The purpose of this paper is to prove strong convergence and T-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.
1. Introduction
In the recent years, numerous papers have been published on the strong convergence and T-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].
Let X 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≤amaxdx,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=maxa,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≤amaxdx,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,Txfor x,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-Txfor x,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]).
Let ann=0∞ and bnn=0∞ be two nonnegative real sequences which converge to a and b, respectively. Let (27)l=limn→∞an-abn-b;
if l=0, then ann=0∞ converges to a faster than bnn=0∞ to b;
if 0<l<∞, then both ann=0∞ and bnn=0∞ have the same convergence rate;
if l=∞, then bnn=0∞ converges to b faster than ann=0∞ to a.
Lemma 3 (see [2]).
Let δ 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 that a∈[0,1) in (29).
2. Main Results
We present our main results as follows.
Let E 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 let k=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).
If q1=q2=q3=⋯=qk=1 in (30), then we obtain the implicit multistep iteration (19) with αn,1=αn, βn,1(l)=βn(l).
If k=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).
If k=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).
If k=2, q1=1, and q2=q3=q4=⋯=qk=0 in (30), we have the implicit Mann scheme (16) with αn,1=αn.
Throughout, the operator T 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.
Let x0∈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 Theorem 5.
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) is T-stable.
Proof.
Let yn∈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) is T-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 are T-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.
Let T:[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) for k=4,5,….
The implicit Kirk-Noor iteration xn(IKN) converges to p=0 faster than the implicit Kirk-Ishikawa iteration xn(IKI) to p=0.
Similarly, using Definition 2, 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 Definition 2, 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 iteration xn(IKM) converges to p=0 faster than the Kirk-Mann iteration xn(KM) to p=0.
For the comparison of implicit Kirk-Ishikawa iteration xn(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 iteration xn(IKI) converges faster than the Kirk-Thianwan iteration xn(KT).
For the comparison of implicit Kirk-Noor iteration xn(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 iteration xn(IKN) has better convergence rate than the Kirk-SP iteration xn(KSP).
For the comparison of implicit Kirk-Multistep iteration xn(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 iteration xn(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 and T-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.
Disclosure
Authors agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.
Competing Interests
The authors hereby declare that there are no competing interests.
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