A Banach space is said to be uniformly convex if for each , there exists a such that for all ,
Definition 2. (see [15]). A Banach space is said to satisfy Opial property if for each sequence in , converging weakly to , we havefor all such that .
Definition 3. A mapping is called a contraction if there exists , such that
Definition 4. A mapping quasi-nonexpansive if for all and and , we have
Definition 5. (see [16]). A mapping is called Suzuki’s generalized nonexpansive mapping if for all , we haveSuzuki [16] proved that the generalized nonexpansive mapping is weaker than nonexpansive mapping and stronger than quasi-nonexpansive mapping and obtained some fixed points and convergence theorems for Suzuki’s generalized nonexpansive mappings. Recently, many authors have studied fixed-point theorems for Suzuki’s generalized nonexpansive mapping (see, e.g., [17]).
Senter and Dotson [7] introduced a class of mappings satisfying condition .
Definition 6. A mapping is said to satisfy condition , if there exists a nondecreasing function with and for all such that , for all , where .
Proposition 1 (see [16]). Let be any mapping. Then,(i)If is nonexpansive, then is a Suzuki’s generalized nonexpansive mapping.(ii)If is a Suzuki’s generalized nonexpansive mapping and has a fixed point, then is a quasi-nonexpansive mapping.(iii)If is a Suzuki generalized nonexpansive mapping, then
Lemma 1 (see [16]). Suppose is Suzuki’s generalized nonexpansive mapping satisfying Opial property. If converges weakly to and , then .
Lemma 2 (see [16]). Let be a uniformly convex Banach space and be a weakly convex compact subset of . Assume that is Suzuki’s generalized nonexpansive mapping. Then, has a fixed point.
Lemma 3 (see [18]). Let be a uniformly convex Banach space and be any real sequence such that for all . Suppose that and be any two sequences of such that , , and hold for some . Then, .
Definition 7. (see [19]). Let be a Banach space and be a nonempty closed convex subset of . Assume that is a bounded sequence in . For , we set . The asymptotic radius of relative to is the set and the asymptotic center of relative to is given by the following set:It is known that, in a uniformly convex Banach space, consists of exactly one point.
Definition 8. (see [20]). Let be a Banach space and . Suppose that and define an iteration procedure which gives a sequence of points in . Assume that converges to the fixed point . Suppose be a sequence in and be a sequence in given by . Then, the iteration procedure defined by is said to be -stable or stable with respect to if
Definition 9. (see [21]). Let be a Banach space and . Then, is called a contractive mapping on if there exist , such that for each ,By using (7), Osilike [21] established several stability results most of which are generalizations of the results of Rhoades [22] and Harder and Hicks [23].
Definition 10. (see [24]). Let be a Banach space and . Then, is called a contractive mapping on if there exist and a monotone increasing function with , such that for each ,
Lemma 4. (see [25]). If is a real number such that , and is the sequence of positive numbers such thatthen for any sequence of positive numbers satisfyingwe have
3. S∗-Iteration Process
Throughout this section, be a nonempty set of a Banach space , and for all , , , and are real sequences in the interval .
We generate the sequence iteratively, taking arbitrary , by
First, we show that -iteration scheme (25) converges faster than all aforementioned iteration schemes for contractive mappings due to Berinde [26] and is stable.
4. Convergence and Stability Results of -Iteration Process
First, we establish convergence results for -iteration process:
Theorem 1. Let be a Banach space and be a nonempty closed convex subset of . Let be a nonexpansive self mapping on , be a sequence defined by (25), and . Then, exists for all .
Proof. Let for all . From (16), we haveThus,Hence, exists for all .
Theorem 2. Let a uniformly convex Banach space and be a nonempty closed convex subset of . Let be a nonexpansive mapping. Suppose that is defined by the iteration process (25) and . Then, the sequence converges to a point of if and only if where .
Proof. Necessity is obvious. Suppose that . As proved in Theorem 1, exists for all , so exists and by assumption. Now, we will prove that is a Cauchy sequence in . For given , there exists such that for all ,In particular, . Hence, there exists such that . Now for all ,which shows that is a Cauchy sequence in . But is a closed subset of , so there exists such that . Now, gives which implies .
Next, we prove that our iteration process is -stable or stable with respect to .
Theorem 3. Let be a Banach space and be a mapping satisfying (21). Suppose has a fixed point . Let be a sequence in satisfying (9). Then, -iteration process (9) is -stable.
Proof. Let be an arbitrary sequence in and the sequence generated by (25) is converging to a unique fixed point and . We will prove that . Assume that andSince and , , , and are in ,Hence by Lemma 4, we have , which gives . On the other hand, suppose that . Then,Taking limit as in (34), we get .
Now, we present an example to compare the rate of convergence of our iteration scheme with others.
Example 1. Let and . Let be a mapping defined by for all . For and , .From Table 1, we can see that all the iteration procedures are converging to . Clearly, our iteration process requires the least number of iteration as compared to other iteration schemes.
In Figure 1, black curve represents our iteration process. The graphical view shows that our iteration process requires less number of iterations as compared to the other iteration processes. The number of iterations in which these processes attain the fixed point is given in Table 2:
5. Some Convergence Results for Suzuki’s Generalized Nonexpansive Mappings
This section contains some weak and strong convergence results for a sequence generated by -iteration process for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces.
Lemma 5. Suppose that be a nonempty closed convex subset of a Banach space . Let be a Suzuki generalized nonexpansive mapping with . For , the sequence generated by -iteration process, exists for all .
Proof. Result follows from Proposition 1 and Theorem 1.
Lemma 6. Suppose that be a nonempty closed convex subset of a uniformly Banach space . Let be a Suzuki’s generalized nonexpansive mapping with . For arbitrarily chosen , the sequence is generated by -iteration process. Then, if and only if is bounded and .
Proof. Suppose that and let . Then, by Lemma 5, exists and is bounded. LetFrom (26) and (37), we haveBy the Proposition 1, we haveThis implies thatFrom equations (26) and (37) and Lemma 3, we haveConversely, assume that is bounded and . Suppose that . Using Proposition 1, we getThis shows that . Since is uniformly convex, is singleton. Thus, we have , that is, .
Theorem 4. (weak convergence theorem). Suppose that be a nonempty closed convex subset of a uniformly Banach space with the Opial property. Let be Suzuki’s generalized nonexpansive mapping. For arbitrarily chosen , let the sequence be generated by -iteration process with . Then, converges weakly to a fixed point of .
Proof. Since , by Lemma 6, the sequence is bounded and . Also, as is uniformly convex so is reflexive, thus by Eberlin’s theorem, there exists a subsequence of say which converges weakly to some . Now, since is closed and convex so by Mazur’s theorem . Hence, by Lemma 1, . We show that converges weakly to . On contrary, suppose that it is not true. Then, there must exist a subsequence of , say , such that converges weakly to with . Using Lemma 1, we have . Now, since exists for all . Using Lemma 6 and Opial property, we havewhich is a contradiction; hence, . This shows that converges weakly to a fixed point of .
Theorem 5. (strong convergence theorem). Suppose that be a nonempty closed convex subset of a uniformly Banach space . Let be a Suzuki’s generalized nonexpansive mapping. For arbitrarily chosen , let the sequence be generated by -iteration process with . Then, converges strongly to a fixed point of .
Proof. Using Lemma 2, we get and hence by Lemma 6, we have . By the compactness of , there exists a subsequence of , say , converging strongly to for some . Now by using Proposition 1, we getTaking limit , we get , that is, . By using Lemma 5, exists for all ; hence, converges strongly to .
Theorem 6. Suppose that be a nonempty closed convex subset of a uniformly Banach space . Let be a Suzuki’s generalized nonexpansive mapping. For arbitrarily chosen , the sequence be generated by -iteration process with . If satisfies condition , then converges strongly to a fixed point of .
Proof. By Lemma 5, exists for all ; hence, exists. Let for some . Now if , then there is nothing to prove. Suppose ; from condition and the hypothesis, we haveAs , by Lemma 5, we have . Hence, (46) implies thatSince is a nondecreasing function, by equation (47), we get . Thus, we have a subsequence of and a sequence in such thatFrom equation (48),Letting , we get . Hence, is a Cauchy sequence in , so it converges to . As is closed, and then converges strongly to . Since exists, we have . This completes the proof.
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 have contributed equally.
Acknowledgments
The authors are very grateful to the Basque Government for its support through Grant IT1207-19.