Abstract
In this article, we considered the class of generalized -nonexpansive (GABN) mappings that properly includes all nonexpansive, Suzuki nonexpansive (SN), generalized -nonexpansive (GAN), and Reich–Suzuki nonexpansive (RSN) mappings. We used the iterative scheme JA for finding fixed points of these mappings in a Banach space setting. We provided both weak and strong convergence results under some mild conditions on the mapping, domain, and on the parameters involved in our iterative scheme. To support these results numerically, we constructed a new example of GABN mappings and proved that the JA iterative scheme converges to its fixed point. Moreover, we proved that JA iterative scheme converges faster to the fixed point corresponding to the some other iterative schemes of the literature. Eventually, we carried out an application of our main outcome to solve a split feasibility of problems (SFPs) in the setting of GABN mappings. Thus, our results were new in the literature and improved well-known results of the literature.
1. Introduction
We know that some time many analytical methods fail to find exact solution of the problems, therefore, fixed point theory recommends some alternate techniques for solving these problems. First, we expressed the solution of the problem in the fixed point of a certain map (the map may be contraction, nonexpansive, or generalized nonexpansive [1]). In this situation, an existence of a solution and the existence of a fixed point have the same meanings. We used some suitable iterative methods to find approximate unique fixed point (see, e.g., [2] and others). The Banach Contraction Principle (BCP) indicates, among other things, that if the operator is a contraction and the subset is a closed subset of a Banach space, there may be a single fixed point. As an operator on a subset of a Banach space is called a contraction if, for all , we have the following equation:where .
A nonexpansive mapping is a mapping that satisfies equation (1) for . Actually, the Picard iteration method was suggested by the BCP proof to determine the estimated value of the unique fixed point of the contraction . If is closed bounded and convex and is a uniformly convex Banach space (UCBS), then has a fixed point (see, e.g., Kirk [3], Browder [4], and Gohde [5]). In applied sciences, nonexpansive mappings have a key role for solving fixed points problems. Therefore, we tried to use some extensions of these mappings.
Suppose is a self-map, that is, and , where is any subset of a Banach space, then is called as follows:(a)SN [6] if (b)GAN [7] (GAN, for short) if (c)RSN [8] (RSN, for short) if
The classes of RSN and GAN self-maps properly include the class of SN self-maps. It is very natural to ask that whether there exists any class of mappings that includes all the classes of mappings mentioned above.
To answer the above question in affirmative, Ullah et al. [1] presented a new class of nonlinear mappings that includes all the above mappings.
Definition 1. (see [1]). A self-map on a subset of a Banach space is said to be GABN when, for all , one can find two real constants such that such thatThe class of GABN mappings includes all these mappings, and thus, the concept of GABN mappings is more difficult but more important than the other mappings mentioned above. The purpose of this work is to carry out some new fixed point convergence results under an effective iterative scheme. Using these results, we show that a SFP can be solved. Eventually, our main results will be supported by a numerical example. Consequently, our results were new in the fixed point literature and improved the well-known results of many authors.
2. Preliminaries
The purpose of this section is to present some earlier results and definitions that will be used in our main outcome.
Definition 2. (see [9]). If denotes any weakly convergent sequence in , then a Banach space is said to be with Opial’s condition if, for all , we have the following equation:where is the weak limit of .
Definition 3. (see [10, 11]). Suppose denotes any bounded sequence in a closed convex subset of a UCBS . In this case, one denotes and defines the asymptotic radius of on the set by , while the asymptotic center of on the set is denoted and defined as . Moreover, the set contains only one element.
Now, we present some propositions and lemmas to understand the characterize of GABN maps.
Proposition 1. Let be a nonempty subset of a Banach space B and be a self-map. Then,(a)When is SN, then it follows that is generalized (0,0)-nonexpansive(b)When is GAN, then it follows that is generalized -nonexpansive(c)When is –RSN, then it follows that is generalized -nonexpansiveThe following key results are from [1].
Lemma 1. Suppose that form a GABN self-map on a subset of a Banach space . Then,(i)If admits at least one fixed point, then for all and for all .(ii)By choosing in , the following estimate holds:(iii)If satisfies Opial’s condition, then one has the following equation:
The following lemma is well-known in the literature and can be found in [12].
Lemma 2. Suppose be a UCBS and for all . Considering and sequences in UCBS satisfying the conditions , , and for any real number , we have .
3. Convergence Theorems in UCBS
Unlike contractions, the Picard iteration is not suitable for nonexpansive mappings [13]. There are simple examples in the literature that shows that if a nonexpansive mapping has a unique fixed point, then it is possible that the Picard may not converge to the fixed point. Thus, in the case of nonexpansive mappings, we studied iterative schemes that are more general than the Picard iteration. This section thus employs the JA iterative scheme of Abdeljawad et al. [14] to provide certain weak and strong convergence results for the class of generalized -nonexpansive mapping in the linear setting of a UCBS. For , we collect some iterative processes of the literature as follows.
In 1953, Mann [15] suggested the following iteration:
In 1974, Ishikawa [16] generalized Mann iteration (6) to the setting of two steps as follows:
In 2000, Noor [16] generalized Ishikawa iteration (7) to the setting of three steps as follows:
In 2007, Agarwal et al. [17] suggested a new two-step iteration called S-iteration process as follows:
In 2014, Abbas and Nazir [18] introduced the following new three-step iteration as follows:
In 2016, Thakur et al. [19] proposed the following new iteration:
In 2018, Ullah and Arshad [20] suggested M-iteration as follows:
Recently, Abdeljawad et al. [14] constructed a novel iteration as follows:
When we compute a fixed point of a certain nonlinear self-map, we try to use a faster iterative scheme in order to obtain a high accurate result in fewer steps of iterations. Agarwal et al. [17] used iterative scheme (9) for contractions and nonexpansive mappings and proved scheme (9) in this case is better than the Mann iteration (6). However, we note that Agarwal iteration (9) is a two-step iteration and it is known that three-step iteration is better than the one- and two-step iterations. Thus, in [18], Abbas and Nazir constructed a three-step iteration and proved that this new three-step iterative scheme converges better from the Agarwal iteration for nonexpansive mappings. After this, Thakur et al. [19] constructed iterative scheme (11) and proved that this scheme is better than the Abbas iteration for SN mappings. Thakur et al. [19] results were recently improved by Ullahn and Arshad [20] and Abdeljwad et al. [14] by constructing iterative schemes (12) and (13), respectively. In this article, we improved the results in [14, 19, 20] to the general context of GABN mappings.
Lemma 3. Suppose that is a GABN self-map on a convex closed subset of a UCBS. If the fixed point set is nonempty and is produced from JA iteration (13), then, subsequently, exists for each choice of in .
Proof. Fixing any point and then applying Lemma 1 (i), one has the following equation:which implies thatHence, is bounded below and nonincreasing, so exists for every .
Theorem 1. Suppose that is a GABN self-map on a convex closed subset of a UCBS. Let be produced from the JA iteration (13). Then, subsequently, the fixed point set is nonempty if and only if is bounded in and satisfies the equation .
Proof. Let and . Then, by Lemma 3, exists and is bounded. We putBy the proof of Lemma 3 together with equation (16), we obtained the following equation:By Lemma 1 (i), we get the following equation:Also, by the proof of Lemma 3, we get the following equation:Thus,So, we obtained . Therefore,From equations (19) and (21), we get the following equation:From (22), we get the following equation:Thus,Since for all , so using equation (16), (18), and (24) together with Lemma 1, we get the following equation:Conversely, suppose that is bounded and and . By Lemma 1 (ii), we have the following equation:So, . As is a UCBS, therefore, consists of a single point. Hence, , that is, the fixed point set of is nonempty.
Theorem 2. Suppose that is a GABN self-map on a convex subset of a UCBS. If the fixed point set is nonempty and is produced from JA iteration (13), then, subsequently, converges (strongly) to some fixed point of if the set is compact.
Proof. Since the subset is convex and compact, we have a subsequence of the iterative sequence that we may denote by such that , where is a point in . We prove that is a fixed point of and strong limit of the iterative sequence and that . For this, we used Theorem 1 and, hence, we get . Hence, applying Lemma 1 (ii), one has the following equation:Thus, , so become a fixed point of . Using Lemma 3, exists. It now clear that is the strong limit of the iterative sequence . Hence, the proof is finished.
The next result does not require the compactness condition. This result is valid in general Banach space setting.
Theorem 3. Suppose that is a GABN self-map on a convex closed subset of a Banach space. If the fixed point set is nonempty and is produced from JA iteration (13), then, subsequently, converges (strongly) to some fixed point of if .
Definition 4. (see [21]). Suppose that is a self-map on a convex closed subset of a UCBS. Then, is said to be with condition (I) if there is a function with and for all and all imply that for all .
Now, the strong convergence using condition (I) of is as follows.
Theorem 4. Suppose that is a GABN self-map on a convex closed subset of a UCBS. If the fixed point set is nonempty and is produced from the JA iteration (13), then, subsequently, converges (strongly) to some fixed point of if is with condition (I).
Proof. It follows from Theorem 1 thatAs satisfies condition , soFrom equation (28), we get the following equation:Using the properties of , we get the following equation:Subsequently, we carry out all the requirements for Theorem 3, and hence, converges strong to some fixed point of .
The last result of this section is the following weak convergence that is obtained under Opial’s condition of the underlying domain.
Theorem 5. Suppose that form a GABN self-map on a convex closed subset of a UCBS . If the fixed point set is nonempty and is produced from JA iteration (13), then, subsequently, converges (weakly) to some fixed point of if is with Opial’s condition.
Proof. We notice that is reflexive because UCBS is always reflexive. Now, since the sequence of iterates is bounded in and so it has a convergent (weakly) subsequence, we denote it here by and its weak limit is denoted here by . However, using Theorem 1, one has . Thus, applying Lemma 1 (iii), the point becomes the fixed point for . We now show that is a weak limit for and this will finish the proof. For this purpose, we assume on the contrary that is not a weak limit for and so we can find a new subsequence, namely, of such that is a weak convergent to some that is different from . As shown earlier, we can show that is a fixed point of . Now, using Opial’s condition of , it follows thatThe above strict inequality suggests a contradiction because . Thus, we must accept that the sequence of iterates converges weakly to .
4. Example
Now, we offer a numerical example of GABN self-map as follows.
Example 1. We construct a self-map using the rule if and when . The first target is to show that is GABN on its domain. We set . Then, we have three cases.
If , then we have the following equation:If , then we have the following equation:If and , then we have the following equation:Hence, is GABN with . However, for and , we have . However,(i)(ii)(iii)From (i)–(iii), we see that does not belong to the classes of SN, GAN, and RSN mappings. We now set and and compare the high accuracy our JA iteration in Table 1 and Figure 1. Clearly, our JA iterative scheme requires less steps to reach the fixed point.
(a)
(b)
5. Application to Split Feasibility Problems
Now, we consider Hilbert spaces, namely, and with and , compact and convex. If the operator bounded and linear, the SFP [22] is defined as follows:
In our study, we target SFP (36) and assume that its solution set is nonempty. In [23], it is shown that any point solves our SFP (36) iff solves essentially the following equation:where the set (resp. the set ) is well-known in the literature and denotes the nearest point projection onto the set (resp. onto the set ) and and the mapping denotes the adjoint operator of . Using the concept of nonexpansive self-maps, Byrne [24] was the first who proved that if with , then the self-mapis eventually nonexpansive, and its iterative scheme is given by the following formula:which converges (weakly) to some point of .
It is not always an easy task to improve a weak convergence result to the setting of strong convergence. In previous literatures, authors used nonexpansive mappings for establishing a weak converges for SFPs. Our alternative approach in this article is to provide a strong convergence for SFPs using the concept of GABN mappings and new general projection type iterative scheme.
Theorem 6. We now assume SFP (36) such that its solution set is nonempty and and is a GABN self-map. In this case, the sequence of iterates (13) converges (strongly) to some solution of SFP (36).
Proof. We take , that is, the map is generalized -nonexpansive. In the view of Theorem 2, converges in the strong sense to some point of . Since , it follows that converges strongly to some solution of SFP (36). This finishes the proof.
6. Conclusions
The following new results are obtained in this research:(a)The JA iterative scheme is used for finding fixed points of GABN mappings(b)Both weak and strong convergence results are obtained under possible mild assumptions(c)We constructed a new example of GABN mappings(d)We proved that JA iterative scheme is more effective than the other schemes of the literature(e)We carried out an application of our work to solve a SFP in the setting of GABN mappings(f)Accordingly, we improved and extended some recent and old results due to various authors in [14, 19, 20] from nonexpansive, SN, and GAN mappings to the general class of GABN mappings [25–27].
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
The authors contributed equally.
Acknowledgments
This work was funded by the University of Jeddah, Jeddah, Saudi Arabia, under grant no. UJ-22-DR-66. The authors, therefore, acknowledge the University of Jeddah for its technical and financial support.