A New Iterative Method for Suzuki Mappings in Banach Spaces

In this paper, an efficient new iterative method for approximating the fixed point of Suzuki mappings is proposed. Some important weak and strong convergence results of the proposed iterative method are established in the setting of Banach space. An example illustrates the theoretical outcome.


Introduction and Preliminaries
roughout the present research, we shall write N and R to denote the set of natural numbers and real numbers set, respectively. We say that a self-map F of a subset M of a Banach space E � (E, ‖.‖) is called a contraction map whenever a real constant 0 ≤ r < 1 exists with the following property: (1) An element q ∈ M is said to be a fixed point of F provided that q � Fq. In this manuscript, the notation Fix(F) will throughout denote the fixed point set of F. e Banach-Caccioppoli fixed point theorem (see, e.g., [1,2] and others) states that any contraction mapping in the setting of complete metric spaces admits a unique fixed point q, and this q is, in fact, the limit of all the sequences u k obtained from the Picard iterates [3], that is, u k+1 � Fu k . However, one of the important classes of mappings in fixed point theory is the class of nonexpansive mappings. Notice that, F is called a nonexpansive mapping whenever (1) holds true for r � 1. In 1965, Browder [4] and Gohde [5] differently proved the same result concerning the existence of fixed points for nonexpansive mappings. Indeed, they suggested that any self-nonexpansive map F of M always admits at least one fixed point whenever M is assumed to be a bounded convex closed subset of some uniformly convex Banach space (UCBS). Nevertheless, the sequence u k defined by Picard iterates may not have a limit in the fixed point set associated with a nonexpansive map in general as shown in the next example. Let M � [0, 1] and set Fu � 1 − u; it is easy to see that F is a self-nonexpansive mapping on M having a unique fixed point 1/2. However, for any u 1 � u ≠ 1/2, we obtain the sequence of Picard iterates as follows: u, 1 − u, u, 1 − u, . . . which is an oscillating sequence and, hence, diverges in Fix(F) � 1/2 { }. For providing comparatively better convergence speed and to overcome such situations, different iterative methods have been suggested by some authors (cf. the works of Mann [6], Ishikawa [7], Noor [8], Agarwal et al. [9], Abbas and Nazir [10], akur et al. [11], and references therein).
In 2008, Suzuki [12] gained a big break through introducing an interesting extension of nonexpansive mappings as follows. We recall that a self-map F: M ⟶ M mapping with (C) property (also called Suzuki mapping) if the following fact is valid: One can easily notice that the Suzuki mappings satisfy the nonexpansive requirement for some elements of the domain. Hence, nonexpansive mappings obviously satisfy (C) property of Suzuki [12]. Interestingly, an example in [12] (see also an example below) nicely shows that there exist many mappings in the class of Suzuki mappings, which do not belong to the class of nonexpansive mappings. Suzuki also extended the celebrating result of Browder [4] and Gohde [5] from the setting of nonexpansive mappings to the framework of Suzuki mappings.
New iterative methods for the investigation of fixed points and solution of functional equations is the busy research topic and has fruitful applications such as in image recovery and signal processing (see, e.g., [13][14][15][16][17][18][19] and others). erefore, it is our purpose to construct a new iterative method for the larger class of nonexpansive mappings called Suzuki mappings. We also show by an example that this new iterative process gives better approximations as compared to other methods. Suppose M is a closed nonempty convex subset of a given Banach space, and assume further that α k , β k , c k ∈ (0, 1), k ∈ N, and F is a self-map of M.
e Mann iterative method [6] is defined as follows: e Ishikawa [7] iterative method is the extension of the Mann method from one step to two steps as follows: e Noor [8] iterative method is the extension of both of the Mann and Ishikawa iterative methods as follows: Agarwal et al.'s [9] method is the slightly modification of the Ishikawa method as follows: Abbas and Nazir's [10] iterative method is a three-step method read as follows: akur et al. [11] proposed a new iterative method as follows: akur et al. [11] showed that method (8) is better than all of the methods, namely, Mann (3), Ishikawa (4), Noor (5), Agarwal (6), and Abbas and Nazir (7). Here, in the current research, we first suggest an efficient new iterative method and prove that it can be used for computations of fixed points of the larger class of nonexpansive maps called Suzuki maps. Furthermore, we shall provide a novel example of the so-called Suzuki mappings and prove that it exceeds the corresponding class of nonexpansive mappings.

Preliminaries
Here, first we present some earlier important definitions, which are needed for our theoretical outcome.
Let E be a given Banach space, and suppose u k ⊆E is weakly convergent to w ∈ E and satisfies the following: Whenever any weakly convergent sequence in E has the abovementioned property, E is called a Banach space endowed with Opial's property (for details, see [20]). We now recall a property I introduced by Sentor and Dotson [21] for F: M ⟶ M (where M is a nonempty subset of a Banach space). We recall that F has condition I [21] if one can find a nondecreasing function, namely, P: [0, ∞) ⟶ [0, ∞), with the properties P(0) � 0, P(a) > 0 for every a > 0, and ‖u − Fu‖ ≥ P(dist(u, Fix(F))) for all u ∈ M.
Let M be any nonempty subset of a general Banach space E, and suppose u k is any given bounded sequence in E. We fix u ∈ E and denote (a) by R(u, u k ), the asymptotic radius of u k at u given by e most well-known fact about the set Z(M, u k ) is that it is always singleton whenever X is UCBS [22]. e fact that the set Z(M, u k ) is convex nonempty whenever M is weakly compact and convex is also well known (see, e.g., [23,24]).
Lemma 1 (See [12]). Assume that M is any nonempty subset of a Banach space, and suppose F: M ⟶ M. If F is a Suzuki mapping, then for every element u ∈ M and for every element q ∈ Fix(F), the fact ‖Fu − Fq‖ ≤ ‖u − q‖ holds.
e following result is known as the demiclosed principle.
Lemma 3 (See [12]). Assume that M is any nonempty subset of a Banach space having the Opial property, and suppose F: M ⟶ M. If F is a Suzuki mapping, then the following condition holds: e fixed-point set endowed with a Suzuki mapping enjoys the following properties.
Lemma 4 (See [12]). Assume that M is any nonempty subset of a Banach space, and suppose F: e following useful lemma can be found in [25].

Main Results
Strongly motivated by those mentioned above, we introduce a new iterative process, namely, JK iteration, as follows: where α k , β k ∈ (0, 1). In the present research section, we establish very interesting and important results for the larger class of the socalled Suzuki maps under the newly suggested method (12). We will present a numerical example to show that the JK iterative process is better than the iterative process by akur et al. (8). Furthermore, in the last section, a novel example of the so-called Suzuki maps which is not nonexpansive shows that Suzuki maps properly include nonexpansive maps. e numerical observations suggest that JK iterative method is for better than the leading method of akur and, hence, many others.
We now state and prove a much needed lemma for our main outcome, which will play a significant role in each result of the sequel.

Lemma 6.
Assume that M is any nonempty closed convex subset of a Banach space X, and suppose F: M ⟶ M is a Suzuki mapping with Fix(F) ≠ ∅. Suppose u k is a sequence given in (12). en, lim k⟶∞ ‖u k − q‖ exists for every fixed point q of F.
Proof. Take q ∈ Fix(F). By Lemma 1, we have ey imply that From the equations mentioned above, we conclude that ‖u k − q‖ is a bounded and nonincreasing sequence of reals, and hence, lim k⟶∞ ‖u k − q‖ exists for every fixed point q of F.

Theorem 1. Assume that M is any nonempty closed convex subset of a UCBC, and suppose F: M ⟶ M is a Suzuki mapping. Assume further that u k is a sequence given in (12). en, Fix(F) ≠ ∅ if and only if u k is bounded, and
Proof. First, we assume that u k is bounded and lim k⟶∞ ‖Fu k − u k ‖ � 0. We shall prove that Fix(F) ≠ ∅. For this, let q ∈ Z(M, u k ). By Lemma 2, we have It follows that Fq ∈ Z(M, u k ). Since in UCBS, asymptotic centers are singleton, we have Fq � q. Hence, the fixed point is nonempty.

Journal of Mathematics
Conversely, we assume that Fix(F) ≠ ∅. Conclusions of Lemma 6 provide that u k is bounded and lim k⟶∞ ‖u k − q‖ exists for every fixed point q of F. Now, if then by observing the proof of Lemma 6 and keeping (16) in mind, we obtain lim sup Appling Lemma 1, we get and by observing the proof of Lemma 6, we see It gives, together with (16), From (17) and (20), we obtain From (21), we have Hence, Now, from (16), (18) and (23) together with Lemma 5, we obtain Now, we are in the position to prove our weak convergence result.

Theorem 2. Assume that E is a UCBS with Opial's property and M is a nonempty convex closed subset of E, and suppose F: M ⟶ M be a Suzuki mapping with Fix(F) ≠ ∅.
Suppose u k is a sequence given in (12). en, u k converges weakly to a fixed point of F.
Proof. By eorem 1, u k is bounded and lim k⟶∞ ‖Fu k − u k ‖ � 0. Since E is uniformly convex, E is reflexive. Hence, one can easily find a subsequence, namely, u k l of u k such that u k l ⇀u for some u ∈ M. By Lemma 3, u ∈ Fix(F). We shall prove that u is the weak limit of u k . Let u not be the weak limit of u k . en, one can find another subsequence, namely, u k m of u k such that u k m ⇀u and v ≠ u. Again by Lemma 3, v ∈ Fix(F). Now, using Lemma 6 and Opial's property, we have Hence, lim k⟶∞ ‖u k − u‖ ≤ lim k⟶∞ ‖u k − v‖ by lim n⟶∞ ‖u k − u‖ < lim k⟶∞ ‖u k − v‖, clearly a contradiction, and so we must accept that u is the only weak limit of u k . Now, we prove the following strong convergence result.

Theorem 3.
Assume that M is any nonempty convex compact subset of a UCBC, and suppose F: M ⟶ M be a Suzuki mapping. Assume further that u k is a sequence given in (12). en, u k converges strongly to a fixed point of F.
Proof. From eorem in [12], we can write Fix(F) ≠ ∅. By eorem 1, lim k⟶∞ ‖Fu k − u k ‖ � 0. Since the domain M is compact, one can easily find a strongly convergent subsequence, namely, u k j of u k having a limit say z. By using Lemma 2, the following holds: Hence, u k j ⟶ Fz whenever j ⟶ ∞, so the uniqueness of limits follows Fz � z. By Lemma 6, lim k⟶∞ ‖u k − z‖ exists. Hence, z is the strong limit of u k . e proof of the following theorem is elementary and, therefore, omitted.

Theorem 4.
Assume that M is any nonempty closed convex subset of a UCBS, and suppose F: M ⟶ M be a Suzuki mapping. If Fix(F) ≠ ∅ and lim inf k⟶∞ dist(u k , Fix(F)) � 0, where u k is a sequence given in (12), then u k converges strongly to a fixed point of F.
We finish this section with a strong convergence theorem under the condition I.

Theorem 5.
Assume that M is any nonempty convex closed subset of a UCBS, and suppose F: M ⟶ M be a Suzuki mapping with Fix(F) ≠ ∅. Assume further that u k is a sequence given in (12). If F fulfils condition (I), then u k converges strongly to a fixed point of F.
Proof. In view of eorem 1, we can conclude that lim inf k⟶∞ ‖Fu k − u k ‖ � 0. Since F fulfils condition (I), one has lim inf k⟶∞ dist(u k , Fix(F)) � 0. e conclusions are now clear from eorem 4.

Numerical Example
is section introduces a novel example of self-Suzuki maps on a closed convex bounded subset of a Banach space. We suggest with many different cases that the novel JK scheme is far better than the earlier iterative methods using this example. Since we are using Suzuki maps in our work, the provided outcome holds simultaneously for nonexpansive maps as well.
One can conclude that F is a Suzuki mapping and, however, not nonexpansive by studying the computations given below. Select u � 8/65 and v � 1/8, and observe that which proves that F is not a nonexpansive on M. Next, we suggest the proof of the Suzuki property of F on M. e proof can be divided as given below.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.