# A New Faster Iterative Scheme for Numerical Fixed Points Estimation of Suzuki’s Generalized Nonexpansive Mappings

Authors: Shanza Hassan; Manuel De la Sen; Praveen Agarwal; Qasim Ali; Azhar Hussain
Journal: Mathematical Problems in Engineering
 (2020)

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## 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 asS∗-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.


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## Body

## 1. Introduction

Most of the nonlinear equations can be transformed into a fixed point problem as follows:(1)ℱu=u,where ℱ is a self-map on a certain distance space X and the solution of the aforementioned equation is considered as a fixed point of the mapping ℱ. Banach [1] proved that if a self-map ℱ on a complete metric space is such that(2)dℱu,ℱv≤qdu,v,for 0≤q<1, then it possesses a unique fixed point u∗. Moreover, the iterative process(3)ℱun=un+1,called the Picard iteration process, converges to u∗. 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 ℱ 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 un is generated by an arbitrary u0∈C as(4)un+1=1−αnun+αnℱun,∀n≥0,where αn is in 0,1.In 1974, Ishikawa [4] developed an iterative scheme to approximate the fixed point of nonexpansive mappings, where un is defined iteratively starting from u0∈C by(5)un+1=1−αnun+αnℱvnvn=1−βnun+βnℱun,for all n≥0, where αnandβn are in 0,1.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 u0∈C, the sequence un is defined by(6)un+1=1−αnun+αnℱvnvn=1−βnun+βnℱwnwn=1−γnun+γnℱun,for all n≥0, where αn,βn, and γn are in 0,1.Agarwal et al. [11], in 2007, proposed the following iterative scheme: for arbitrary u0∈C, a sequence un is generated by(7)un+1=1−αnℱun+αnℱvnvn=1−βnun+βnℱun,for all n≥0, where αnandβn are in 0,1. 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 un is formulated from arbitrary u0∈C by(8)un+1=1−αnℱvn+αnℱwnvn=1−βnℱun+βnℱwnwn=1−γnun+γnℱun,for all n≥0, where αn,βn, and γn are in 0,1.Later in 2016, Thakur et al. [13] developed the following iterative procedure, where a sequence un is generated iteratively by arbitrary u0∈C and(9)un+1=1−αnℱwn+αnℱvnvn=1−βnwn+βnℱwnwn=1−γnun+γnℱun,for all n≥0, where αn,βn, and γn are in 0,1.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 arbitraryu0∈C and(10)un+1=ℱvnvn=ℱ1−αnwn+αnℱwnwn=1−βnun+βnℱun,for all n≥0, where αn,βn, and γn are in 0,1.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,ℰ is a nonempty closed convex subset of a uniformly convex Banach space X, ℕ denotes the set of all positive integers and Fℱ denotes the set of all fixed points of ℱ, that is,(11)Fℱ≔y:ℱy=y.Definition 1.
(see [14]). A Banach space X is said to be uniformly convex if for each ϵ∈0,2, there exists a δ>0 such that for all u,v∈X,(12)u≤1v≤1u−v>ϵimpliesu+v2≤δ.Definition 2.
(see [15]). A Banach space X is said to satisfy Opial property if for each sequence un in X, converging weakly to u∈X, we have(13)limsupn⟶∞un−u<limsupn⟶∞un−v,for all v∈X such that u≠v.Definition 3.
A mappingℱ:ℰ⟶ℰ is called a contraction if there exists α∈0,1, such that(14)ℱu−ℱv≤αu−v,for allu,v∈ℰ.Definition 4.
A mappingℱ:ℰ⟶ℰ quasi-nonexpansive if for all u∈ℰ and p∈Fℱ and α∈0,1, we have(15)ℱu−p≤αu−p.Definition 5.
(see [16]). A mapping ℱ:ℰ⟶ℰ is called Suzuki’s generalized nonexpansive mapping if for all u,v∈ℰ, we have(16)12u−ℱu≤αu−vimpliesℱu−ℱv≤u−v.
Suzuki [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 I.Definition 6.
A mappingℱ:ℰ⟶ℰ is said to satisfy condition I, if there exists a nondecreasing function f:0,∞⟶0,∞ with f0=0 and fδ>0 for all δ>0 such that u−ℱu≥fdu,Fℱ, for all u∈ℰ, where du,Fℱ=infp∈Fℱu−p.Proposition 1 (see [16]).
Letℱ:ℰ⟶ℰ be any mapping. Then,(i)
Ifℱ is nonexpansive, then ℱ is a Suzuki’s generalized nonexpansive mapping.(ii)
Ifℱ is a Suzuki’s generalized nonexpansive mapping and has a fixed point, then ℱ is a quasi-nonexpansive mapping.(iii)
Ifℱ is a Suzuki generalized nonexpansive mapping, then(17)u−ℱv≤3ℱu−u+u−v,∀u,v∈ℰ.Lemma 1 (see [16]).
Supposeℱ:ℰ⟶ℰ is Suzuki’s generalized nonexpansive mapping satisfying Opial property. If un converges weakly to u and limn⟶∞ℱun−un=0, then ℱu=u.Lemma 2 (see [16]).
LetX be a uniformly convex Banach space and ℰ be a weakly convex compact subset of X. Assume that ℱ:ℰ⟶ℰ is Suzuki’s generalized nonexpansive mapping. Then, ℱ has a fixed point.Lemma 3 (see [18]).
LetX be a uniformly convex Banach space and tn be any real sequence such that 0<p≤un≤q<1 for all n≥1. Suppose that un and vn be any two sequences of X such that limsupn⟶∞un≤r, limsupn⟶∞vn≤r, and limsupn⟶∞tnun+1−tnvn=r hold for some r≥0. Then, limsupn⟶∞un−vn=0.Definition 7.
(see [19]). Let X be a Banach space and ℰ be a nonempty closed convex subset of X. Assume that un is a bounded sequence in X. For u∈X, we set ru,un=limsupn⟶∞un−u. The asymptotic radius of un relative to ℰ is the set rℰ,un=infru,un:u∈ℰ and the asymptotic center of un relative to ℰ is given by the following set:(18)Aℰ,un=u∈ℰ:ru,un=rℰ,un.It is known that, in a uniformly convex Banach space, Aℰ,un consists of exactly one point.Definition 8.
(see [20]). Let X be a Banach space and ℱ:X⟶X. Suppose that u0∈X and un+1=fℱ,un define an iteration procedure which gives a sequence of points un in X. Assume that xn converges to the fixed point p. Suppose vn be a sequence in X and ϵn be a sequence in ℝ+=0,∞ given by ϵn=vn+1−fℱ,vn. Then, the iteration procedure defined by un+1=fℱ,un is said to be ℱ-stable or stable with respect to ℱ if(19)limn⟶∞ϵn=0ifflimn⟶∞vn=p.Definition 9.
(see [21]). Let X be a Banach space and ℱ:X⟶X. Then, ℱ is called a contractive mapping on X if there exist L≥0, a∈0,1 such that for each u,v∈X,(20)ℱu−ℱv≤Lu−ℱu+au−v.
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 X be a Banach space and ℱ:X⟶X. Then, ℱ is called a contractive mapping on X if there exist b∈0,1 and a monotone increasing function φ:ℝ+⟶ℝ+ with φ0=0, such that for each u,v∈X,(21)ℱu−ℱv≤φu−ℱu+bu−v.Lemma 4.
(see [25]). If λ is a real number such that 0≤λ<1, and ϵn is the sequence of positive numbers such that(22)limn⟶∞ϵn=0,then for any sequence of positive numbers vn satisfying(23)vn+1≤λvn+ϵn,forn=1,2,…,we have(24)limn⟶∞vn=0.


## 3. S∗-Iteration Process

Throughout this section,C be a nonempty set of a Banach space X, and for all n≥0, αn, βn, γn and ζn are real sequences in the interval 0,1.We generate the sequenceun iteratively, taking arbitrary u0∈C, by(25)un+1=ℱ1−αnvn+αnℱvnvn=ℱ1−βnwn+βnℱwnwn=ℱ1−γnxn+γnℱxnxn=ℱ1−ζnun+ζnℱun.First, we show thatS∗-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 ofS∗-Iteration Process

First, we establish convergence results forS∗-iteration process:Theorem 1.
LetX be a Banach space and ℰ be a nonempty closed convex subset of X. Let ℱ be a nonexpansive self mapping on ℰ, un be a sequence defined by (25), and Fℱ≠ϕ. Then, limn⟶∞un−p exists for all p∈Fℱ.Proof.
Letp∈Fℱ for all n∈ℕ. From (16), we have(26)xn−p=ℱ1−ζnun+ζnℱun−p≤1−ζnun+ζnℱun−p≤1−ζnun−p+ζnℱun−p≤1−ζnun−p+ζnun−p=un−p,(27)wn−p=ℱ1−γnxn+γnFxn−p≤1−γnxn+γnℱxn−p≤1−γnxn−p+γnℱxn−p≤1−γnxn−p+γnxn−p=xn−p≤un−p,(28)vn−p=ℱ1−βnwn+βnℱwn−p≤1−βnwn+βnℱwn−p≤1−βnwn−p+βnℱwn−p≤1−βnwn−p+βnwn−p=wn−p≤un−p.
Thus,(29)un+1−p=ℱ1−αnvn+αnℱvn−p≤1−αnvn+αnℱvn−p≤1−αnvn−p+αnℱvn−p≤1−αnun−p+αnvn−p≤1−αnun−p+αnun−p=un−p.
Hence,limn⟶∞un−p exists for all p∈Fℱ.Theorem 2.
LetX a uniformly convex Banach space and ℰ be a nonempty closed convex subset of X. Let ℱ:ℰ⟶ℰ be a nonexpansive mapping. Suppose that un is defined by the iteration process (25) and Fℱ≠ϕ. Then, the sequence un converges to a point of Fℱ if and only if liminfn⟶∞dun,Fℱ=0 where du,Fℱ=infu−p:p∈Fℱ.Proof.
Necessity is obvious. Suppose thatliminfn⟶∞dun,Fℱ=0. As proved in Theorem 1, limn⟶∞un−u exists for all u∈Fℱ, so limn⟶∞dun,Fℱ exists and liminfn⟶∞dun,Fℱ=0 by assumption. Now, we will prove that un is a Cauchy sequence in ℰ. For given ϵ>0, there exists N∈ℕ such that for all n≥N,(30)dun,Fℱ<ϵ2.
In particular,infuN−p:p∈Fℱ<ϵ/2. Hence, there exists p∗∈Fℱ such that uN−p∗<ϵ/2. Now for all m,n≥N,(31)um+n−un≤um+n−p∗+un−p∗≤2uN−p∗<ϵ,which shows that un is a Cauchy sequence in ℰ. But ℰ is a closed subset of X, so there exists p∈ℰ such that limn⟶∞un=p. Now, limn⟶∞dun,Fℱ=0 gives dp,Fℱ which implies p∈Fℱ.
Next, we prove that our iteration process isℱ-stable or stable with respect to ℱ.Theorem 3.
LetX be a Banach space and ℱ:X⟶X be a mapping satisfying (21). Suppose ℱ has a fixed point p. Let un be a sequence in X satisfying (9). Then, S∗-iteration process (9) is ℱ-stable.Proof.
Lettn be an arbitrary sequence in X and the sequence generated by (25) is un+1=fℱ,un converging to a unique fixed point p and ϵn=tn+1−fℱ,tn. We will prove that limn⟶∞ϵn=0⟺limn⟶∞tn=p. Assume that limn⟶∞ϵn=0 and(32)tn+1−p=tn+1−fℱ,tn+fℱ,tn−p≤tn+1−fℱ,tn+fℱ,tn−p≤tn+1−ℱ1−αnsn+αnℱsn+ℱ1−αnsn+αnℱsn−p≤ϵn+b1−αn1−bsn−p=ϵn+b1−αn1−bℱ1−βnrn+βnℱrn−p≤ϵn+b21−αn1−b1−βn1−brn−p=ϵn+b21−αn1−b1−βn1−bℱ1−γnvn+γnℱvn−p≤ϵn+b31−αn1−b1−βn1−b1−γn1−bvn−p=ϵn+b31−αn1−b1−βn1−b1−γn1−bℱ1−ζntn+ζnℱtn−p≤ϵn+b41−αn1−b1−βn1−b1−γn1−b1−ζn1−btn−p.
Sinceb∈0,1 and αn, βn, γn, and ζn are in 0,1,(33)b41−αn1−b1−βn1−b1−γn1−b1−ζn1−b<1.Hence by Lemma 4, we have limn⟶∞tn−p=0, which gives limn⟶∞tn=p. On the other hand, suppose that limn⟶∞tn=p. Then,(34)ϵn=tn+1−fℱ,tn=tn+1−p+p−fℱ,tn≤tn+1−p+b1−αn+bαnsn−p=tn+1−p+b1−αn+bαnℱ1−βnrn+βnℱrn−p≤tn+1−p+b21−αn+bαn1−βn1−brn−p=tn+1−p+b21−αn+bαn1−βnℱ1−γnvn+γnℱvn−p≤tn+1−p+b31−αn1−b1−βn1−b1−γn1−bvn−p=tn+1−p+b31−αn1−b1−βn1−b1−γn1−bℱ1−ζntn+ζnℱtn−p≤tn+1−p+b41−αn1−b1−βn1−b1−γn1−b1−ζn1−btn−p.Taking limit as n⟶∞ in (34), we get limn⟶∞ϵn=0.
Now, we present an example to compare the rate of convergence of our iteration scheme with others.Example 1.
LetX=ℝ and C=1,50. Let ℱ:C⟶C be a mapping defined by ℱu=u2−9u+54 for all u∈C. For u1=30 and αn=βn=γn=3/4, n=1,2,3,….From Table 1, we can see that all the iteration procedures are converging to p∗=6. Clearly, our iteration process requires the least number of iteration as compared to other iteration schemes.
In Figure1, 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:Table 1
Comparison of the rate of convergence with various iteration schemes.
StepPicardIshikawaNoorAgarwalAbbasThakurK. UllahS∗-iter.130.0000000030.0000000030.0000000030.0000000030.0000000030.0000000030.0000000030.00000000226.1533936625.0119824023.4891033224.0503308222.6107900821.3066758517.140342937.939900241322.4191761020.2547559017.4668190718.4372719415.8281562713.588995977.9202415346.000499545418.8373796515.8509087812.3265857313.3938203610.258206418.1129739556.0388186846.000000071515.4696624212.013305158.7275766179.3725555877.0018379256.2256746266.0004692296.000000000612.413037249.0688620336.9585711606.9939357186.1191542106.0151302216.0000056146.00000000079.8166266257.2820400266.3102146266.1862067866.0112132586.0009604946.0000000676.00000000087.8750567416.4668031466.0979255676.0283693666.0010243036.0000607496.0000000016.00000000096.7187058286.1600652386.0306808436.0041338826.0000933046.0000038416.0000000006.000000000106.2187342406.0537250406.0095903086.0005981886.0000084976.0000002426.0000000006.000000000116.0583865346.0179028376.0029956086.0000864726.0000007746.0000000166.0000000006.000000000126.0148623086.0059514316.0009354926.0000124986.0000000716.0000000016.0000000006.000000000136.0037328236.0019768486.0002921226.0000018066.0000000056.0000000006.0000000006.000000000146.000934296.0006564626.0000912176.0000002616.0000000016.0000000006.0000000006.000000000156.0002336416.0002179766.0000284836.0000000376.0000000006.0000000006.0000000006.000000000166.0000584156.0000723766.0000088946.0000000056.0000000006.0000000006.0000000006.000000000176.0000146036.0000240326.0000027786.0000000016.0000000006.0000000006.0000000006.000000000186.0000036516.0000079796.0000008666.0000000006.0000000006.0000000006.0000000006.000000000196.0000009126.0000026496.0000002706.0000000006.0000000006.0000000006.0000000006.000000000206.0000002276.0000008806.0000000846.0000000006.0000000006.0000000006.0000000006.000000000216.0000000576.0000002936.0000000266.0000000006.0000000006.0000000006.0000000006.000000000226.0000000146.0000000976.0000000086.0000000006.0000000006.0000000006.0000000006.000000000236.0000000036.0000000326.0000000036.0000000006.0000000006.0000000006.0000000006.000000000246.0000000016.0000000106.0000000016.0000000006.0000000006.0000000006.0000000006.000000000256.0000000006.0000000036.0000000006.0000000006.0000000006.0000000006.0000000006.000000000266.0000000006.0000000016.0000000006.0000000006.0000000006.0000000006.0000000006.000000000276.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.000000000286.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.000000000296.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.000000000306.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.0000000006.000000000Figure 1
Graphical representation of convergence of iterative schemes.Table 2
Number of iterations in which fixed point attains.
Iterative methodNumber of iterationsPicard25Ishikawa27Noor25Agarwal18Abbas15Thakur13K. Ullah9S∗-iter.5


## 5. Some Convergence Results for Suzuki’s Generalized Nonexpansive Mappings

This section contains some weak and strong convergence results for a sequence generated byS∗-iteration process for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces.Lemma 5.
Suppose thatℰ be a nonempty closed convex subset of a Banach space X. Let ℱ:ℰ⟶ℰ be a Suzuki generalized nonexpansive mapping with Fℱ≠ϕ. For u0∈ℰ, the sequence un generated by S∗-iteration process, limn⟶∞un−p exists for all p∈Fℱ.Proof.
Result follows from Proposition1i and Theorem 1.Lemma 6.
Suppose thatℰ be a nonempty closed convex subset of a uniformly Banach space X. Let ℱ:ℰ⟶ℰ be a Suzuki’s generalized nonexpansive mapping with Fℱ≠ϕ. For arbitrarily chosen u0∈ℰ, the sequence un is generated by S∗-iteration process. Then, Fℱ≠ϕ if and only if un is bounded and limn⟶∞ℱun−un=0.Proof.
Suppose thatFℱ≠ϕ and let p∈ℰ. Then, by Lemma 5, limn⟶∞un−p exists and un is bounded. Let(35)limn⟶∞un−p=r.From (26) and (37), we have(36)limsupn⟶∞un−p≤limsupn⟶∞un−p=r.By the Proposition 1iii, we have(37)limsupn⟶∞ℱxn−p≤limsupn⟶∞un−p=r,(38)un+1−p=ℱ1−αnvn+αnℱvn−p≤1−αnvn+αnℱvn−p≤vn−p=ℱ1−βnwn+βnℱwn−p≤wn−p=ℱ1−γnxn+γnℱxn−p≤1−γnun−p+γnxn−p=un−p−γnun−p+γnxn−p.
This implies that(39)un+1−p−un−pγn≤xn−p−un−p,un+1−p−un−p≤un+1−p−un−pγn≤xn−p−un−p,un+1−p≤xn−p,(40)r≤liminfn⟶∞xn−p,(41)limn⟶∞xn−p=r,limn⟶∞1−ζnun+ζnℱun−p=r,limn⟶∞‖1−ζnun−p+ζnℱun−p‖=r.
From equations (26) and (37) and Lemma 3, we have(42)limn⟶∞ℱun−un=0.
Conversely, assume thatun is bounded and limn⟶∞ℱun−un=0. Suppose that p∈Aℰ,un. Using Proposition 1iii, we get(43)rℱp,un=limsupn⟶∞un−ℱp≤limsupn⟶∞3ℱun−un+un−p≤limsupn⟶∞un−p=rp,un.
This shows thatℱp∈Aℰ,un. Since X is uniformly convex, Aℰ,un is singleton. Thus, we have ℱp=p, that is, Fℱ≠ϕ.Theorem 4.
(weak convergence theorem). Suppose thatℰ be a nonempty closed convex subset of a uniformly Banach space X with the Opial property. Let ℱ:ℰ⟶ℰ be Suzuki’s generalized nonexpansive mapping. For arbitrarily chosen u0∈ℰ, let the sequence un be generated by S∗-iteration process with Fℱ≠ϕ. Then, un converges weakly to a fixed point of ℱ.Proof.
SinceFℱ≠ϕ, by Lemma 6, the sequence un is bounded and limn⟶∞ℱun−un=0. Also, as X is uniformly convex so X is reflexive, thus by Eberlin’s theorem, there exists a subsequence of un say uni which converges weakly to some q1∈X. Now, since ℰ is closed and convex so by Mazur’s theorem q1∈ℰ. Hence, by Lemma 1, q1∈Fℱ. We show that un converges weakly to q1. On contrary, suppose that it is not true. Then, there must exist a subsequence of un, say unj, such that unj converges weakly to q2∈ℰ with q1≠q2. Using Lemma 1, we have q2∈Fℱ. Now, since limn⟶∞un−p exists for all p∈Fℱ. Using Lemma 6 and Opial property, we have(44)limn⟶∞un−q1=limi⟶∞uni−q1<limi⟶∞uni−q2=limn⟶∞un−q2=limj⟶∞unj−q2<limj⟶∞unj−q1=limn⟶∞un−q1,which is a contradiction; hence, q1=q2. This shows that un converges weakly to a fixed point of ℱ.Theorem 5.
(strong convergence theorem). Suppose thatℰ be a nonempty closed convex subset of a uniformly Banach space X. Let ℱ:ℰ⟶ℰ be a Suzuki’s generalized nonexpansive mapping. For arbitrarily chosen u0∈ℰ, let the sequence un be generated by S∗-iteration process with Fℱ≠ϕ. Then, un converges strongly to a fixed point of ℱ.Proof.
Using Lemma2, we get Fℱ≠ϕ and hence by Lemma 6, we have limn⟶∞ℱun−un=0. By the compactness of ℰ, there exists a subsequence of un, say uni, converging strongly to p for some p∈ℰ. Now by using Proposition 1iii, we get(45)uni−ℱp≤3ℱuni−uni+uni−p.
Taking limiti⟶∞, we get ℱp=p, that is, p∈Fℱ. By using Lemma 5, limn⟶∞un−p exists for all p∈FT; hence, un converges strongly to p.Theorem 6.
Suppose thatℰ be a nonempty closed convex subset of a uniformly Banach space X. Let ℱ:ℰ⟶ℰ be a Suzuki’s generalized nonexpansive mapping. For arbitrarily chosen u0∈ℰ, the sequence un be generated by S∗-iteration process with Fℱ≠ϕ. If ℱ satisfies condition I, then un converges strongly to a fixed point of ℱ.Proof.
By Lemma5, limn⟶∞un−p exists for all p∈Fℱ; hence, limn⟶∞dun,Fℱ exists. Let limn⟶∞un−p=δ for some δ≥0. Now if δ=0, then there is nothing to prove. Suppose δ>0; from condition I and the hypothesis, we have(46)fdun,Fℱ≤ℱun−un.
AsFℱ≠ϕ, by Lemma 5, we have limn⟶∞ℱun−un=0. Hence, (46) implies that(47)limn⟶∞fdun,Fℱ=0.
Sincef is a nondecreasing function, by equation (47), we get limn⟶∞dun,Fℱ=0. Thus, we have a subsequence uni of un and a sequence yi in Fℱ such that(48)uni−yi<12i,for alli∈ℕ.From equation (48),(49)uni+1−yi≤uni−yi<12i,yi+1−yi≤yi+1−ui+1+ui+1−yi≤12i+1+12i<12i−1.
Lettingi⟶∞, we get 1/2i−1⟶0. Hence, yi is a Cauchy sequence in Fℱ, so it converges to p. As Fℱ is closed, p∈Fℱ and then uni converges strongly to p. Since limn⟶∞un−p exists, we have un⟶p∈Fℱ. This completes the proof.



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