# On Faster Implicit Hybrid Kirk-Multistep Schemes for Contractive-Type Operators

Authors: O. T. Wahab; K. Rauf
Journal: International Journal of Analysis
 (2016)

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## Abstract

The purpose of this paper is to prove strong convergence andT-stability results of some modified hybrid Kirk-Multistep iterations for contractive-type operator in normed linear spaces. Our results show through analytical and numerical approach that the modified hybrid schemes are better in terms of convergence rate than other hybrid Kirk-Multistep iterative schemes in the literature.


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## Body

## 1. Introduction

In the recent years, numerous papers have been published on the strong convergence andT-stability of various iterative approximations of fixed points for contractive-type operators. See [1–9]. The Picard iterative scheme defined for x0∈X(1)xn=Txn-1,n≥1,was the first iteration to be proved by Banach [10] for a self-map T in a complete metric space (X,d) satisfying (2)dTx,Ty≤cdx,y(called strict contraction), for all x,y∈X and c∈(0,1). Picard iteration (1) which obeys (2) is said to have a fixed point in FT, where FT is the set of all fixed points. The Picard iteration will no longer converge to a fixed point of the operator if contractive condition (2) is weaker. Hence, there is a need to consider other iterative procedures.Mann [11] defined a more general iteration in a Banach space E satisfying quasi-nonexpansive operators. For x0∈E, the Mann iteration is given by (3)xn=1-αnxn-1+αnTxn-1,n≥1,where {αn} is a sequence of positive numbers in [0,1]. Putting αn=1 in (3) yields Picard iteration (1).A double Mann iteration, called Ishikawa iteration, was introduced by Ishikawa [12]. It is defined for x0∈E as (4)xn=1-αnxn-1+αnTyn-1yn-1=1-βnxn-1+βnTxn-1,n≥1,where {αn} and {βn} are sequences of positive numbers in [0,1].The three-step iteration, which is more general than Mann and Ishikawa iterations, was defined by Noor [13]. For x0∈E, the Noor iteration is given as (5)xn=1-αnxn-1+αnTyn-1yn-1=1-βnxn-1+βnTzn-1zn-1=1-γnxn-1+γnTxn-1,n≥1,where {αn}, {βn}, and {γn} are sequences in [0,1] with ∑αn=∞.Rhoades and Soltuz [14] defined a multistep iteration in a normed linear space E as follows. For x0∈E, (6)xn=1-αnxn-1+αnTyn-11yn-1l=1-βnlxn-1+βnlTyn-1l+1,l=1,2,…,k-2yn-1k-1=1-βnxn-1+βnTxn-1,n≥1,where {αn} and {βn(l)}, l=1,2,…,k-1, are sequences of positive numbers in [0,1] with ∑αn=∞. Iteration (6) generalized (3), (4), and (5); for example, if k=3 in (6), we recover the form (5); if k=2, we have (4); on putting k=2 and βn(l)=0 for each l, we have (3).Another two-step scheme, which is independent of (4), was introduced by Thianwan [15]. Let x0∈E; the sequence xn⊂E is defined as (7)xn=1-αnyn-1+αnTyn-1yn-1=1-βnxn-1+βnTxn-1,n≥1,where {αn} and {βn} are sequences of positive numbers in [0,1] with ∑αn=∞.The three-step iteration of (7) called SP iteration was introduced by Phuengrattana and Suantai [16] and it was defined as follows. For x0∈E, (8)xn=1-αnyn-11+αnTyn-11yn-11=1-βnyn-12+βnTyn-12yn-12=1-γnxn-1+γnTxn-1,n≥1,where {αn},{βn},{γn}⊂[0,1] with ∑αn=∞. The two-step iteration of (8) can be easily obtained when γn=0.In [17], Gürsoy et al. defined a generalized scheme of forms (7) and (8) in a Banach space. For x0∈E, (9)xn=1-αnyn-11+αnTyn-11yn-1l=1-βnlyn-1l+1+βnlTyn-1l+1,l=1,2,3,…,k-2yn-1k-1=1-βnk-1xn-1+βnk-1Txn-1,n≥1,where {αn},{βn(l)},⊂[0,1], l=1,2,…,k-1, with ∑αn=∞.Several results have been proved for the strong convergence of the explicit iteration as well as the SP-iterative scheme of fixed points for different types of contractive-like operators in various spaces. See [3, 6, 18, 19]. The stability results of explicit and SP-iterative schemes have been discussed in [2, 4, 9, 20, 21].Kirk’s iterative procedure was defined by Kirk [22]. For x0∈E, where E is a Banach space and T is a self-map of E, (10)xn+1=∑i=0qαiTixn,n=0,1,2,…,where q is a fixed integer with q≥1, αi≥0 for each i and ∑i=0qαi=1.In order to reduce the cost of computations, Olatinwo [8] introduced two hybrid schemes, namely, Kirk-Mann and Kirk-Ishikawa iterative schemes in a normed linear space. For x0∈E,(11)xn+1=∑i=0qαn,iTixn;∑i=0qαn,i=1,n=0,1,2,…,xn+1=αn,0xn+∑i=1qαn,iTiyn,∑i=0qαn,i=1,yn=∑i=0rβn,iTixn,∑i=0rβn,i=1;n=0,1,2…,respectively, where q and r are fixed integers with q≥r and αn,i and βn,i are sequences in [0,1] satisfying αn,i≥0, αn,0≠0, βn,i≥0, and βn,0≠0.The Kirk-Noor iteration was introduced by Chugh and Kumar [23] as follows: for x0∈E, (12)xn+1=αn,0xn+∑i=1qαn,iTiyn,∑i=0qαn,i=1,yn=βn,0xn+∑i=1rβn,iTizn,∑i=0rβn,i=1,zn=∑i=0sγn,iTixn,∑i=0sγn,i=1;n=0,1,2…,where q, r, and s are fixed integers with q≥r≥s, and αn,i, βn,i, and γn,i are sequences in [0,1] satisfying αn,i≥0, αn,0≠0, βn,i≥0, βn,0≠0, γn,i≥0, and γn,0≠0.In an attempt to generalize (11) and (12), Gürsoy et al. [17] introduced the Kirk-Multistep iteration in an arbitrary Banach space E. For x0∈E, (13)xn+1=αn,0xn+∑i=1q1αn,iTiyn,∑i=0q1αn,i=1,ynl=βn,0lxn+∑i=1ql+1βn,ilTiynl+1,∑i=0ql+1βn,il=1,l=11k-2ynk-1=∑i=0qkβn,ik-1Tixn,∑i=0qkβn,ik-1=1;k≥2,n=0,1,2,…,where q1,q2,q3,…,qk are fixed integers with q1≥q2≥⋯≥qk; αn,in=1∞ and βn,i(l)n=1∞ are sequences in [0,1] satisfying αn,i≥0, αn,0≠0, βn,i(l)≥0, and βn,0(l)≠0 for each l.The Kirk-SP iteration was defined by Hussain et al. [24] as follows: for x0∈E, (14)xn+1=αn,0yn+∑i=1qαn,iTiyn,∑i=0qαn,i=1,yn=βn,0zn+∑i=1rβn,iTizn,∑i=0rβn,i=1,zn=∑i=0sγn,iTixn,∑i=0sγn,i=1;n=0,1,2…,where q, r, and s are fixed integers with q≥r≥s and αn,i, βn,i, and γn,i are sequences in [0,1] satisfying αn,i≥0, αn,0≠0, βn,i≥0, βn,0≠0, γn,i≥0, and γn,0≠0. The Kirk-Thianwan iteration can be obtained if s=0 in (14).The Kirk-Multistep-SP, which generalized both Kirk-SP and Kirk-Thianwan schemes, was introduced by Akewe et al. [1] and it was defined as follows: for x0∈E, (15)xn+1=αn,0yn1+∑i=1q1αn,iTiyn1,∑i=0q1αn,i=1,ynl=βn,0lynl+1+∑i=1ql+1βn,ilTiynl+1,∑i=0ql+1βn,il=1,l=11k-2ynk-1=∑i=0qkβn,ik-1Tixn,∑i=0qkβn,ik-1=1;k≥2,n=0,1,2…,where q1,q2,q3,…,qk are fixed integers with q1≥q2≥⋯≥qk; αn,in=1∞ and βn,i(l)n=1∞ are sequences in [0,1] satisfying αn,i≥0, αn,0≠0, βn,i(l)≥0, and βn,0(l)≠0 for each l.Another modified form of explicit iteration is the implicit iteration. The implicit Mann iteration and implicit Ishikawa iteration were discussed by Ćirić et al. [7] and Xue and Zhang [19], respectively. For x0∈C, C being a closed subset of normed linear space, the implicit Mann and implicit Ishikawa iterations are, respectively, (16)xn=αnxn-1+1-αnTxn,n≥1,(17)xn=αnyn-1+1-αnTxnyn-1=βnxn-1+1-βnTyn-1,n≥1,where {αn} and {βn} are sequences in [0,1].The implicit Noor iteration was defined by Chugh et al. [5] as follows: for x0∈C(18)xn=αnyn-1+1-αnTxnyn-1=βnzn-1+1-βnTyn-1zn-1=γnxn-1+1-γnTzn-1,n≥1,where {αn}, {βn}, and {γn} are sequences in [0,1] with ∑1-αn=∞. The implicit Noor iteration (18) is more general than (16) and (17).The most generalized Banach operator used by several authors is the one proved by Zamfirescu [25].LetX be a complete metric space and let T be a self-map of X. The operator T is Zamfirescu operator if for each pair of points x,y∈X, at least one of the following is true: (19)Z1:dTx,Ty≤adx,yZ2:dTx,Ty≤bdx,Tx+dy,TyZ3:dTx,Ty≤cdx,Ty+dy,Tx,where a, b, and c are nonnegative constants satisfying a∈[0,1), b,c≤1/2.The equivalence form of (19) is (20)dTx,Ty≤amax⁡dx,y,12dx,Tx+dy,Ty,12dx,Ty+dy,Txfor x,y∈X and a∈[0,1).Berinde [2] observed that condition (20) implies (21)dTx,Ty≤2hdx,Tx+hdx,y,where h=max⁡a,a/(2-a).In [26], Rhoades used a more general contractive condition than (21): for x,y∈X, there exists a∈[0,1) such that (22)dTx,Ty≤amax⁡dx,y,12dx,Tx+dy,Ty,dx,Ty,dy,Tx.Osilike [20] extended and generalized the contractive condition (22): for x,y∈X, there exists a∈[0,1) and L≥0 such that (23)dTx,Ty≤Ldx,Tx+adx,y.Imoru and Olatinwo [21] employed a more general class of operators T than (23) satisfying the following contractive conditions: (24)dTx,Ty≤adx,y+φdx,Txforx,y∈X,where a∈[0,1) and φ:R+→R+ is a monotone increasing function with φ(0)=0.The equivalence form of (24) in a normed linear space is (25)Tx-Ty≤ax-y+φx-Txforx,y∈XWe will need the following definitions and lemmas to prove our main results.Definition 1 (see [9]).
Let(X,d) be a metric space and T:X→X a self-mapping. Suppose that FT=p∈X:Tp=p is the set of fixed points of T. Let xnn=0∞⊂X be the sequence generated by an iterative procedure involving T which is defined by (26)xn+1=fT,αnxn,n≥0,where x0∈X is the initial approximation and fT,αnxn is a function such that αn∈[0,1]. Suppose that xn converges to a fixed point p of T. Let ynn=0∞⊂X and set ϵn=dyn+1,fT,αnyn,n=0,1,2,…. Then, iterative procedure (26) is said to be T-stable or stable with respect to T if and only if limn→∞ϵn=0 implies limn→∞yn=p.Definition 2 (see [2]).
Letann=0∞ and bnn=0∞ be two nonnegative real sequences which converge to a and b, respectively. Let (27)l=limn→∞⁡an-abn-b;(1)
ifl=0, then ann=0∞ converges to a faster than bnn=0∞ to b;(2)
if0<l<∞, then both ann=0∞ and bnn=0∞ have the same convergence rate;(3)
ifl=∞, then bnn=0∞ converges to b faster than ann=0∞ to a.Lemma 3 (see [2]).
Letδ be a real number such that δ∈[0,1) and ϵnn=0∞ is a sequence of nonnegative numbers such that limn→∞ϵn=0; then, for any sequence of positive numbers unn=0∞ satisfying (28)un+1≤δun+ϵn,∀n∈Nwe have limn→∞un=0.Lemma 4 (see [8]).
Let(E,·) be a normed linear space and T:E→E a map satisfying (25). Let φ:R+→R+ be a subadditive, monotone increasing function such that φ(0)=0, φ(Lu)=Lφ(u), for u∈R+, L≥0. Then, for all i∈N, L≥0 and for all x,y∈E(29)Tix-Tiy≤∑j=1iijai-jφjx-Tx+aix-y.Note thata∈[0,1) in (29).


## 2. Main Results

We present our main results as follows.LetE be an arbitrary Banach space and T:E→E a self-map. Let x0∈E; we define the following iteration, namely, implicit hybrid Kirk-Multistep iterative scheme, as follows: (30)xn=αn,0xn-11+∑i=1q1αn,iTixn,∑i=0q1αn,i=1xn-1l=βn,0lxn-1l+1+∑i=1ql+1βn,ilTixn-1l,∑i=0ql+1βn,il=1,l=11k-2xn-1k-1=βn,0k-1xn-1+∑i=1qkβn,ik-1Tixn-1k-1,∑i=0qkβn,ik-1=1,k≥2,n≥1,where q1,q2,q3,…,qk are fixed integers with q1≥q2≥q3≥⋯≥qk; αn,in=1∞ and βn,i(l)n=1∞ are sequences in [0,1] satisfying αn,i≥0, αn,0≠0, βn,i(l)≥0, and βn,0(l)≠0 for each l with ∑n=1∞1-αn,0=∞.If we letk=3 in (30), we obtain the implicit Kirk-Noor iteration defined by (31)xn=αn,0xn-11+∑i=1q1αn,iTixn,∑i=0q1αn,i=1xn-11=βn,01xn-12+∑i=1q2βn,i1Tixn-11,∑i=0q2βn,i1=1xn-12=βn,02xn-1+∑i=1q3βn,i2Tixn-12,∑i=0q3βn,i2=1,n≥1.By setting k=2 in (30), we have the implicit Kirk-Ishikawa iteration and we can also obtain the implicit Kirk-Mann iteration when k=2 and q2=0 in (30).Ifq1=q2=q3=⋯=qk=1 in (30), then we obtain the implicit multistep iteration (19) with αn,1=αn, βn,1(l)=βn(l).Ifk=3, q1=q2=q3=1, and q4=q5=⋯=qk=0 in (30), we have the implicit Noor scheme (18) with αn,1=αn, βn,1(1)=βn(1), and βn,1(2)=βn(2).Ifk=2, q1=q2=1, and q3=q4=⋯=qk=0 in (30), we have the implicit Ishikawa scheme (17) with αn,1=αn, βn,1(1)=βn(1).Ifk=2, q1=1, and q2=q3=q4=⋯=qk=0 in (30), we have the implicit Mann scheme (16) with αn,1=αn.Throughout, the operatorT will be assumed as fixed and a fixed point p∈FT with the condition (29) is unique.Theorem 5.
Let(E,·) be a normed linear space. Assume T is self-map of E satisfying the contractive condition (29) with FT≠ϕ. Then, for x0∈E, the sequence xn defined by (30) with ∑n=1∞1-αn,0=∞ converges strongly to the fixed point p∈FT.Proof.
Letx0∈X and p∈FT; then using (29) and (30) we have (32)xn-p≤αn,0xn-11-p+∑i=1q1αn,iTixn-Tip≤αn,0xn-11-p+∑i=1q1αn,i∑j=1iijai-jφjp-Tp+aixn-p=αn,0xn-11-p+∑i=1q1αn,iaixn-p.This implies that (33)xn-p≤αn,01-∑i=1q1αn,iaixn-11-p.Also, from (30), we have (34)xn-11-p≤βn,01xn-12-p+∑i=1q2βn,i1Tixn-11-Tip≤βn,01xn-12-p+∑i=1q2βn,i1∑j=1iijai-jφjp-Tp+aixn-11-p=βn,01xn-12-p+∑i=1q2βn,i1aixn-11-p.This becomes (35)xn-11-p≤βn11-∑i=1q2βn,i1aixn-12-p.From (30) again, we have (36)xn-12-p≤βn,02xn-13-p+∑i=1q3βn,i2Tixn-12-Tip≤βn,02xn-13-p+∑i=1q3βn,i2∑j=1iijai-jφjp-Tp+aixn-12-p=βn,02xn-13-p+∑i=1q3βn,i2aixn-12-pwhich implies (37)xn-12-p≤βn21-∑i=1q3βn,i2aixn-13-p.Continuing this way up to k-1 in (30), we have (38)xn-1k-1-p≤βn,0k-1xn-1-p+∑i=1qkβn,ik-1Tixn-1k-1-Tip≤βn,0k-1xn-1-p+∑i=1qkβn,ik-1∑j=1iijai-jφjp-Tp+aixn-1k-1-p=βn,0k-1xn-1-p+∑i=1qkβn,ik-1aixn-1k-1-pimplying that (39)xn-1k-1-p≤βnk-11-∑i=1qkβn,ik-1aixn-1-p.Substituting (35)–(39) into (33) it becomes (40)xn-p≤αn,01-∑i=1q1αn,iaiβn11-∑i=1q2βn,i1aiβn21-∑i=1q3βn,i2ai×⋯βnk-11-∑i=1qkβn,ik-1aixn-1-p.Let λn=αn,0/1-∑i=1q1αn,iai; then (41)1-λn=1-αn,01-∑i=1q1αn,iai=1-∑i=1q1αn,iai-αn,01-∑i=1q1αn,iai≥1-∑i=1q1αn,iai+αn,0.Therefore, (42)λn≤∑i=1q1αn,iai+αn,0=∑i=0q1αn,iai<∑i=0q1αn,i=1.Similarly, we can easily obtain the following from (40): (43)βn,011-∑i=1q2βn,i1ai≤∑i=1q2βn,i1ai<∑i=1q2βn,i1=1βn,021-∑i=1q3βn,i2ai≤∑i=1q3βn,i2ai<∑i=1q3βn,i2=1βn,031-∑i=1q4βn,i3ai≤∑i=1q4βn,i3ai<∑i=1q4βn,i3=1⋮βn,0k-11-∑i=1qkβn,ik-1ai≤∑i=1qkβn,ik-1ai<∑i=1qkβn,ik-1=1.Applying (42) and (43) in (40) and letting ai≤a<1 for each i, we have (44)xn-p≤∑i=1q1αn,iai+αn,0xn-1-p≤1-αn,0a+αn,0xn-1-p=1-1-αn,01-axn-1-p≤1-1-αn,01-a1-1-αn-1,01-axn-2-p⋮≤∏r=1n1-1-αr,01-ax0-p≤e-1-a∑r=1n1-αr,0x0-p.As n→∞, ∑r=1∞(1-αr,0)=∞. Hence, limn→∞xn-p=0.
Therefore, implicit Kirk-Multistep scheme (30) converges strongly to p∈FT.Corollary 6.
Let(E,·) be a normed linear space. Assume T is self-map of E satisfying the contractive condition (29) with FT≠ϕ. Then, for x0∈E, the implicit Kirk-Noor, the implicit Kirk-Ishikawa, and the implicit Kirk-Mann schemes with ∑n=1∞‍(1-αn,0)=∞ converge strongly to the fixed point p∈FT.Remark 7.
The strong convergence results for implicit Noor, implicit Ishikawa, and implicit Mann schemes are obvious from Theorem5.Theorem 8.
Let(E,·) be a normed linear space and T is a self-map of E satisfying contractive condition (29) with FT≠ϕ. Then, for x0∈E and p∈FT, the sequence xn defined by (30) isT-stable.Proof.
Letyn∈E be an arbitrary sequence and let ϵn=yn-αn,0zn-1(1)-∑i=1q1αn,iTiyn, where (45)zn-11=βn,01zn-12+∑i=1q2βn,i1Tizn-11,∑i=0q2βn,i1=1,zn-1l=βn,0lzn-1l+1+∑i=1ql+1βn,ilTizn-1l,∑i=0ql+1βn,il=1,l=11k-2zn-1k-1=βn,0k-1yn-1+∑i=1qkβn,ilTizn-1k-1,∑i=0qkβn,ik-1=1.Suppose limn→∞ϵn=0 and p∈FT; by (29) we have (46)yn-p≤yn-αn,0zn-11-∑i=1q1αn,iTiyn+αn,0zn-11+∑i=1q1αn,iTiyn-p≤ϵn+αn,0zn-11-p+∑i=1q1αn,iTiyn-p≤ϵn+αn,0zn-11-p+∑i=1q1αn,iaiyn-p.This implies that (47)yn-p≤ϵn1-∑i=1q1αn,iai+αn,01-∑i=1q1αn,iaizn-11-p.From inequalities (42) and (43), one can easily obtain the following: (48)zn-11-p≤zn-12-p≤zn-13-p≤⋯≤zn-1k-1-p≤yn-1-p.Then, inequality (47) becomes (49)yn-p≤ϵn1-∑i=1q1αn,iai+αn,01-∑i=1q1αn,iaiyn-1-p.Letting δ=αn,0/1-∑i=1q1‍αn,iai<1 and by Lemma 3, we have (50)yn-p=0.Conversely, suppose yn-p=0 for p∈FT; then (51)ϵn=yn-αn,0zn-11-∑i=1q1αn,iTiyn≤yn-p+p-αn,0zn-11+∑i=1q1αn,iTiyn≤yn-p+αn,0zn-11-p+∑i=1q1αn,iTiyn-Tip≤yn-p+αn,0yn-1-p+∑i=1q1αn,iaiyn-p≤1+∑i=1q1αn,iaiyn-p+αn,0yn-1-p.Since yn-p→0, then limn→∞ϵn=0.
Therefore, iterative scheme (30) isT-stable.Corollary 9.
Let(E,·) be a normed linear space and T is a self-map of E satisfying the contractive condition (29) with FT≠ϕ. Then, for x0∈E and p∈FT, the sequence xn defined by implicit Kirk-Mann, implicit Kirk-Ishikawa, and implicit Kirk-Noor schemes areT-stable.Remark 10.
The stability results for implicit Mann, implicit Ishikawa, and implicit Noor schemes with contractive condition (29) are special cases of Corollary 9.


### 2.1. Comparison of Several Iterative Schemes

We compare our iterative schemes with others by using the following example.Example 11.
LetT:[0,1]→[0,1] and Tx=x/2 with x0≠0 and fixed point p=0 using αn,0=βn,0(l)=1-2/n, αn,i=βn,i(l)=4/n, for each l, n≥25, and q1=q2=q3=q4=2.For the implicit Kirk-Mann iteration (IKM), we have(52)xn=αn,0xn-1+∑i=12αn,iTixn=1-2nxn-1+2nxn+1nxn.This implies that (53)xnIKM=2n-42n-3xn-1=∏r=25n2r-42r-3x0.Also, for implicit Kirk-Ishikawa iteration (IKI), we have (54)xn=2n-42n-3xn-11with (55)xn-11=2n-42n-3xn-1.Hence, (56)xnIKI=2n-42n-32xn-1=∏r=25n2r-42r-32x0.Similarly, implicit Kirk-Noor iteration (IKN) implies (57)xnIKN=2n-42n-33xn-1=∏r=25n2r-42r-33x0while the implicit multistep Kirk iteration (IMK) gives (58)xnIMK=∏r=25n2r-42r-3kx0.Now, using Definition 2, we compare the implicit Kirk type iterations as follows: for k≥4, we have (59)xnIMK-0xnIKN-0=∏r=25n2r-42r-3k-3=∏r=25n1-12r-3k-3with (60)0≤limn→∞⁡∏r=25n1-12r-3k-3≤limn→∞⁡∏r=25n1-1rk-3=limn→∞⁡2425·2526⋯n-2n-1·n-1nk-3=limn→∞⁡24nk-3=0,∀k≥4.Remark 12.
The implicit multistep Kirk iteration (IMK) converges faster than the implicit Kirk-Noor iteration (IKN) fork=4,5,….Also,(61)xnIKN-0xnIKI-0=∏r=25n2r-42r-3=∏r=25n1-12r-3with (62)0≤limn→∞⁡∏r=25n1-12r-3≤limn→∞⁡∏r=25n1-1r=limn→∞⁡2425·2526⋯n-2n-1·n-1n=limn→∞⁡24n=0.Remark 13.
The implicit Kirk-Noor iterationxn(IKN) converges to p=0 faster than the implicit Kirk-Ishikawa iteration xn(IKI) to p=0.Similarly, using Definition2, we have that (63)limn→∞⁡xnIKI-0xnIKM-0=0which implies that the implicit Kirk-Ishikawa iteration xn(IKI) converges faster than the implicit Kirk-Mann iteration xn(IKM).For the Kirk-Mann iteration (KM), we have the following estimate:(64)xn=αn,0xn-1+∑i=12αn,iTixn-1=1-2nxn-1+2nxn-1+1nxn-1.This implies that (65)xnKM=1+nnxn-1=∏r=25n1+rrx0.The estimates for Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations are, respectively, (66)xnKT=∏r=25n1+rr2x0,xnKSP=∏r=25n1+rr3x0,xnKMSP=∏r=25n1+rrkx0.We compare Kirk-Mann (KM), Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations with our iterative schemes as follows.Again, using Definition2, we have (67)xnIKM-0xnKM-0=∏r=25n2r-42r-3rr+r=∏r=25n2r3/2-4r2r3/2-3r1/2-r=∏r=25n1-3r-r1/22r3/2-3r1/2-rwith (68)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-r≤limn→∞⁡∏r=25n1-1r=limn→∞⁡2425·2526⋯n-2n-1·n-1n=limn→∞⁡24n=0.Remark 14.
The implicit Kirk-Mann iterationxn(IKM) converges to p=0 faster than the Kirk-Mann iteration xn(KM) to p=0.For the comparison of implicit Kirk-Ishikawa iterationxn(IKI) and Kirk-Thianwan iteration xn(KT), we have (69)xnIKI-0xnKT-0=∏r=25n2r-42r-3rr+r2=∏r=25n1-3r-r1/22r3/2-3r1/2-r2with (70)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-r2≤limn→∞⁡∏r=25n1-1r2=limn→∞⁡2425·2526⋯n-2n-1·n-1n2=limn→∞⁡24n2=0.Remark 15.
The implicit Kirk-Ishikawa iterationxn(IKI) converges faster than the Kirk-Thianwan iteration xn(KT).For the comparison of implicit Kirk-Noor iterationxn(IKN) and Kirk-SP iteration xn(KSP), we have (71)xnIKN-0xnKSP-0=∏r=25n2r-42r-3rr+r3=∏r=25n1-3r-r1/22r3/2-3r1/2-r3with (72)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-r3≤limn→∞⁡∏r=25n1-1r3=limn→∞⁡2425·2526⋯n-2n-1·n-1n3=limn→∞⁡24n3=0.Remark 16.
The implicit Kirk-Noor iterationxn(IKN) has better convergence rate than the Kirk-SP iteration xn(KSP).For the comparison of implicit Kirk-Multistep iterationxn(IKM) and Kirk-Multistep-SP iteration xn(KMSP), we have (73)xnIKM-0xnKMSP-0=∏r=25n2r-42r-3rr+rk=∏r=25n1-3r-r1/22r3/2-3r1/2-rkwith (74)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-rk≤limn→∞⁡∏r=25n1-1rk=limn→∞⁡2425·2526⋯n-2n-1·n-1nk=limn→∞⁡24nk=0.Remark 17.
The implicit Kirk-Multistep iterationxn(IKM) has better convergence rate than the Kirk-Multistep-SP iteration xn(KMSP) for k≥4.


## 2.1. Comparison of Several Iterative Schemes

We compare our iterative schemes with others by using the following example.Example 11.
LetT:[0,1]→[0,1] and Tx=x/2 with x0≠0 and fixed point p=0 using αn,0=βn,0(l)=1-2/n, αn,i=βn,i(l)=4/n, for each l, n≥25, and q1=q2=q3=q4=2.For the implicit Kirk-Mann iteration (IKM), we have(52)xn=αn,0xn-1+∑i=12αn,iTixn=1-2nxn-1+2nxn+1nxn.This implies that (53)xnIKM=2n-42n-3xn-1=∏r=25n2r-42r-3x0.Also, for implicit Kirk-Ishikawa iteration (IKI), we have (54)xn=2n-42n-3xn-11with (55)xn-11=2n-42n-3xn-1.Hence, (56)xnIKI=2n-42n-32xn-1=∏r=25n2r-42r-32x0.Similarly, implicit Kirk-Noor iteration (IKN) implies (57)xnIKN=2n-42n-33xn-1=∏r=25n2r-42r-33x0while the implicit multistep Kirk iteration (IMK) gives (58)xnIMK=∏r=25n2r-42r-3kx0.Now, using Definition 2, we compare the implicit Kirk type iterations as follows: for k≥4, we have (59)xnIMK-0xnIKN-0=∏r=25n2r-42r-3k-3=∏r=25n1-12r-3k-3with (60)0≤limn→∞⁡∏r=25n1-12r-3k-3≤limn→∞⁡∏r=25n1-1rk-3=limn→∞⁡2425·2526⋯n-2n-1·n-1nk-3=limn→∞⁡24nk-3=0,∀k≥4.Remark 12.
The implicit multistep Kirk iteration (IMK) converges faster than the implicit Kirk-Noor iteration (IKN) fork=4,5,….Also,(61)xnIKN-0xnIKI-0=∏r=25n2r-42r-3=∏r=25n1-12r-3with (62)0≤limn→∞⁡∏r=25n1-12r-3≤limn→∞⁡∏r=25n1-1r=limn→∞⁡2425·2526⋯n-2n-1·n-1n=limn→∞⁡24n=0.Remark 13.
The implicit Kirk-Noor iterationxn(IKN) converges to p=0 faster than the implicit Kirk-Ishikawa iteration xn(IKI) to p=0.Similarly, using Definition2, we have that (63)limn→∞⁡xnIKI-0xnIKM-0=0which implies that the implicit Kirk-Ishikawa iteration xn(IKI) converges faster than the implicit Kirk-Mann iteration xn(IKM).For the Kirk-Mann iteration (KM), we have the following estimate:(64)xn=αn,0xn-1+∑i=12αn,iTixn-1=1-2nxn-1+2nxn-1+1nxn-1.This implies that (65)xnKM=1+nnxn-1=∏r=25n1+rrx0.The estimates for Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations are, respectively, (66)xnKT=∏r=25n1+rr2x0,xnKSP=∏r=25n1+rr3x0,xnKMSP=∏r=25n1+rrkx0.We compare Kirk-Mann (KM), Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations with our iterative schemes as follows.Again, using Definition2, we have (67)xnIKM-0xnKM-0=∏r=25n2r-42r-3rr+r=∏r=25n2r3/2-4r2r3/2-3r1/2-r=∏r=25n1-3r-r1/22r3/2-3r1/2-rwith (68)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-r≤limn→∞⁡∏r=25n1-1r=limn→∞⁡2425·2526⋯n-2n-1·n-1n=limn→∞⁡24n=0.Remark 14.
The implicit Kirk-Mann iterationxn(IKM) converges to p=0 faster than the Kirk-Mann iteration xn(KM) to p=0.For the comparison of implicit Kirk-Ishikawa iterationxn(IKI) and Kirk-Thianwan iteration xn(KT), we have (69)xnIKI-0xnKT-0=∏r=25n2r-42r-3rr+r2=∏r=25n1-3r-r1/22r3/2-3r1/2-r2with (70)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-r2≤limn→∞⁡∏r=25n1-1r2=limn→∞⁡2425·2526⋯n-2n-1·n-1n2=limn→∞⁡24n2=0.Remark 15.
The implicit Kirk-Ishikawa iterationxn(IKI) converges faster than the Kirk-Thianwan iteration xn(KT).For the comparison of implicit Kirk-Noor iterationxn(IKN) and Kirk-SP iteration xn(KSP), we have (71)xnIKN-0xnKSP-0=∏r=25n2r-42r-3rr+r3=∏r=25n1-3r-r1/22r3/2-3r1/2-r3with (72)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-r3≤limn→∞⁡∏r=25n1-1r3=limn→∞⁡2425·2526⋯n-2n-1·n-1n3=limn→∞⁡24n3=0.Remark 16.
The implicit Kirk-Noor iterationxn(IKN) has better convergence rate than the Kirk-SP iteration xn(KSP).For the comparison of implicit Kirk-Multistep iterationxn(IKM) and Kirk-Multistep-SP iteration xn(KMSP), we have (73)xnIKM-0xnKMSP-0=∏r=25n2r-42r-3rr+rk=∏r=25n1-3r-r1/22r3/2-3r1/2-rkwith (74)0≤limn→∞⁡∏r=25n1-3r-r1/22r3/2-3r1/2-rk≤limn→∞⁡∏r=25n1-1rk=limn→∞⁡2425·2526⋯n-2n-1·n-1nk=limn→∞⁡24nk=0.Remark 17.
The implicit Kirk-Multistep iterationxn(IKM) has better convergence rate than the Kirk-Multistep-SP iteration xn(KMSP) for k≥4.


## 3. Conclusion

We have established and proved strong convergence andT-stability results for implicit Kirk-Multistep, implicit Kirk-Noor, implicit Kirk-Ishikawa, and implicit Kirk-Mann iterative schemes of fixed points with contractive-type operators in normed linear spaces. These iterative schemes have better convergence rate when compared with other iterative schemes, namely, multistep Kirk-SP iteration, Kirk-Multistep scheme, Kirk-SP iteration, Kirk-Thianwan scheme, Kirk-Noor scheme, Kirk-Ishikawa scheme, Kirk-Mann scheme, implicit Noor iteration, implicit Ishikawa iteration, implicit Mann iteration, and many more iterative schemes of fixed point in the literature.



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*Source: 3791506-2016-09-19.xml*