# The Convergence of Three-Step Iterative Schemes for GeneralizedΦ−Hemi-Contractive Mappings and the Comparison of Their Rate of Convergence

Authors: Linxin Li; Dingping Wu
Journal: Journal of Mathematics
 (2021)

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## Abstract

Charles proved the convergence of Picard-type iteration for generalizedΦ−accretive nonself-mappings in a real uniformly smooth Banach space. Based on the theorems of the zeros of strongly Φ−quasi-accretive mappings and fixed points of strongly Φ−hemi-contractions, we extend the results to Noor iterative process and SP iterative process for generalized Φ−hemi-contractive mappings. Finally, we analyze the rate of convergence of four iterative schemes, namely, Noor iteration, iteration of Corollary 2, SP iteration, and iteration of Corollary 4.


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## Body

## 1. Introduction and Preliminaries

In 2009, Charles [1] proved the convergence of Picard-type iteration for generalized Φ−accretive nonself-mappings in a real uniformly smooth Banach space. In this paper, we consider that the Noor iteration process and SP iteration process will be extended from the results of Charles [1].In [2], Noor et al. proved the convergence of Noor iteration to fixed point of a pseudocontractive self-map defined in a real uniformly smooth Banach space. In [3], Qin and Yao showed the weak convergence of a Mann-like algorithm for nonexpansive and accretive operators. In [4], the author considers some generalized nonexpansive mappings on convex metric spaces and gives some sufficient and necessary conditions for a Noor-type iteration to approximate a common fixed point of an infinite family of uniformly quasi-supfn−Lipschitzian mappings and an infinite family of gn−expansive mappings in convex metric spaces.In 1953, the most general Mann iterative scheme now studied is the following:x0∈K,(1)xn+1=1−cnxn+cnTxn,n=0,1,2,…,where cnn=1∞⊂0,1 is a real sequence satisfying appropriate conditions. In 1974, Ishikawa [5] introduced the Ishikawa iteration process as follows: for a convex subset D of a Banach space E and a mapping T from D into itself, for any given x0∈D, the sequence xn in D is defined by(2)xn+1=1−αnxn+αnTyn,yn=1−βnxn+βnTxn,n≥0,where αn and βn are two sequences in [0, 1] satisfying the conditions 0≤αn≤βn≤1, for all n,limn⟶∞βn=0 and ∑n=0∞αnβn=∞.In 2000, Noor [6, 7] gave the following three-step iterative schemes for solving nonlinear operator equations in uniformly smooth Banach spaces.LetD be a nonempty convex subset of E and let T:D⟶D be a mapping. For a given xn in D, compute the sequence xnn=0∞ by the iterative schemes:(3)xn+1=1−αnxn+αnTyn,yn=1−βnxn+βnTzn,zn=1−γnxn+γnTxn,n≥0,which is called the Noor iterative process, where αn,βn,andγn are three real sequences in [0,1] satisfying some certain conditions.In 2011, Phuengrattana and Suantai defined the SP iteration schemes [8] as follows.LetD be a nonempty convex subset of E and let T:D⟶D be a mapping. For a given xn in D, define the sequence xnn=0∞ by(4)xn+1=1−αnyn+αnTyn,yn=1−βnzn+βnTzn,zn=1−γnxn+γnTxn,n≥0,where αn,βn,andγn are three real sequences in [0,1] satisfying some certain conditions.Definition 1.
(see [1]). Given a gauge function ϕ, the mapping Jϕ:E⟶2E∗ defined by(5)Jϕx≔u∗∈E∗:x,u∗=xu∗;u∗=ϕxis called the duality map with gauge function ϕ, where E is any normed space. In the particular case, ϕt=t, the duality map J=Jϕ is called the normalized duality map.Proposition 1.
(see [9, 10]). If a Banach space E has a uniformly Gateaux differentiable norm, then J:E⟶E∗ is uniformly continuous on bounded subsets of E from the strong topology of E to the weak∗ topology of E∗.Definition 2.
(see [11]). Let E be an arbitrary real normed linear space. A mapping T:DT⊆E⟶E is called strongly hemi-contractive if FT≠∅, and there exists t>1 such that, for all r>0,(6)x−x∗≤1+rx−x∗−rtTx−x∗,holds for all x∈DT,x∗∈FT. If t=1, then T is called hemi-contractive. Finally, T is called generalized Φ−hemi-contractive with strictly increasing continuous function Φ:0,∞⟶0,∞ such that Φ0=0 if, for all x∈DT,x∗∈FT≠∅, there exists jx−x∗∈Jx−x∗ such that(7)I−Tx−I−Tx∗,jx−x∗≥Φx−x∗.
It follows from inequality (7) that T is generalized Φ−hemi-contractive if and only if(8)Tx−x∗,jx−x∗≤x−x∗2−Φx−x∗,∀n≥0.Definition 3.
(see [1, 12]). Let NT=x∈E:Tx=0≠∅. The mapping T:DT⊆E⟶E is called generalized Φ−quasi-accretive with strictly increasing continuous function Φ:0,∞⟶0,∞ such that Φ0=0 if, for all x∈E,x∗∈NT≠∅, there exists jx−x∗∈Jx−x∗ such that(9)Tx−Tx∗,jx−x∗≥Φx−x∗.Definition 4.
(see [13, 14]). A mapping T:D⟶D is called generalized Lipschitz if there exists a constant L>0 such that Tx−Ty≤L1+x−y,∀x,y∈D.Proposition 2.
(see [1]). If FT=x∈E:Tx=x≠∅, the mapping T:E⟶E is strongly hemi-contractive if and only if I−Tis strongly quasi-accretive, T is strongly ϕ−hemi-contractive if and only if I−T is strongly ϕ−quasi-accretive, and T is generalized Φ−hemi-contractive if and only if I−T is generalized Φ−quasi-accretive.Proposition 3.
(see [1]). Let E be a uniformly smooth real Banach space, and let J:E⟶2E∗ be a normalized duality mapping. Then,(10)x+y2≤x2+2y,Jx+y,for all x,y∈E.Proposition 4.
(see [1]). Let λn and γn be sequences of nonnegative numbers and αn be a sequence of positive numbers satisfying the conditions ∑n=1∞αn=∞ and γn/αn⟶0, as n⟶∞. Let the recursive inequality(11)λn+1≤λn−αnψλn+γn,n=1,2,…,be given, where ψ:0,∞⟶0,∞ is strictly increasing continuous function such that it is positive on 0,∞ and ψ0=0. Then, λn⟶0, as n⟶∞.


## 2. Main Results

In this section, we will consider to extend the result of Charles [1] to the Noor iteration process and SP iteration process under the following assumptions.First, we extend the result of Charles [1] to the Noor iteration process.Theorem 1.
SupposeD is a nonempty closed convex subset of a real uniformly smooth Banach space E. Suppose T:D⟶D is a bounded generalized Lipschitz Φ−hemi-contractive mapping and x∗∈FT≠∅. For arbitrary x0∈D,xn is a Noor iterative sequence defined by (3), where αn,βn,γn⊆0,1,limn⟶∞αn=limn⟶∞βn=0, γn=oβn, and ∑αn=∞. Then, there exists a constant d0>0 such that if 0<αn,βn,γn≤d0,xn converges strongly to the unique fixed point x∗ of T.Proof.
SinceT:D⟶D is a bounded generalized Lipschitz Φ−hemi-contractive mapping, there exists a strictly increasing continuous function Φ:0,∞⟶0,∞ with Φ0=0 such that(12)I−Tx−I−Tx∗,jx−x∗≥Φx−x∗,i.e.,(13)Tx−x∗,jx−x∗≤x−x∗2−Φx−x∗,∀n≥0,for any x,y∈D, x∗∈FT and exist a constant L>0 such that(14)Tx−Ty≤L1+x−y,∀x,y∈D.
Letr be sufficiently large such that x1∈Brx∗. Define G:=Brx∗¯∩D. Then, since T is bounded, we have that I−TG is bounded.
Asj is uniformly continuous on bounded subsets of E, for ε0:=Φr/4/40L1+2r, there exists a δ:r/2>δ>0 such that x,y∈DT,x−y<δ implies jx−jy<ε0. Set d0=minδ/2L+1+2Lr,Φr/4/8r2.Claim 1.
xn is bounded.
Suffices to show thatxn is in G for all n≥1. The proof is by induction. By our assumption, x1∈G. Suppose xn∈G. We prove that xn+1∈G. Assume for contradiction that xn+1∉G. Then, since xn+1∈D,∀n≥1, we have that xn+1−x∗>r. We have the following estimates:(15)xn−x∗−zn−x∗≤γnxn−Txn≤γnxn−x∗+Txn−x∗≤d0L+1+Lr<δ,zn−x∗=1−γnxn+γnTxn−x∗≤1−γnxn−x∗+γnTxn−x∗≤1−γnxn−x∗+γnL1+xn−x∗≤r+d0L1+r≤2r.
Using (15), we obtain(16)Tzn−x∗≤L1+zn−x∗≤L1+2r.
Using (16), we obtain(17)yn−x∗=1−βnxn+βnTzn−x∗≤1−βnxn−x∗+βnTzn−x∗≤r+d0L1+zn−x∗≤2rand(18)yn−x∗−xn−x∗≤βnTzn−xn≤βnTzn−x∗+xn−x∗≤d0⋅L+1+2Lr<δ.
Using (17), we obtain(19)Tyn−x∗≤L1+yn−x∗≤L1+2r.
Using (19), we obtain(20)xn+1−x∗−xn−x∗≤αnTyn−xn≤αnTyn−x∗+xn−x∗≤d0L+1+2Lr<δ.
Then,(21)xn−x∗≥xn+1−x∗−αnTyn−xn≥r−d0L+1+2Lr≥r2,zn−x∗≥xn−x∗−γnTxn−xn≥r2−d0L+1+Lr≥r4.yn−x∗≥xn−x∗−βnTzn−xn≥r2−d0L+1+2Lr≥r4,
Therefore,(22)jxn+1−x∗−jxn−x∗<ε0,jxn−x∗−jyn−x∗<ε0,jxn−x∗−jzn−x∗<ε0.
Using Proposition3 and the above formulas, we obtain(23)zn−x∗2=1−γnxn+γnTxn−x∗2=1−γnxn−1−γnx∗+γnTxn−γnx∗2≤1−γn2xn−x∗2+2γnTxn−x∗,jxn−x∗+2γnTxn−x∗,jzn−x∗−jxn−x∗≤r2+2γnL1+rε0,(24)yn−x∗2=1−βnxn+βnTzn−x∗2≤1−βn2xn−x∗2+2βnTzn−x∗,jzn−x∗+2βnTzn−x∗,jyn−x∗−jxn−x∗+2βnTzn−x∗,jxn−x∗−jzn−x∗≤1−βn2xn−x∗2+2βnzn−x∗2−Φzn−x∗+2βnTzn−x∗jyn−x∗−jxn−x∗+2βnTzn−x∗jxn−x∗−jzn−x∗≤1−βn2r2+2βnzn−x∗2−Φzn−x∗+4βnL1+2rε0,(25)xn+1−x∗2=1−αnxn+αnTyn−x∗2≤1−αn2xn−x∗2+2αnTyn−x∗,jyn−x∗+2αnTyn−x∗,jxn−x∗−jyn−x∗+2αnTyn−x∗,jxn+1−x∗−jxn−x∗≤1−αn2xn−x∗2+2αnyn−x∗2−Φyn−x∗+2αnTyn−x∗jxn−x∗−jyn−x∗+2αnTyn−x∗jxn+1−x∗−jxn−x∗≤1−αn2r2+4αnL1+2rε0+2αnyn−x∗2−Φyn−x∗.
Substitute (23) into (24) and then substitute (24) into (25); since 0<αn,βn,γn≤d0 and d0=minδ/2L+1+2Lr,Φr/4/8r2, we have(26)xn+1−x∗2≤1−αn2r2+4αnL1+2rε0+2αn1−βn2r2+4βnL1+2rε0+2αn2βnr2+2γnL1+rε0−Φr4−Φr4≤r2+2αn2βn2r2+10L1+2rε0−2βnΦr4−Φr4≤r2+2αnΦr/42−2βnΦr4−Φr4≤r2,i.e., xn+1−x∗≤r, a contradiction. Therefore, xn+1∈G. Thus, by induction, xn is bounded. Then, yn,zn,Txn,Tyn,andTzn are also bounded.Claim 2.
xn⟶x∗.
LetAn=jxn+1−x∗−jxn−x∗,Bn=jxn−x∗−jyn−x∗,Cn=jzn−x∗−jxn−x∗. Note that xn+1−xn⟶0,xn−yn⟶0,andzn−xn⟶0 as n⟶∞, and hence, by the uniform continuity of j on bounded subsets of E, we have that(27)An⟶0,Bn⟶0,Cn⟶0,asn⟶∞.
LetM1=maxsupxn−x∗,supTxn−x∗,supTyn−x∗,supTzn−x∗; by (24)–(26), we obtain that(28)zn−x∗2≤1−γn2xn−x∗2+2γnM1Cn≤xn−x∗2+2γnM1Cn,(29)yn−x∗2≤1−βn2xn−x∗2+2βnzn−x∗2−Φzn−x∗+2βnM1Cn+2βnM1Bn,and(30)xn+1−x∗2≤1−αn2xn−x∗2+2αnyn−x∗2−Φyn−x∗+2αnTyn−x∗jxn+1−x∗−jxn−x∗+2αnTyn−x∗jxn−x∗−jyn−x∗≤1−αn2xn−x∗2+2αnM1An+2αnM1Bn+2αnyn−x∗2−Φyn−x∗.
Taking (28) into (29) and taking (29) into (30),(31)xn+1−x∗2≤1−αn2xn−x∗2+2αnM1An+2αnM1Bn+2αn1+βn2xn−x∗2−Φyn−x∗+2αn2βn2γnM1Cn−Φzn−x∗+M1Cn+M1Bn≤xn−x∗2+2αnαn2M12+βn2M12−Φyn−x∗+2αnM1An+2αnM1Bn+4αnβnM1Cn+4αnβnM1Bn+2αn4βnγnM1Cn−2βnΦzn−x∗≤xn−x∗2+2αnQn−2βnΦzn−x∗−Φyn−x∗,where Qn=αn/2M12+βn2M12+4βnγnM1Cn+2βnM1Cn+2βnM1Bn+M1An+M1Bn⟶0 as n⟶∞.
SetinfΦzn−x∗/Φxn−x∗+1=W; since Φ is a strictly increasing continuous function, then W exists. Thus,(32)Φzn−x∗≥WΦxn−x∗+W≥WΦxn−x∗.
SetinfΦyn−x∗/Φxn−x∗+1=N; since Φ is a strictly increasing continuous function, then N exists. Thus,(33)Φyn−x∗≥NΦxn−x∗+N≥NΦxn−x∗.
Then,(34)xn+1−x∗2≤xn−x∗2+2αnQn−2βnΦzn−x∗−Φyn−x∗≤xn−x∗2−2αn2βnW−NΦxn−x∗+2αnQn.
Letλn=xn−x∗2,Sn=22βnW−N, and ρn=2αnQn; then, from inequality (34), we obtain that λn+1≤λn−αnSnΦλn+ρn, where ρn/αnSn⟶0 as n⟶∞. Therefore, the conclusion of the theorem follows from Proposition 4. Uniqueness of x∗ is derived from the definition of T.By Definition4, we know that the generalized Lipschitz mapping is the extension of nonexpansive mapping, so the following corollary follows trivially.Corollary 1.
SupposeD is a nonempty closed convex subset of a real uniformly smooth Banach space E. Suppose T:D⟶D is a bounded generalized Φ−hemi-contractive mapping and x∗∈FT≠∅. For arbitrary x0∈D,xn is a Noor iterative sequence defined by (3), where αn,βn,γn⊆0,1,limn⟶∞αn=limn⟶∞βn=0, γn=oβn, and ∑αn=∞. Then, there exists a constant d0>0 such that if 0<αn,βn,γn≤d0,xn converges strongly to the unique fixed point x∗ of T.The following corollary follows trivially from Definition2 and Definition 3.Corollary 2.
SupposeE is a real uniformly smooth Banach space. Suppose T:E⟶E is a bounded generalized Φ−accretive mapping and the solution x∗ of the equation Tx=0 exists. For arbitrary x0∈E,xn in E is defined as(35)xn+1=1−αnxn−αnTyn,yn=1−βnxn−βnTzn,zn=1−γnxn−γnTxn,n≥0,where αn,βn,γn⊆0,1,limn⟶∞αn=limn⟶∞βn=0, γn=oβn, and ∑αn=∞. Then, there exists a constant d0>0 such that if 0<αn,βn,γn≤d0,xn converges strongly to the unique solution Tx=0.Now, we extend the result of Charles [1] to the SP iteration process as follows.Theorem 2.
SupposeD is a nonempty closed convex subset of a real uniformly smooth Banach space E. Suppose T:D⟶D is a bounded generalized Lipschitz Φ−hemi-contractive mapping and x∗∈FT≠∅. For arbitrary x0∈D,xn is a SP iterative sequence defined by (4), where αn,βn,γn⊆0,1,limn⟶∞αn=limn⟶∞βn=0, γn=oβn, and ∑γn=∞. Then, there exists a constant d0>0 such that if 0<αn,βn,γn≤d0,xn converges strongly to the unique fixed point x∗ of T.Proof.
SinceT:D⟶D is a bounded generalized Lipschitz Φ−hemi-contractive mapping, there exists a strictly increasing continuous function Φ:0,∞⟶0,∞ with Φ0=0 such that(36)I−Tx−I−Tx∗,jx−x∗≥Φx−x∗,i.e.,(37)Tx−x∗,jx−x∗≤x−x∗2−Φx−x∗,∀n≥0,for any x,y∈D and x∗∈FT, and there exist a constant L>0 such that(38)Tx−Ty≤L1+x−y,∀x,y∈D.
Letr be sufficiently large such that x1∈Brx∗. Define G:=Brx∗¯∩D. Then, since T is bounded, we have that I−TG is bounded.
Asj is uniformly continuous on bounded subsets of E, for ε0:=Φr/4/6L+3r1+L, there exists a δ:r/4>δ>0 such that x,y∈DT,x−y<δ implies jx−jy<ε0. Set d0=δ/2L+1+L3r.Claim 3.
xn is bounded.
Suffices to show thatxn is in G for all n≥1. The proof is by induction. By our assumption, x1∈G. Suppose xn∈G. We prove that xn+1∈G. Assume for contradiction that xn+1∉G. Then, since xn+1∈D,∀n≥1, we have that xn+1−x∗>r. We have the following estimates:(39)xn−x∗−zn−x∗≤γnxn−Txn≤γnxn−x∗+Txn−x∗≤d0L+1+Lr<δ,zn−x∗=1−γnxn+γnTxn−x∗≤1−γnxn−x∗+γnTxn−x∗≤1−γnxn−x∗+γnL1+xn−x∗≤r+d0L1+r≤2r.
Using (39), we obtain(40)Tzn−zn≤Tzn−x∗+zn−x∗≤L+1+L2r.
Using (40), we obtain(41)zn−x∗−yn−x∗≤βnzn−Tzn≤βnzn−x∗+Tzn−x∗≤βn2r+L1+2r<δ,yn−x∗=1−βnzn+βnTzn−x∗≤1−βnzn−x∗+βnTzn−x∗≤2r+d0L1+2r≤3r.
Using (41), we obtain(42)yn−x∗−xn+1−x∗≤αnyn−x∗+Tyn−x∗≤αnyn−x∗+L1+yn−x∗≤d03r+L1+3r<δ,Tyn−yn=Tyn−x∗−yn−x∗≤L1+yn−x∗+yn−x∗≤L+1+L3r.
Then,(43)yn−x∗≥xn+1−x∗−αnTyn−yn≥r−d0L+1+L3r≥r2,zn−x∗≥yn−x∗−γnTzn−zn≥r2−d0L+1+L2r≥r4.
Therefore,(44)jxn+1−x∗−jyn−x∗<ε0,jyn−x∗−jzn−x∗<ε0,jzn−x∗−jxn−x∗<ε0.
Using Proposition3 and the above formulas, we obtain(45)zn−x∗2=1−γnxn+γnTxn−x∗2=1−γnxn−1−γnx∗+γnTxn−γnx∗2≤xn−x∗2−2γnI−Txn,jxn−x∗+2γnI−Txn,jzn−x∗−jxn−x∗≤xn−x∗2−2γnΦxn−x∗+2γnL+1+Lrε0≤r2+2γnL+1+Lrε0,(46)yn−x∗2=1−βnzn+βnTzn−x∗2=zn−x∗−βnI−Tzn2≤zn−x∗2−2βnI−Tzn,jzn−x∗−2βnI−Tzn,jyn−x∗−jzn−x∗≤zn−x∗2−2βnΦzn−x∗+2βnzn−Tznjyn−x∗−jzn−x∗≤zn−x∗2−2βnΦzn−x∗+2βnL+1+L2rε0,(47)xn+1−x∗2=1−αnyn+αnTyn−x∗2≤yn−x∗−αnI−Tyn2≤yn−x∗2−2αnI−Tyn,jyn−x∗−2αnI−Tyn,jxn+1−x∗−jyn−x∗≤yn−x∗2−2αnΦyn−x∗+2αnyn−Tynjxn+1−x∗−jyn−x∗≤yn−x∗2−2αnΦyn−x∗+2αnL+1+L3rε0.
Substitute (45) into (46) and then substitute (46) into (47); since 0<αn,βn,γn≤d0 and d0=δ/2L+1+L3r, we have(48)xn+1−x∗2≤r2+2γnL+1+Lrε0−2βnΦr4+2βnL+1+L2rε0−2αnΦr2+2αnL+1+L3rε0≤r2+2αnL+1+L3rε0+βnαnL+1+L2rε0+2αnγnαnL+1+Lrε0−Φr2−βnαnΦr4≤r2+2αnΦr/42−Φr4−βnαnΦr4≤r2,i.e., xn+1−x∗≤r, a contradiction. Therefore, xn+1∈G. Thus, by induction, xn is bounded. Then, yn,zn,Txn,Tyn,andTzn are also bounded.Claim 4.
xn⟶x∗.
LetAn=jxn+1−x∗−jyn−x∗,Bn=jyn−x∗−jzn−x∗,Cn=jzn−x∗−jxn−x∗; note that xn+1−yn⟶0,yn−zn⟶0,andzn−xn⟶0, as n⟶∞, and hence, by the uniform continuity of j on bounded subsets of E, we have that(49)An⟶0,Bn⟶0,Cn⟶0,asn⟶∞.
LetM1=maxsupyn−Tyn,supzn−Tzn,supxn−Txn,supTzn−x∗; by (46)–(48), we obtain that(50)zn−x∗2≤xn−x∗2−2γnΦxn−x∗+2γnM1Cn,(51)yn−x∗2≤zn−x∗2−2βnΦzn−x∗+2βnM1Bn,(52)xn+1−x∗2≤yn−x∗2−2αnΦyn−x∗+2αnM1An.
Taking (50) into (51) and taking (51) into (52),(53)xn+1−x∗2≤xn−x∗2−2γnΦxn−x∗+2γnM1Cn−2βnΦzn−x∗−2αnΦyn−x∗+2βnM1Bn+2αnM1An≤xn−x∗2−2γnΦxn−x∗−2αnΦyn−x∗+2γnαnγnM1An+βnγnM1Bn+M1Cn−2βnΦzn−x∗,where Qn=αn/γnM1An+βn/γnM1Bn+M1Cn⟶0 as n⟶∞.
SetinfΦzn−x∗/Φxn−x∗+1=W; since Φ is a strictly increasing continuous function, then W exists. Thus,(54)Φzn−x∗≥WΦxn−x∗+W≥WΦxn−x∗.
SetinfΦyn−x∗/Φxn−x∗+1=N; since Φ is a strictly increasing continuous function, then N exists. Thus,(55)Φyn−x∗≥NΦxn−x∗+N≥NΦxn−x∗.
Then,(56)xn+1−x∗2≤xn−x∗2+2γnQn−2γnΦxn−x∗−2βnWΦxn−x∗−2αnNΦxn−x∗≤xn−x∗2+2γnQn−2γnΦxn−x∗1+βnγnW+αnγnN.
Letλn=xn−x∗2,Sn=21+βn/γnW+αn/γnN and ρn=2γnQn; then, from inequality (56), we obtain that λn+1≤λn−γnSnΦλn+ρn, where ρn/γnSn⟶0 as n⟶∞. Therefore, the conclusion of the theorem follows from Proposition 4. Uniqueness of x∗ is derived from the definition of T.By Definition4, we know that the generalized Lipschitz mapping is the extension of nonexpansive mapping, so the following corollary follows trivially.Corollary 3.
SupposeD is a nonempty closed convex subset of a real uniformly smooth Banach space E. Suppose T:D⟶D is a bounded generalized Φ−hemi-contractive mapping and x∗∈FT≠∅. For arbitrary x0∈D,xn be a SP iterative sequence defined by (4), where αn,βn,γn⊆0,1,limn⟶∞αn=limn⟶∞βn=0,  γn=oβn, and ∑γn=∞. Then, there exists a constant d0>0 such that if 0<αn,βn,γn≤d0,xn converges strongly to the unique fixed point x∗ of T.The following corollary follows trivially from Definition2 and Definition 3.Corollary 4.
SupposeE is a real uniformly smooth Banach space. Suppose T:E⟶E is a bounded generalized Φ−accretive mapping and the solution x∗ of the equation Tx=0 exists. For arbitrary x0∈E,xn in E is defined as(57)xn+1=1−αnyn−αnTyn,yn=1−βnzn−βnTzn,zn=1−γnxn−γnTxn,n≥0,where αn,βn,γn⊆0,1,limn⟶∞αn=limn⟶∞βn=0,  γn=oβn, and ∑γn=∞. Then, there exists a constant d0>0 such that if 0<αn,βn,γn≤d0,xn converges strongly to the unique solution Tx=0.


## 3. Experiments

In this section, we analyze the rate of convergence of four iterative schemes, namely, Noor iteration, iteration of Corollary2, SP iteration, and iteration of Corollary 4, iterative schemes for complex space by using Visual Studio. The results obtained are extensions of some recent results of Rana et al. [15] and Chugh et al. [16].We takeαn=s,βn=s′,andγn=s″ and derive the fixed points of the following polynomial functions:Quadratic functions=z2+cCubic functions=z3+cBiquadratic functions=z4+cRecently, Rana et al. [15] drew a comparative analysis of Picard, Mann, and Ishikawa iterative schemes by starting with z=0,0 and c=0.1 in complex space. In this paper, we will continue the comparative study in complex space by taking the same z and c, for Noor iteration, iteration of Corollary 2, SP iteration, and iteration of Corollary 4 and, hence, extend the results of Rana et al. [15] and Chugh et al. [16].Quadratic functions are provided in Tables1 and 2, and their corresponding graphs are shown in Figures 1(a) and 1(b), respectively. Cubic functions are provided in Tables 3 and 4, and their corresponding graphs are shown in Figures 1(c) and 1(d). Biquadratic functions are provided in Tables 5–8, and their corresponding graphs are shown in Figures 1(a)–1(d).Figure 1
(a, b) Graphs of quadratic functions in Tables1 and 2, respectively. (c, d) Graphs of cubic functions in Tables 3 and 4, respectively.
(a)(b)(c)(d)Table 1
Iterations of quadratic functions=0.6,s′=0.3,s″=0.1.
s=0.6,s′=0.3,s″=0.1NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.0605411−0.06054110.0756347−0.075634720.0874702−0.08747020.0992939−0.099293930.100248−0.1002480.107705−0.10770540.106495−0.1064950.11082−0.1108250.109594−0.1095940.11199−0.1119960.111142−0.1111420.112432−0.11243270.111918−0.1119180.1126−0.112680.112308−0.1123080.112663−0.11266390.112503−0.1125030.112687−0.112687100.112602−0.1126020.112696−0.112696110.112652−0.1126520.1127−0.1127120.112676−0.1126760.112701−0.112701130.112689−0.1126890.112701−0.112701140.112695−0.1126950.112702−0.112702150.112698−0.1126980.112702−0.112702160.1127−0.11270.112702−0.112702170.112701−0.1127010.112702−0.112702180.112701−0.1127010.112702−0.112702190.112701−0.1127010.112702−0.112702200.112702−0.1127020.112702−0.112702……………250.112702−0.1127020.112702−0.112702Table 2
Iterations of quadratic functions=0.6,s′=0.8,s″=0.7.
s=0.6,s′=0.8,s″=0.7NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.0642255−0.06422550.104921−0.10492120.091647−0.0916470.111992−0.11199230.103513−0.1035130.112636−0.11263640.108682−0.1086820.112696−0.11269650.110942−0.1109420.112701−0.11270160.111931−0.1119310.112702−0.11270270.112364−0.1123640.112702−0.11270280.112554−0.1125540.112702−0.11270290.112637−0.1126370.112702−0.112702100.112673−0.1126730.112702−0.112702110.112689−0.1126890.112702−0.112702120.112696−0.1126960.112702−0.112702130.112699−0.1126990.112702−0.112702140.112701−0.1127010.112702−0.112702150.112701−0.1127010.112702−0.112702160.112701−0.1127010.112702−0.112702170.112702−0.1127020.112702−0.112702……………250.112702−0.1127020.112702−0.112702Table 3
Iterations of cubic functions=0.6,s′=0.3,s″=0.1.
s=0.6,s′=0.3,s″=0.1NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.0600162−0.05998380.0748305−0.074769520.0842313−0.08377040.0940842−0.093220130.0941177−0.09309290.0991753−0.097648340.0981802−0.09672410.100534−0.098701450.0998545−0.09813470.100898−0.098951260.100545−0.0986820.100996−0.099010470.100831−0.09889440.101022−0.099024580.100948−0.09897670.101029−0.099027890.100997−0.09900860.101031−0.0990286100.101017−0.0990210.101031−0.0990288110.101025−0.09902580.101031−0.0990288120.101029−0.09902770.101031−0.0990288130.10103−0.09902840.101031−0.0990289140.101031−0.09902870.101031−0.0990289150.101031−0.09902880.101031−0.0990289160.101031−0.09902880.101031−0.0990289170.101031−0.09902880.101031−0.0990289180.101031−0.09902880.101031−0.0990289190.101031−0.09902890.101031−0.0990289……………250.101031−0.09902890.101031−0.0990289Table 4
Iterations of cubic functions=0.6,s′=0.8,s″=0.7.
s=0.6,s′=0.8,s″=0.7NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.0603104−0.0596960.0982125−0.096996220.0846007−0.08342030.100946−0.098990430.0943985−0.09283720.101029−0.099028140.0983532−0.09657310.101031−0.099028850.0999499−0.09805490.101031−0.099028960.100595−0.09864260.101031−0.099028970.100855−0.09887570.101031−0.099028980.10096−0.09896810.101031−0.099028990.101003−0.09900480.101031−0.0990289100.10102−0.09901930.101031−0.0990289110.101027−0.09902510.101031−0.0990289120.101029−0.09902730.101031−0.0990289130.10103−0.09902830.101031−0.0990289140.101031−0.09902860.101031−0.0990289150.101031−0.09902880.101031−0.0990289160.101031−0.09902880.101031−0.0990289170.101031−0.09902880.101031−0.0990289180.101031−0.09902880.101031−0.0990289190.101031−0.09902890.101031−0.0990289……………250.101031−0.09902890.101031−0.0990289Table 5
Iterations of biquadratic functions=0.6,s′=0.1,s″=0.1.
s=0.6,s′=0.1,s″=0.1NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.06−0.060.0676001−0.067600120.0840101−0.08401010.0895219−0.089521930.0936363−0.09363630.0966522−0.096652240.0975019−0.09750190.0989758−0.098975850.0990556−0.09905560.0997335−0.099733560.0996802−0.09968020.0999807−0.099980770.0999314−0.09993140.100061−0.10006180.100032−0.1000320.100088−0.10008890.100073−0.1000730.100096−0.100096100.100089−0.1000890.100099−0.100099110.100096−0.1000960.1001−0.1001120.100099−0.1000990.1001−0.1001130.1001−0.10010.1001−0.1001……………250.1001−0.10010.1001−0.1001Table 6
Iterations of biquadratic functions=0.6,s′=0.3,s″=0.1.
s=0.6,s′=0.3,s″=0.1NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.0600005−0.06000050.0748011−0.074801120.0840163−0.08401630.0936851−0.093685130.0936439−0.09364390.0984714−0.098471440.0975076−0.09750760.0996866−0.099686650.099059−0.0990590.0999953−0.099995360.0996821−0.09968210.100074−0.10007470.0999324−0.09993240.100094−0.10009480.100033−0.1000330.100099−0.10009990.100073−0.1000730.1001−0.1001100.10009−0.100090.1001−0.1001110.100096−0.1000960.1001−0.1001120.100099−0.1000990.1001−0.1001130.1001−0.10010.1001−0.1001……………250.1001−0.10010.1001−0.1001Table 7
Iterations of biquadratic functions=0.6,s′=0.5,s″=0.4.
s=0.6,s′=0.5,s″=0.4NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.0600038−0.06000380.0880149−0.088014920.0840261−0.08402610.0986334−0.098633430.0936535−0.09365350.0999221−0.099922140.0975142−0.09751420.100079−0.10007950.0990629−0.09906290.100098−0.10009860.0996841−0.09968410.1001−0.100170.0999334−0.09993340.1001−0.100180.100033−0.1000330.1001−0.100190.100074−0.1000740.1001−0.1001100.10009−0.100090.1001−0.1001110.100096−0.1000960.1001−0.1001120.100099−0.1000990.1001−0.1001130.1001−0.10010.1001−0.1001……………250.1001−0.10010.1001−0.1001Table 8
Iterations of biquadratic functions=0.6,s′=0.8,s″=0.7.
s=0.6,s′=0.8,s″=0.7NoorCo2.3SPCo2.6nF (x)F (x)F (x)F (x)10.0600246−0.06002460.0976546−0.097654620.0840529−0.08405290.10004−0.1000430.093674−0.0936740.100099−0.10009940.0975268−0.09752680.1001−0.100150.0990697−0.09906970.1001−0.100160.0996876−0.09968760.1001−0.100170.0999351−0.09993510.1001−0.100180.100034−0.1000340.1001−0.100190.100074−0.1000740.1001−0.1001100.10009−0.100090.1001−0.1001110.100096−0.1000960.1001−0.1001120.100099−0.1000990.1001−0.1001130.1001−0.10010.1001−0.1001……………250.1001−0.10010.1001−0.1001Figure 2
(a–d) Graphs of biquadratic functions given in Tables5–8, respectively.
(a)(b)(c)(d)


## 4. Conclusion

Keeping in mind comparative analysis drawn by [15, 16], Tables 1–8, we conclude that (i)
In case of quadratic polynomial,Fx of Noor and Co2.3 iterative schemes and SP and Co2.6 iterative schemes are opposite to each other; they show equivalence,and the speed of convergence of iterative schemes is compared as follows: SP>Noor; Co2.6>Co2.3.(ii)
In case of cubic polynomial, the speed of convergence of iterative schemes is compared as follows: SP>Noor; Co2.6>Co2.3; Noor>Co2.3; SP>Co2.6.(iii)
In case of biquadratic polynomial,Fx of Noor and Co2.3 iterative schemes and SP and Co2.6 iterative schemes are opposite to each other; they show equivalence, and the speed of convergence of iterative schemes is compared as follows: SP>Noor; Co2.6>Co2.3 (Figure 2).(iv)
In the case of biquadratic polynomial, the most important discovery is that, as long as the value ofs is set, the convergence rate of Noor and the convergence rate of Co2.3 iterative schemes will not change.



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