# A New Iterative Method for Suzuki Mappings in Banach Spaces

Authors: Junaid Ahmad; Kifayat Ullah; Muhammad Arshad; Zhenhua Ma
Journal: Journal of Mathematics
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction and Preliminaries

Throughout the present research, we shall writeN 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)Fu−Fv≤ru−v,for allu,v∈M.An elementq∈M is said to be a fixed point of F provided that q=Fq. In this manuscript, the notation FixF will throughout denote the fixed point set of F. The 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 uk obtained from the Picard iterates [3], that is, uk+1=Fuk. 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 uk 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 u1=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 FixF=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], Thakur 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:(2)12u−Fu≤u−v⟹Fu−Fv≤u−v,for each two elementsu,v∈M.One can easily notice that the Suzuki mappings satisfy the nonexpansive requirement for some elements of the domain. Hence, nonexpansive mappings obviously satisfyC 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., [13–19] and others). Therefore, 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 αk,βk,γk∈0,1, k∈ℕ, and F is a self-map of M.The Mann iterative method [6] is defined as follows:(3)u1=u∈M,uk+1=1−αkuk+αkFuk.The Ishikawa [7] iterative method is the extension of the Mann method from one step to two steps as follows:(4)u1=u∈M,vk=1−βkuk+βkFuk,uk+1=1−αkuk+αkFvn.The Noor [8] iterative method is the extension of both of the Mann and Ishikawa iterative methods as follows:(5)u1=u∈M,wk=1−γkuk+γkFuk,vk=1−βkuk+βkFwk,uk+1=1−αkuk+αnFvk.Agarwal et al.’s [9] method is the slightly modification of the Ishikawa method as follows:(6)u1=u∈M,vk=1−βkuk+βkFuk,uk+1=1−αkFuk+αkFvn.Abbas and Nazir’s [10] iterative method is a three-step method read as follows:(7)u1=u∈M,wk=1−γkuk+γkFuk,vk=1−βkFuk+βkFwk,xk+1=1−αkFvk+αkFwk.Thakur et al. [11] proposed a new iterative method as follows:(8)u1=u∈M,wk=1−βkuk+βkFuk,vk=F1−αkuk+αkwk,uk+1=Fvk.Thakur 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.


## 2. Preliminaries

Here, first we present some earlier important definitions, which are needed for our theoretical outcome.LetE be a given Banach space, and suppose uk⊆E is weakly convergent to w∈E and satisfies the following:(9)liminfk⟶∞uk−w<liminfk⟶∞uk−z,for allz∈E−w.Whenever any weakly convergent sequence inE 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 P0=0, Pa>0 for every a>0, and u−Fu≥Pdistu,FixF for all u∈M.LetM be any nonempty subset of a general Banach space E, and suppose uk is any given bounded sequence in E. We fix u∈E and denote(a)
byℛu,uk, the asymptotic radius of uk at u given by ℛu,uk:=limsupk⟶∞u−uk(b)
byℛM,uk, the asymptotic radius associated with M of uk given by ℛM,uk=infℛu,uk:u∈M(c)
byZM,uk, the asymptotic center associated with M of uk given by ZM,uk=u∈E:ℛu,uk=ℛM,ukThe most well-known fact about the setZM,uk is that it is always singleton whenever X is UCBS [22]. The fact that the set ZM,uk is convex nonempty whenever M is weakly compact and convex is also well known (see, e.g., [23, 24]).Lemma 1 (See [12]).
Assume thatM 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∈FixF, the fact Fu−Fq≤u−q holds.Lemma 2 (See [12]).
Assume thatM is any nonempty subset of a Banach space, and suppose F:M⟶M. If F is a Suzuki mapping, then for every two elements u,v∈M, we have the following property:(10)u−Fv≤3u−Fu+u−v.The following result is known as the demiclosed principle.Lemma 3 (See [12]).
Assume thatM 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:(11)uk⊆M,uk⇀w,uk−Fuk⟶0⇒Fw=w.The fixed-point set endowed with a Suzuki mapping enjoys the following properties.Lemma 4 (See [12]).
Assume thatM is any nonempty subset of a Banach space, and suppose F:M⟶M. If F is a Suzuki mapping, then the set FixF is closed. Furthermore, if E is a strictly convex Banach space and M is convex, then FixF is convex too.The following useful lemma can be found in [25].Lemma 5.
Let0<u≤γk≤v<1 for each k∈ℕ and η≥0. If yk and zk are any sequences in a UCBS, E with limsupk⟶∞yk≤η, limsupk⟶∞zk≤η, and limk⟶∞γkyk+1−γkzk=η. Then, limk⟶∞yk−zk=0.


## 3. Main Results

Strongly motivated by those mentioned above, we introduce a new iterative process, namely, JK iteration, as follows:(12)u1=u∈M,wk=1−βkuk+βkFuk,vk=Fwk,uk+1=F1−αkFwk+αkFvk,k∈ℕ,where αk,βk∈0,1.In the present research section, we establish very interesting and important results for the larger class of the so-called 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 Thakur 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. The numerical observations suggest that JK iterative method is for better than the leading method of Thakur 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 thatM is any nonempty closed convex subset of a Banach space X, and suppose F:M⟶M is a Suzuki mapping with FixF≠∅. Suppose uk is a sequence given in (12). Then, limk⟶∞uk−q exists for every fixed point q of F.Proof.
Takeq∈FixF. By Lemma 1, we have(13)wk−q=1−βkuk+βkFuk−q≤1−βkuk−q+βkFuk−q≤1−βkuk−q+βkuk−q≤uk−q,vk−q=Fwk−q≤wk−q.
They imply that(14)uk+1−q=F1−αkFwk+αkFvk−q≤1−αkFwk+αkFyk−q≤1−αkFwk−q+αkFvk−q≤1−αkwk−q+αkvk−q≤1−αkwk−q+αkwk−q,=wk−q≤uk−q.
From the equations mentioned above, we conclude thatuk−q is a bounded and nonincreasing sequence of reals, and hence, limk⟶∞uk−q exists for every fixed point q of F.Theorem 1.
Assume thatM is any nonempty closed convex subset of a UCBC, and suppose F:M⟶M is a Suzuki mapping. Assume further that uk is a sequence given in (12). Then, FixF≠∅ if and only if uk is bounded, and limk⟶∞Fuk−uk=0.Proof.
First, we assume thatuk is bounded and limk⟶∞Fuk−uk=0. We shall prove that FixF≠∅. For this, let q∈ZM,uk. By Lemma 2, we have(15)ℛFq,uk=limsupk⟶∞uk−Fq≤3limsupk⟶∞uk−Fuk+limsupk⟶∞uk−q,=limsupk⟶∞uk−q,=ℛq,uk.
It follows thatFq∈ZM,uk. Since in UCBS, asymptotic centers are singleton, we have Fq=q. Hence, the fixed point is nonempty.
Conversely, we assume thatFixF≠∅. Conclusions of Lemma 6 provide that uk is bounded and limk⟶∞uk−q exists for every fixed point q of F. Now, if(16)limn⟶∞uk−q=η,then by observing the proof of Lemma 6 and keeping (16) in mind, we obtain(17)limsupk⟶∞wk−q≤limsupn⟶∞uk−q=η.
Appling Lemma1, we get(18)limsupk⟶∞Fuk−q≤limsupk⟶∞uk−q=η,and by observing the proof of Lemma 6, we see(19)uk+1−q≤wk−q.
It gives, together with (16),(20)η≤liminfk⟶∞wk−q.
From (17) and (20), we obtain(21)η=limk⟶∞wk−q.
From (21), we have(22)η=limk⟶∞wk−q,=limk⟶∞1−βkuk+βkFuk−q,=limk⟶∞1−βkuk−q+βkFuk−q.
Hence,(23)η=limk⟶∞1−βkuk−q+βkFuk−q.
Now, from (16), (18) and (23) together with Lemma 5, we obtain(24)limk⟶∞Fuk−uk=0.
Now, we are in the position to prove our weak convergence result.Theorem 2.
Assume thatE 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 FixF≠∅. Suppose uk is a sequence given in (12). Then, uk converges weakly to a fixed point of F.Proof.
By Theorem1, uk is bounded and limk⟶∞Fuk−uk=0. Since E is uniformly convex, E is reflexive. Hence, one can easily find a subsequence, namely, ukl of uk such that ukl⇀u for some u∈M. By Lemma 3, u∈FixF. We shall prove that u is the weak limit of uk. Let u not be the weak limit of uk. Then, one can find another subsequence, namely, ukm of uk such that ukm⇀u and v≠u. Again by Lemma 3, v∈FixF. Now, using Lemma 6 and Opial’s property, we have(25)limk⟶∞uk−u=liml⟶∞ukl−u<liml⟶∞ukl−v,=limn⟶∞uk−v,=limm⟶∞ukm−v<limm⟶∞ukm−u,=limn⟶∞uk−u.
Hence,limk⟶∞uk−u≤limk⟶∞uk−vbylimn⟶∞uk−u<limk⟶∞uk−v, clearly a contradiction, and so we must accept that u is the only weak limit of uk.
Now, we prove the following strong convergence result.Theorem 3.
Assume thatM is any nonempty convex compact subset of a UCBC, and suppose F:M⟶M be a Suzuki mapping. Assume further that uk is a sequence given in (12). Then, uk converges strongly to a fixed point of F.Proof.
From Theorem in [12], we can write FixF≠∅. By Theorem 1, limk⟶∞Fuk−uk=0. Since the domain M is compact, one can easily find a strongly convergent subsequence, namely, ukj of uk having a limit say z. By using Lemma 2, the following holds:(26)ukj−Fz≤3ukj−Fukj+ukj−z.
Hence,ukj⟶Fz whenever j⟶∞, so the uniqueness of limits follows Fz=z. By Lemma 6, limk⟶∞uk−z exists. Hence, z is the strong limit of uk.
The proof of the following theorem is elementary and, therefore, omitted.Theorem 4.
Assume thatM is any nonempty closed convex subset of a UCBS, and suppose F:M⟶M be a Suzuki mapping. If FixF≠∅ and liminfk⟶∞distuk,FixF=0, where uk is a sequence given in (12), then uk converges strongly to a fixed point of F.We finish this section with a strong convergence theorem under the conditionI.Theorem 5.
Assume thatM is any nonempty convex closed subset of a UCBS, and suppose F:M⟶M be a Suzuki mapping with FixF≠∅. Assume further that uk is a sequence given in (12). If F fulfils condition (I), then uk converges strongly to a fixed point of F.Proof.
In view of Theorem1, we can conclude that liminfk⟶∞Fuk−uk=0. Since F fulfils condition (I), one has liminfk⟶∞distuk,FixF=0. The conclusions are now clear from Theorem 4.


## 4. Numerical Example

This 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.Example 1.
Consider a closed convexM=0,1 of a Banach space ℝ. Set a self-map F:M⟶M as follows:(27)Fu=1−u,ifu<18,u+45,ifu≥18.
One can conclude thatF 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(28)Fu−Fv=Fu−Fv=1−865−3340,=1352600>1520=u−v,which proves that F is not a nonexpansive on M.
Next, we suggest the proof of the Suzuki property ofF on M. The proof can be divided as given below.Case 1.
Selectu∈0,1/8; then, 1/2u−v=1−2u/2∈3/8,1/2. For 1/2u−Fu≤u−v, one has 1−2u/2≤v−u, i.e., 1/2≤v; therefore, v∈1/2,1. So, one has(29)Fu−Fv=v+45−1−u=v+5u−15<18,u−v=u−v>18−12=38.
Hence,1/2u−Fu≤u−v⇒Fu−Fv≤u−v.Case 2.
Selectu∈1/8,1; then, 1/2u−Fu=1/2u+4/5−u=4−4u/10∈0,7/20. For 1/2u−Fu≤u−v, one has 4−4u/10≤v−u, and so the following possible situations occur:(a)
Wheneveru<v, 4−4u/10≤v−u⇒v≥4+6u/10⇒v∈19/40,1⊂1/8,1. So,(30)Fu−Fv=u+45−v+45=15u−v≤u−v.Therefore,1/2u−Fu≤u−v⇒Fu−Fv≤u−v.(b)
Wheneveru>v, 4−4u/10≤u−v⇒v≤u−4−4u/10=14u−4/10⇒v∈−9/40,1. Since v∈0,1, v≤14u−4/10⇒u∈4/14,1. So, the case is u∈4/14,1 and v∈0,1.
Whenu∈4/14,1 and v∈1/8,1 are already included in a, we assume u∈4/14,1 and v∈0,1/8; then,(31)Fu−Fv=u+45−1−v,=u+5v−15.
For convenience, first we assumeu∈4/14,1/2 and v∈0,1/8, then Fu−Fv≤1/8 and u−v>18/112. Hence, Fu−Fv≤u−v.
Next, we consideru∈1/2,1 and v∈0,1/8, then Fu−Fv≤1/8 and u−v>3/8. Hence, Fu−Fv≤u−v. So, 1/2u−Fu≤u−v⇒Fu−Fv≤u−v. Hence, F is a Suzuki mapping on M. Let αk=2k/7k+9 and βk=1/3k+71/2, k∈ℕ. A strong convergence of JK (12), leading Thakur (8), and leading two-step Agarwal (6) iterates to q=1 can be seen in the Table 1 and Figure 1. Clearly, JK iterates converge faster to q=1.Table 1
Sequences defined by JK, Thakur, and Agarwal iterative methods for mappingF defined in Example 1.
kJK (12)Thakur (8)Agarwal (6)10.90.90.920.99820715730.99650596440.982529822130.99998143170.99988610290.997152573340.99999992650.99999644220.9995552826510.99999989200.9999325257610.99999999670.9999899750710.99999999990.99999853408110.99999978839110.999999969710110.999999995711110.999999999412110.999999999913111Figure 1
Convergence behavior of JK (12), Thakur (8), and Agarwal (6) iterates to converge to q=1, a unique fixed point of the self-map F in Example 1 where u1=0.9.Now, to show further effectiveness of JK iteration, we set stopping criteriauk−q<10−15, where q=1 is a fixed point of F. The observations are given below.Table 2
αk=k+1/5k+11/11 and βk=k/2k+3.
Iterates number to converge an iteration to1u1SThakurJK0.10201180.15201180.20201180.50201080.80191080.9019107Table 3
αk=k/6k+7 and βk=1/2k+31/5.
Iterates number to converge an iteration to1u1SThakurJK0.10211190.15211190.20211190.50211190.80211190.9020109Table 4
αk=k/k+51/7 and βk=1/7k+3.
Iterates number to converge an iteration to1u1SThakurJK0.10191170.15191170.20191170.50191170.80191070.9019107Remark 1.
One can observe from Tables1–4 that the new JK iterative method converges faster than the fastest Thakur and fastest two-step Agarwal methods for the larger class of Suzuki mappings.



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