# On the JK Iterative Process in Banach Spaces

Authors: Junaid Ahmad; Hüseyin Işık; Faeem Ali; Kifayat Ullah; Eskandar Ameer; Muhammad Arshad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the recent progress, different iterative procedures have been constructed in order to find the fixed point for a given self-map in an effective way. Among the other things, an effective iterative procedure called the JK iterative scheme was recently constructed and its strong and weak convergence was established for the class of Suzuki mappings in the setting of Banach spaces. The first purpose of this research is to obtain the strong and weak convergence of this scheme in the wider setting of generalizedα-nonexpansive mappings. Secondly, by constructing an example of generalized α-nonexpansive maps which is not a Suzuki map, we show that the JK iterative scheme converges faster as compared the other iterative schemes. The presented results of this paper properly extend and improve the corresponding results of the literature.


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## Body

## 1. Introduction

A mappingS on a subset U of a Banach space is called contraction provided that for all z,z′∈U follows that(1)Sz−Sz′≤δz−z′,where δ∈0,1 is fixed. A point v0 is called a fixed point for S if v0=Sv0. Normally, we denote the set of all fixed points of S by FS, that is, FS=v0∈U:Sv0=v0. The Banach–Caccioppoli fixed point theorem (BCFPT) [1, 2] provides the existence of a unique fixed point for every self-contraction of a complete metric space.We say that a self-mapS:U⟶U is nonexpansive on the set U provided that(2)Sz−Sz′≤z−z′,forallz,z′∈U.We may observe that every contraction of a subsetU of a Banach space is nonexpansive but the converse may not hold in general. Unlike contractions, every self-nonexpansive mapping of a complete metric space does not admit a fixed point. After many years of BCFPT, Browder [3], Gohde [4] and Kirk [5] independently obtained that a self-nonexpansive mapping of a closed bounded convex subset of a uniformly convex Banach space (UCBS, for short) always has a fixed point.In 2008, Suzuki [6] provided a new type of generalization of nonexpansive mappings and proved some related fixed point results for this class of mappings in Banach spaces. Notice that a self-map S:U⟶U is mapping with the C property (also called Suzuki mapping) if any z,z′∈U follows that(3)12z−Sz≤z−z′⇒Sz−Sz′≤z−z′.In 2011, Aoyama and Kohsaka [7] provided the idea of α-nonexpansive mappings. A self-map S:U⟶U is called α-nonexpansive if any z,z′∈U follows that(4)Sz−Sz′2≤αz−Sz′2+αz′−Sz2+1−2αz−z′2,where α∈0,1.In 2017, Pant and Shukla [8] defined a very general class of nonexpansive mappings which properly contains the class of Suzuki mappings and partially extends the class of α-nonexpansive mappings. A self-map S:U⟶U is called generalized α-nonexpansive if any z,z′∈U follows that(5)12z−Sz≤z−z′⇒Sz−Sz′≤αz−Sz′+αz′−Sz+1−2αz−z′,where α∈0,1.Fixed point approximation for nonexpansive mappings under a suitable iterative method is a very active field of research and provides many interesting and important applications in applied sciences (cf. [9–12] and others). Finding the fixed points for nonexpansive and generalized nonexpansive under Picard iteration is not possible in general. A simple situation of such a case which is the rotation of the unit disk about the origin in a plane is a best example of a nonexpansive mapping which has a unique fixed point but Picard iteration does not converge to this point. In order to find fixed points of nonexpansive and hence generalized nonexpansive mappings and secondly to obtain relatively high accuracy, some authors introduced different types of iterative procedures (cf. Mann [13], Ishikawa [14], Noor [15], Agarwal et al. [16], Abbas and Nazir [17], Thakur et al. [18] and references therein). Suppose that U is a closed nonempty convex subset of a given Banach space, and assume further that ξk,ηk,μk∈0,1, k∈ℕ, and S is a self-map of U.The Mann [13] iteration process is stated as follows:(6)p1=p∈U,pk+1=1−ξkpk+ξkSpk.The Ishikawa [14] iterative process may be viewed as a two-step Mann iteration, which is given by(7)p1=p∈U,qk=1−ηkpk+ηkSpk,pk+1=1−ξkpk+ξkSqk.In 2000, Noor [15] suggested a three-step iterative process which is more general than the Mann and Ishikawa iteration processes as follows:(8)p1=p∈U,rk=1−μkpk+μkSpk,qk=1−ηkpk+ηkSrk,pk+1=1−ξkpk+ξkSqk.In 2007, Agarwal et al. [16] suggested a new iteration process, which converges faster than the Mann iteration for contraction mappings in Banach spaces:(9)p1=p∈U,qk=1−ηkpk+ηkSpk,pk+1=1−ξkSpk+ξkSqk.In 2014, Abbas and Nazir [17] proposed a new three-step iteration which converges faster than all of the Picard, Mann, Ishikawa, and Agarwal iterative processes for nonexpansive mappings, as follows:(10)p1=p∈U,rk=1−μkpk+μkSpk,qk=1−ηkSpk+ηkSrk,pk+1=1−ξkSqk+ξkSrk.In 2016, Thakur et al. [18] suggested the following iteration process, which converges faster than all of the above iterative processes for Suzuki mappings:(11)p1=p∈U,rk=1−ηkpk+ηkSpk,qk=S1−ξkpk+ξkrk,pk+1=Sqk.Very recently, Ahmad et al. [19] introduced a new iterative process named JK iteration, as follows:(12)p1=p∈U,rk=1−ηkpk+ηkSpk,qk=Srk,pk+1=S1−ξkSrk+ξkSqk.They observed that JK iteration (12) can be used for fixed points of Suzuki mappings. Moreover, they proved by providing a novel example of Suzuki mappings that the JK iteration process converges faster than all of the above iterative processes including the leading Thakur iteration (11). In this paper, firstly, we improve and extend the main results of Ahmad et al. [19] from the context of Suzuki mappings to the more general framework of generalized α-nonexpansive mappings. We then provide a novel example of generalized α-nonexpansive mappings and show that its JK iterative process is better than the mentioned iterative processes. Our results can be used for finding the solutions of split feasibility problems, solutions of differential and integral equations provided that the operator is generalized α-nonexpansive.


## 2. Preliminaries

We now provide some definitions.Definition 1 [20].
A Banach spaceW is said to be endowed with Opial’s property if every weakly convergence sequence pk⊆W having a weak limit v0∈W follows that(13)limsupk⟶∞pk−v0<limsupk⟶∞pk−u0,foreverychoiceofu0≠v0.Definition 2 [21].
A self-mappingS of a subset U of a Banach space is said to be endowed with condition I if one has a nondecreasing function η:0,∞⟶0,∞ having η0=0 and ην>0 for every ν∈0,∞ and z−Sz≥ηdz,FS for every z∈U; here, dz,FS stands for the distance of z from FS.Definition 3.
Suppose thatW is any given Banach space and pk⊆W is bounded. Assume that ∅≠U⊆W is closed and convex. Then, the asymptotic radius of the sequence pk relative to the set U is given by rU,pk=inflimsupk⟶∞pk−z:z∈U. Moreover, the asymptotic center of pk with respect to U is given by AU,pk=z∈U:limsupk⟶∞pk−z=rU,pk.Remark 4.
The most well-known fact about the setAU,pk is that it is always singleton whenever W is UCBS [22]. The fact that the set AU,pk is convex and nonempty also known in the case when U is weakly compact and convex [23, 24].Now, we combine some elementary properties of generalizedα-nonexpansive mappings, which can be found in [8].Proposition 5.
Suppose thatU is any nonempty subset of a Banach space W and S:U⟶U.(a)
IfS is Suzuki mapping, then, S is generalized α-nonexpansive(b)
IfS is generalized α-nonexpansive having nonempty fixed point set, then, for any v0∈FS, Sz−Sv0≤z−v0 for all z∈U(c)
IfS is generalized α-nonexpansive, then, the set FS is closed in U. Also, FS is convex in the case when W is strictly convex and U is convex(d)
IfS is a generalized α-nonexpansive mapping, then, for every choice of z,z′∈U, the following holds:(14)z−Sz′≤3+α1−αz−Sz+z−z′(e)
Suppose thatS is generalized α-nonexpansive and W is endowed with the Opial property. If pk is weakly convergent to l0 and limk⟶∞pk−Spk=0, it follows that l0∈FSThe following useful lemma can be found in [25].Lemma 6.
Suppose that0<i≤yk≤j<1 for each k∈ℕ and λ≥0. If pk and qkare any sequences in a UCBS W endowed with limsupk⟶∞pk≤λ, limsupk⟶∞qk≤λ, and limk⟶∞ykpk+1−ykqk=λ, then limk⟶∞pk−qk=0.


## 3. Main Results

The aim of this section is at giving some important weak and strong convergence of JK (12) for the class of generalized α-nonexpansive mappings. We start the section with a key lemma.Lemma 7.
Suppose thatW is UCBS, ∅≠U⊆W is closed convex, and S:U⟶U is a generalized α-nonexpansive having FS≠∅. If pk is a JK iteration sequence as provided in (12). Then, limk⟶∞pk−v0 exists for each v0∈FS.Proof.
If we choosev0∈FS, then, using (12) along with Proposition 5 (b), we have(15)rk−v0=1−ηkpk+ηkSpk−v0≤1−ηkpk−v0+ηkSpk−v0≤1−ηkpk−v0+ηkpk−v0≤pk−v0.
Hence,(16)pk+1−v0=S1−ξkSrk+ξkSqk−v0≤1−ξkSrk+ξkSqk−v0≤1−ξkSrk−v0+ξkSqk−v0≤1−ξkrk−v0+ξkqk−v0=1−ξkrk−v0+ξkSrk−v0≤1−ξkrk−v0+ξkrk−v0=rk−v0≤pk−v0.
Consequently, we conclude thatpk−v0 is nonincreasing and bounded; accordingly, we must have that limk⟶∞pk−v0 exists for every element v0 of FS. ☐Theorem 8.
Suppose thatW is UCBS, ∅≠U⊆W is closed convex, and S:U⟶U is a generalized α-nonexpansive. If pk is a JK iteration sequence as provided in (12). Then, FS≠∅ if and only if pk is bounded and limk⟶∞pk−Spk=0.Proof.
Firstly, we may takeFS≠∅. According to Lemma 7, one concludes that pk is bounded and limk⟶∞pk−v0 exists for every element v0 of FS. We now suppose(17)limk⟶∞pk−v0=λ.
We need to obtain thatlimk⟶∞pk−Spk=0. Then, using Lemma 7, we have(18)rk−v0≤pk−v0⇒limsupk⟶∞rk−v0≤limsupk⟶∞pk−v0=λ.
Sincev0∈FS, so by Proposition 5 (b), we infer(19)Spk−v0≤pk−v0⇒limsupk⟶∞Spk−v0≤limsupk⟶∞pk−v0=λ.
Now from (16), we get(20)pk+1−v0≤rk−v0.
Using this together with (17), we obtain(21)λ≤liminfk⟶∞rk−v0.
From (18) and (21), we deduce(22)λ=limk⟶∞rk−v0.
Using (22), we get(23)λ=limk⟶∞rk−v0=limk⟶∞1−ηkpk+ηkSpk−v0=limk⟶∞1−ηkpk−v0+ηkSpk−v0.
Thus,(24)λ=limk⟶∞1−ηkpk−v0+ηkSpk−v0.
Using (17), (19), and (24) and keeping Lemma 6 in mind, one concludes that(25)limk⟶∞pk−Spk=0.
Conversely, we may suppose thatpk is bounded in U such that limk⟶∞pk−Spk=0. The aim is to prove that FS≠∅. If we take any v0∈AU,pk, then, using Proposition 5 (d), it follows that(26)ASv0,pk=limsupk⟶∞pk−Sv0≤3+α1−αlimsupk⟶∞pk−Spk+limsupk⟶∞pk−v0=limsupk⟶∞pk−v0=Av0,pk.
It follows thatSv0∈AU,pk. Since W is UCBS, AU,pk contains only one element, that is, we must have Sv0=v0. Hence, v0∈FS, that is, the fixed point FS is nonempty. ☐Now, we are in the position to prove our weak convergence result.Theorem 9.
Suppose thatW is UCBS, ∅≠U⊆W is closed convex, and S:U⟶U is a generalized α-nonexpansive having FS≠∅. If pk is a JK iteration sequence as provided in (12) and W has Opial’s property, then, pk converges weakly to a point of FS.Proof.
By Theorem8, pk is bounded. The uniform convexity of W follows reflexivity of W, that is, pk has a weakly convergent subsequence pkt with a weak limit, namely, l0. According to Theorem 8, limm⟶∞pkm−Spkm=0. Hence, using Proposition 5 (e), we get l0∈FS. We claim that l0 is the weak limit of pk. We may suppose on the contrary that l0 is not the weak limit of pk, that is, pk has another weakly convergent subsequence pks with a weak limit, namely, l0′≠l0. According to Theorem 8, lims⟶∞pks−Spks=0. Hence, using Proposition 5 (e), we get l0′∈FS. Now using Lemma 7 and Opial’s property, we have(27)limsupk⟶∞pk−l0=limsupm⟶∞pkm−l0<limsupm⟶∞pkm−l0′=limsupk⟶∞pk−l0′=limsups⟶∞pks−l0′<limsups⟶∞pks−l0=limsupk⟶∞pk−l0.
Consequently, we obtainedlimsupk⟶∞pk−l0<limsupk⟶∞pk−l0, which suggests a contradiction. Therefore, we conclude that l0 is the weak limit of the sequence pk. ☐Now, we prove the following strong convergence result.Theorem 10.
Suppose thatW is UCBS, ∅≠U⊆W is compact convex, and S:U⟶U is a generalized α-nonexpansive having FS≠∅. If pk is a JK iteration sequence as provided in (12), then, pk converges strongly to a point of FS.Proof.
Sincepk⊆U and U is compact, so we can find a subsequence, namely, pkm of pk such that limm⟶∞pkm−u0=0 for some element u0∈U. Moreover, since FS≠∅, so according to the Theorem 8, limm⟶∞pkm−Spkm=0. Applying Proposition 5 (d), we get(28)pkm−Su0≤3+α1−αpkm−Spkm+pkm−u0.
Consequently,pkm⟶Su0 provided that m⟶∞. But W is a Banach space, and so, the limit of a convergent sequence is always unique. Thus, Su0=u0. Lemma 7 provides us that limk⟶∞pk−u0 exists. Hence, u0 is the strong limit of pk. ☐We now state and then prove another strong convergence theorem as follows.Theorem 11.
Suppose thatW is UCBS, ∅≠U⊆W is closed convex, and S:U⟶U is a generalized α-nonexpansive having FS≠∅. If pk is a JK iteration sequence as provided in (12), then, pk converges strongly to a point FS whenever liminfk⟶∞dpk,FS=0Proof.
According to Lemma7, limk⟶∞pk−v0 exists, for every choice of fixed point v0 of S. It follows that limk⟶∞dpk,FS exists. Accordingly, we have(29)limk⟶∞dpk,FS=0.
The above strong limit suggests the existence of two subsequencespks, vs in pk and FS, respectively, with the property pks−vs≤1/2s for every natural constant s. According to the proof of Lemma 7, the iterative sequence pk is nonincreasing. Accordingly, we have(30)pks+1−vs≤pks−vs≤12s.
Using the above and triangle inequality, one has(31)vs+1−vs≤vs+1−pks+1+pks+1−vs≤12s+1+12s≤12s−1⟶0,providedthats⟶∞.
Accordingly, we obtainedlims⟶∞vs+1−vs=0, that is, vs form the Cauchy sequence in the closed set FS⊆U. It follows that lims⟶∞vs=u0 for some u0∈FS. Cosequently, u0∈FS. By Lemma 7, limk⟶∞pk−v0 exists, that is, u0 is also the strong limit of pk. ☐We finish this section with a strong convergence theorem under the conditionI.Theorem 12.
Suppose thatW is UCBS, ∅≠U⊆W is closed convex, and S:U⟶U is a generalized α-nonexpansive having FS≠∅. If pk is a JK iteration sequence as provided in (12). Then, pk converges strongly to a point of FS whenever S is endowed with condition I.Proof.
According to Theorem8, one can conclude that liminfk⟶∞pk−Spk=0. Applying the condition I of S, one obtain liminfk⟶∞dpk,FS=0. It now follows from Theorem 11 that pk is strongly convergent in the set FS. ☐


## 4. Numerical Example

The aim of this section is to provide a new example of generalizedα-nonexpansive mappings that exceeds the class of Suzuki mappings. We connect the mentioned iterative schemes with this example to show the effectiveness of our obtained results.Example 13.
We take a setU=5,10 and set a self-map on U by the following rule:(32)Sz=z+52,ifz<10,5,ifz=10.
We show thatS is generalized α-nonexpansive having α=1/2, but not Suzuki mapping. This example thus exceeds the class of Suzuki mappings.Case 1.
Whenz,z′∈10, we have(33)12z−Sz′+12z′−Sz+1−212z−z′≥0=Sz−Sz′.Case 2.
Whenz,z′∈5,10, we have(34)12z−Sz′+12z′−Sz+1−212z−z′=12z′−z+52+12z−z′+52≥12z′−z+52−z−z′+52=122z′−z−5−2z+z′+52=123z′−3z2=34z−z′≥12z−z′=Sz−Sz′.Case 3.
Whenz′∈10 and z∈5,10, we have(35)12z−Sz′+12z′−Sz+1−212z−z′=12z−5+12z′−z+52≥12z−5=z−52=Sz−Sz′.The above cases clearly suggest thatS is generalized 1/2 nonexpansive mapping having FS=5. Choose z=8.8 and z′=10; then, z−z′=1.2, Sz−Sz′=1.9, and 1/2z−Sz=0.95. Thus, it is seen that, 1/2z−Sz<z−z′ but Sz−Sz′>z−z′. Thus, S exceeds the class of Suzuki mappings.Now, we by choosingξk=0.70, ηk=0.65, and μk=0.80, we may observe in Table 1 and Figure 1 that JK (12) iterative process converges faster to 5∈FS as compared the other processes.Table 1
Strong convergence comparison of JK (12), Thakur (11), Abbas (10), Agarwal (9), Noor (8), Ishikawa (7), and Mann (6) iterates for the self-map S in Example 13.
kJKThakurAbbasAgarwalNoorIshikawaMann16.50006.50006.50006.50006.50006.50006.500025.16455.28975.39835.57945.73615.80445.975035.01805.05595.10575.22385.36135.43135.633845.00205.01085.02815.08645.17735.23135.411955.00025.00215.00755.03345.08705.12405.267865.00005.00045.00205.01295.04275.06655.174075.00005.00015.00055.00505.02105.03575.113185.00005.00005.00015.00195.01035.01915.073595.00005.00005.00005.00075.00505.01035.0478105.00005.00005.00005.00035.00255.00555.0311115.00005.00005.00005.00015.00125.00295.0202125.00005.00005.00005.00005.00065.00165.0131135.00005.00005.00005.00005.00035.00085.0085145.00005.00005.00005.00005.00015.00055.0055155.00005.00005.00005.00005.00015.00025.0036165.00005.00005.00005.00005.00005.00015.0023175.00005.00005.00005.00005.00005.00015.0015185.00005.00005.00005.00005.00005.00005.0010195.00005.00005.00005.00005.00005.00005.0006205.00005.00005.00005.00005.00005.00005.0004215.00005.00005.00005.00005.00005.00005.0003225.00005.00005.00005.00005.00005.00005.0002235.00005.00005.00005.00005.00005.00005.0001245.00005.00005.00005.00005.00005.00005.0001255.00005.00005.00005.00005.00005.00005.0000Figure 1
Convergence behavior of the sequences developed by remarkable iterative processes.Now we show the further effectiveness of the JK iteration (12) in the class of generalized α-nonexpansive mappings. Using S defined in Example 13, we suggest some different values for the parameters and p1. To do this, we set the stopping criteria pk−5<10−15. The obtained results are provided in Table 2. The bold numbers show that JK iteration (12) requires less iteration numbers as compared to the leading three-step Thakur (11) and leading two-step Agarwal (9).Remark 14.
The main outcome of this paper extended the corresponding results of Ahmad et al. [19] from the class of Suzuki maps to the setting of generalized α-nonexpansive maps. We have observed in Tables 1 and 2 as well as in Figure 1 that the JK iterative scheme (12) is still more effective than the other iterative schemes even in the general setting of generalized α-nonexpansive maps.Table 2
Influence of parameters: comparison of various iterative schemes.
IterationsInitial points5.16.06.97.88.99.9Forξk=1/k+5, ηk=k/k+3Agarwal293131323232Thakur182020202021JK141515151616Forξk=1/k, ηk=1/k+231/4Agarwal434748484949Thakur222424242525JK182020202021Forξk=1−1/4k+61/5,ηk=1/k4Agarwal444849495050Thakur222425252525JK182020202021Forξk=k+1/2k+11/2,ηk=1/3k+75/30Agarwal454849505050Thakur242525252525JK171819191919Forξk=1−1/4k+31/4,ηk=1/k7Agarwal444849495050Thakur222425252525JK181920202020



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