# Convergence Rate of Some Two-Step Iterative Schemes in Banach Spaces

Authors: O. T. Wahab; R. O. Olawuyi; K. Rauf; I. F. Usamot
Journal: Journal of Mathematics
 (2016)

---

## Abstract

This article proves some theorems to approximate fixed point of Zamfirescu operators on normed spaces for some two-step iterative schemes, namely, Picard-Mann iteration, Ishikawa iteration, S-iteration, and Thianwan iteration, with their errors. We compare the aforementioned iterations using numerical approach; the results show that S-iteration converges faster than other iterations followed by Picard-Mann iteration, while Ishikawa iteration is the least in terms of convergence rate. These results also suggest the best among two-step iterative fixed point schemes in the literature.


---

## Body

## 1. Introduction

Fixed point theory takes a large amount of literature, since it provides useful tools to solve many problems that have applications in different fields like engineering, economics, chemistry, game theory, and so forth. However, to find fixed points is not an easy task; that is why we use iterative methods for computing them. By time, many iterative methods have been developed and it is impossible to cover them all.In the last four decades, numerous papers were published on the iterative approximation of fixed points of self- and non-self-contractive type operators in metric spaces, Hilbert spaces, or several classes of Banach spaces, while, for strict contractive type operators, the Picard iteration can be used to approximate the unique fixed point, for operators satisfying slightly weaker contractive type conditions, instead of Picard iteration, which does not generally converge; it was necessary to consider other fixed point iteration procedures. The Krasnoselskij iteration, the Mann iteration, and the Ishikawa iteration are certainly the most studied of these fixed point iteration procedures. Other iterations which have been studied also are Implicit Mann, Implicit Ishikawa, Thianwan, S-iteration, and hybrid Picard-Mann iterations. Recently, Wahab and Rauf [1] obtained some results on a faster implicit hybrid Kirk-multistep schemes for contractive type operators, to mention but a few.Our aim in this paper is to establish the convergence and convergence rate of some two-step iterative schemes with errors using Zamfirescu operator in Banach spaces.


## 2. Preliminaries

The Picard iteration process is defined by the sequence(1)xn+1=Txn,n≥0, and the concept of Picard iteration process with error is defined as follows:(2)xn+1=Txn+un,n≥0, where un satisfy ∑n=1∞‍un<∞.In [2], Banach proved the convergence of (1) with the aid of the following contractive mapping:(3)dTx,Ty≤adx,y, where T:X→X, (X,d) is a metric space, a∈0,1, and x,y∈X. Banach’s theorem is given as follows.Theorem 1.
Let(X,d) be a metric space and T:X→X be a contraction map on X. Then T has a unique fixed point x∈X.When condition (3) is weaker, Picard iteration (1) will no longer converge to a fixed point. So, other iterations such as Mann iteration, Picard-Mann iteration, Ishikawa iteration, S-iteration, and Thianwan iteration would be considered.Mann [3] defined an iteration process by the sequence, for x0∈C, C of a closed subset of normed space:(4)xn+1=1-αnxn+αnTxn,n≥0, where αn is a sequence in 0,1.The concept of Mann iteration with error was discussed in [4] and it is given by the sequence(5)xn+1=1-αnxn+αnTxn+un,n≥0, where αn is a sequence in 0,1 and un satisfy ∑n=1∞‍un<∞.Ishikawa [5] defined another iteration process given by the sequence(6)xn+1=1-αnyn+αnTxn,yn=1-βnxn+βnTxn,n≥0,where αn and βn are sequences in 0,1. The Ishikawa iteration is a double Mann iteration and has better approximation than Mann iteration (5). The Ishikawa iteration with errors is given as(7)xn+1=1-αnyn+αnTxn+en,yn=1-βnxn+βnTxn+fn,n≥0,where en and fn satisfy ∑n=1∞‍en<∞ and ∑n=1∞‍fn<∞. See [4].In an attempt to reduce computational cost, Agarwal et al. [6] defined another iteration called S-iteration which is independent of Mann and Ishikawa iterations and it is defined by the sequence(8)xn+1=1-αnTxn+αnTyn,yn=1-βnxn+βnTxn,n≥0, where αn and βn are sequences in 0,1.The S-iteration with errors can be given as(9)xn+1=1-αnTxn+αnTyn+en,yn=1-βnxn+βnTxn+fn,n≥0,where en and fn satisfy ∑n=1∞‍en<∞ and ∑n=1∞‍fn<∞.In [7], Thianwan defined a new iteration process given by the sequence(10)xn+1=1-αnyn+αnTyn,yn=1-βnxn+βnTxn,n≥0, where αn and βn are sequences in 0,1.The Thianwan iteration with errors is given by(11)xn+1=1-αnyn+αnTyn+en,yn=1-βnxn+βnTxn+fn,n≥0, where en and fn satisfy ∑n=1∞‍en<∞ and ∑n=1∞‍fn<∞.Khan [8] gave a different perspective to iteration procedure. He introduced the following hybrid Picard-Mann iterative scheme for a single nonexpansive mapping T which is defined as(12)xn+1=Tyn,yn=1-αnxn+αnTxn,n≥0, where αn is a sequence in 0,1 with ∑‍αn=∞. While the Picard-Mann iteration process with errors is given as(13)xn+1=Tyn+en,yn=1-αnxn+αnTxn+fn,n≥0, where en and fn satisfy ∑n=1∞‍en<∞ and ∑n=1∞‍fn<∞.The most generalized operator used to approximate fixed point is the one proved by Zamfirescu [9]. The Zamfirescu operator was obtained from the Banach [2], Kannan [10], and Chatterjea [11] contractive mappings as follows. The operator T is called a Kannan mapping if there exists b∈0,1/2 such that(14)dTx,Ty≤bdx,Tx+dy,Ty∀x,y∈X. Another similar definition due to Chatterjea mapping is as follows: there exists c∈0,1/2 such that(15)dTx,Ty≤cdx,Ty+dy,Tx∀x,y∈X.By combining (3), (14), and (15) conditions, we have the Zamfirescu operator given, for x,y∈X, by(16)z1:dTx,Ty≤adx,y,a∈0,1,z2:dTx,Ty≤bdx,Tx+dy,Ty,b∈0,12,z3:dTx,Ty≤cdx,Ty+dy,Tx,c∈0,12. The equivalence of (16) is given as follows: (17)dTx,Ty≤hmaxdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx, where h=maxa,b,c and x,y∈X.The following Lemma is useful in the proof of our results.Lemma 2 (see [12]).
Letδ be a real number such that δ∈0,1 and unn=0∞ is a sequence of nonnegative numbers such that limn→∞un=0; then for any sequence of positive numbers xnn=0∞ satisfying(18)xn+1≤δxn+un∀n∈N, we have(19)limn→∞xn=0.The Zamfirescu operator was used to prove the strong convergence of (1) as follows.Theorem 3.
LetC be a nonempty subset of a normed space E,·. Let T:C→C be z-operator. If FT≠ϕ,  ∑n=1∞‍αn=∞, un=0. Then xn converges strongly to a fixed point of T.Proof.
By Lemma2, T has a unique fixed point in C, say q. Let x,y∈C; since T is a z-operator, at least one of each conditions z1–z3 is satisfied. If z2 holds, then (20)dTx,Ty≤bdx,Tx+dy,Tyis equivalent to (21)Tx-Ty≤bx-Tx+y-Ty=bx-Tx+y-x+x-Tx+Tx-Ty≤bx-Tx+y-x+x-Tx+Tx-Ty. This implies (22)1-bTx-Ty≤bx-y+2bx-Tx. Therefore,(23)Tx-Ty≤b1-bx-y+2b1-bx-Tx. Similarly, if z3 holds, we obtain(24)Tx-Ty≤c1-cx-y+2c1-cx-Tx. Let(25)δ=maxa,b1-b,c1-c. Then we have 0≤δ<1 and in view of (25) it results in that inequalities (23) and (24) become(26)Tx-Ty≤δx-y+2δx-Tx∀x,y∈C. Now, for x=q and y=xn in (26), we obtain (27)Tq-Txn≤δq-xn, which implies(28)xn+1-q≤δxn-q. By Lemma 2, we conclude that xn converges strongly to q.


## 3. Analytical Results

We present our main results usingz-operators (16) on normed spaces for the two-step iterative schemes with errors defined in the last section.Throughout,N denotes the set of all positive integers, C is nonempty convex subset of a complete normed space E,·, and T:C→C is a self-map.Theorem 4.
LetC be a nonempty subset of a normed space E,·. Let T:C→C be z-operator (16). Let xn be defined with iterative process (5). If FT≠ϕ, ∑n=1∞‍αn=∞, un=0. Then xn converges strongly to a fixed point of T.Proof.
Letx0∈C; by using (5) and (16), we have (29)xn+1-q=1-αnxn+αnTxn+un-q=1-αnxn+αnTxn+un-q-αnq+αnq=1-αnxn-q+αnTxn-q+un≤1-αnxn-q+αnTxn-q+un≤1-αnxn-q+αnTxn-q+un.Substituting (28) in (29), (30)xn+1-q≤1-αnxn-q+αnδxn-q+un=1-αn+αnδxn-q+un=1-1-δαnxn-q+un,n=0,1,2,…. Hence, by Lemma 2 and using the fact that 0≤δ<1, ∑n=1∞‍αn=∞, and un=0, it results in that (31)limn→∞xn+1-q=0. Therefore, (32)xn+1⟶q∈FT.Theorem 5.
LetC be a nonempty subset of a normed E,·. Let T:C→C be z-operator (16). Let xn be defined with iterative process (13). If FT≠ϕ, ∑n=1∞‍αn=∞, gn=0. Then xn converges strongly to a fixed point of T.Proof.
From (26), we have (33)Tx-Ty≤δx-y+2δx-Tx∀x,y∈C. Also from (28), we have (34)Txn-q≤δxn-q. Now, using (13) and (28), we have (35)xn+1-q≤Tyn-q+en≤Tyn-q+en. This implies(36)xn+1-q≤δyn-q+en. Also, (37)yn-q≤1-αnxn+αnTxn-q+fn≤1-αnxn+αnTxn-q-αnq+αnq+fn=1-αnxn-q+αnTxn-q+fn≤1-αnxn-q+αnTxn-q+fn.Substituting (37) in (36), we have(38)xn+1-q≤δ1-αnxn-q+αnTxn-q+fn+en. Using (28) in (38), we have (39)xn+1-q≤δ1-αnxn-q+αnδxn-q+fn+en=δ-αnδxn-q+αnδ2xn-q+fn+en=δ-αnδ+αnδ2xn-q+fn+en=δ1-1-δαnxn-q+gn≤1-1-δαnxn-q+gn,n=0,1,2,…, where gn≡fn+en.
By Lemma2 and using the fact that 0≤δ<1,  ∑n=1∞‍αn=∞ and gn=0, this becomes (40)limn→∞xn+1-q=0. Hence, (41)xn+1⟶q∈FT.Theorem 6.
LetC be a nonempty subset of a normed space E,·. Let T:C→C be z-operator. Let xn be defined with iterative process (7). If FT≠ϕ, ∑n=1∞‍αn=∞, gn=0. Then, xn converges strongly to a fixed point of T.Proof.
From (26), we have (42)Tx-Ty≤δx-y+2δx-Tx∀x,y∈C. Also from (28), we have (43)Txn-q≤δxn-q. Now, by combining (7) and (28), we have (44)xn+1-q≤1-αnyn+αnTxn-q+en=1-αnyn-q+αnTxn-q+en. This gives(45)xn+1-q≤1-αnyn-q+αnTxn-q+en. Also, from (7), we have (46)yn-q≤1-βnxn+βnTxn-q+fn=1-βnxn-q+βnTxn-q+fn. This also gives(47)yn-q≤1-βnxn-q+βnTxn-q+fn. By combining (47) and (45), we have (48)xn+1-q≤1-αn1-βnxn-q+βnTxn-q+αnTxn-q+en+fn=1-αn1-βnxn-q+1-αnβnTxn-q+αnTxn-q+en+fn≤1-αn1-βnxn-q+1-αnβn+αnTxn-q+en+fn.Using (28) in (48), we have (49)xn+1-q≤1-αn1-βnxn-q+1-αnβn+αnδxn-q+en+fn=1-αn1-βn+1-αnβn+αnδxn-q+en+fn=1-αn1-βn+1-αnδβn+αnδxn-q+en+fn≤1-1-δ2αnxn-q+en+fn,n=0,1,2,…. By letting gn=fn+en and by Lemma 2, using the fact that 0≤δ<1, ∑n=1∞‍αn=∞, and gn=0, we get (50)limn→∞xn+1-q=0. Therefore, (51)xn+1⟶q∈FT.Theorem 7.
LetC be a nonempty subset of a normed space E,·. Let T:C→C be z-operator (16). Let xn be defined with iterative process (9). If FT≠ϕ, ∑n=1∞‍αn=∞, gn=0. Then xn converges strongly to a fixed point of T.Proof.
From (26), we have (52)Tx-Ty≤δx-y+2δx-Tx∀x,y∈C. Also from (28), we have (53)Txn-q≤δxn-q. Applying (28) to (9), we obtain (54)xn+1-q≤1-αnTxn+αnTyn-q+en=1-αnTxn-q+αnTyn-q+en≤1-αnδxn-q+αnδyn-q+en≤1-αnδxn-q+αnδyn-q+en.Also, from (9), we have (55)yn-q≤1-βnxn+βnTxn-q+fn=1-βnxn-q+βnTxn-q+fn≤1-βnxn-q+βnδxn-q+fn≤1-βn+βnδxn-q+fn.Substituting (55) in (54), we have(56)xn+1≤1-αnδ+αnδ1-βn+βnδxn-q+en+fn. Inequality (56) becomes (57)xn+1-q≤1-1-δαnxn-q+en+fn,n=0,1,2,…. By letting gn=fn+en and by Lemma 2, using the condition of the theorem, we have (58)limn→∞xn+1-q=0. Hence, (59)xn+1⟶q∈FT.Theorem 8.
LetC be a nonempty subset of a normed space E,·. Let T:C→C be z-operator (16). Let xn be defined with iterative process (11). If FT≠ϕ, ∑n=1∞‍αn=∞, un=0. Then xn converges strongly to a fixed point of T.Proof.
From (26), we have (60)Tx-Ty≤δx-y+2δx-Tx∀x,y∈C. Also from (28), we have (61)Txn-q≤δxn-q. Now, applying (28) to xn+1 in (11), we get (62)xn+1-q≤1-αnyn+αnTyn-q+en=1-αnyn-q+αnTyn-q+en≤1-αnyn-q+αnδyn-q+en. This implies(63)xn+1-q≤1-αn+αnδyn-q+en. Also, applying (28) to yn in (11), we get (64)yn-q≤1-βnxn+βnTxn-q+fn=1-βnxn-q+βnTxn-q+fn≤1-βnxn-q+βnδxn-q+fn. This gives(65)yn-q≤1-βn+βnδxn-q+fn. By combining (63) and (65), we have (66)xn+1-q≤1-αn+αnδ1-βn+βnδxn-q+en+fn=1-1-δαn1-1-δβnxn-q+en+fn≤1-1-δαnxn-q+gn,n=0,1,2,…. Thus, applying Lemma 2 and the condition of the theorem, we obtain (67)limn→∞xn+1-q=0. Therefore, (68)xn+1⟶q∈FT.


## 4. Numerical Results

In this section, we support our analytical results with the following two numerical examples.Example 9.
Let the functionf:0,4→0,4 be defined by fx=x+21/3 with fixed point q=1.521379707. Initial guess is x0=1.99, αn=βn=1/4.Example 10.
Let the functionf:0,4→0,4 be defined by fx=2x+31/2 with fixed point q=3.000000000. Initial guess is x0=4, αn=1/n+2, βn=1/n+1.The results for Examples9 and 10 with various iterations are presented in Tables 1 and 2, respectively.Table 1
The results for various iterations of Example9.
n Mann Picard-Mann Ishikawa Thianwan S-iteration 0 1.990000000 1.990000000 1.990000000 1.990000000 1.990000000 1 1.687057835 1.572582235 1.813283737 1.809910017 1.582703394 2 1.580420632 1.527152378 1.703481407 1.699251872 1.52692284 3 1.542481649 1.522032849 1.63509312 1.631121832 1.52251127 4 1.52892988 1.521453636 1.592432357 1.58912183 1.521534023 5 1.524082164 1.521388075 1.565793786 1.563209142 1.521400741 6 1.522347139 1.521380654 1.549149294 1.547213668 1.521382574 7 1.521726048 1.521379814 1.53874517 1.537336782 1.521380098 8 1.525103699 1.521379719 1.532240114 1.531236794 1.52137976 9 1.521424097 1.521379708 1.528172254 1.527468964 1.521379714 10 1.521395599 1.521379707 1.52562821 1.525141481 1.521379708 11 1.521385396 1.521379707 1.524037063 1.523703668 1.521379707 12 1.521381741 1.521379707 1.523041857 1.522815428 1.521379707 13 1.521380435 1.521379707 1.522419375 1.522266655 1.521379707 14 1.521379968 1.521379707 1.522030019 1.521927661 1.521379707 15 1.52137980 1.521379707 1.521786478 1.521718232 1.521379707 16 1.521379740 1.521379707 1.521634144 1.521588448 1.521379707 17 1.521379731 1.521379707 1.521538858 1.521508914 1.521379707 18 1.521379715 1.521379707 1.521479257 1.521459531 1.521379707 19 1.521379710 1.521379707 1.521441976 1.521429016 1.521379707 20 1.521379708 1.521379707 1.521418656 1.52141017 1.521379707 21 1.521379707 1.521379707 1.52140407 1.521398527 1.521379707 22 1.521379707 1.521379707 1.521394946 1.521391334 1.521379707 23 1.521379707 1.521379707 1.521389239 1.52138689 1.521379707 24 1.521379707 1.521379707 1.521385669 1.521384145 1.521379707 25 1.521379707 1.521379707 1.521383436 1.521382449 1.521379707 26 1.521379707 1.521379707 1.52138204 1.521381401 1.521379707 27 1.521379707 1.521379707 1.521381166 1.521380753 1.521379707 28 1.521379707 1.521379707 1.521380620 1.521380353 1.521379707 29 1.521379707 1.521379707 1.521380278 1.521380106 1.521379707 30 1.521379707 1.521379707 1.521380064 1.521379954 1.521379707 31 1.521379707 1.521379707 1.521379930 1.521379860 1.521379707Table 2
The results for various iterations of Example10.
n Mann Picard-Mann Ishikawa Thianwan S-iteration 0 4.000000000 4.000000000 4.000000000 4.000000000 4.000000000 1 3.658312395 3.211950309 3.316624790 3.210186229 3.210186229 2 3.360737671 3.05436713 3.174706708 3.10855584 3.061581272 3 3.178630457 3.015059648 3.11619388 3.070267725 3.019321892 4 3.082897312 3.004347136 3.085110108 3.050716647 3.006219262 5 3.036738167 3.001287744 3.066152455 3.03905625 3.002026325 6 3.015723592 3.000388341 3.053529001 3.031402901 3.000664646 7 3.006547498 3.000118657 3.044594314 3.026040224 3.000218903 8 3.002666794 3.000036622 3.037979605 3.022099423 3.000072291 9 3.001066599 3.000011394 3.032910338 3.019096558 3.000023918 10 3.000420156 3.000003568 3.028917572 3.016742078 3.000007924 11 3.000163391 3.000001123 3.025701936 3.014852836 3.000002628 12 3.000062842 3.000000355 3.023063963 3.013307708 3.000000872 13 3.00002394 3.000000113 3.020865992 3.012023608 3.000000029 14 3.000009044 3.000000036 3.019010164 3.010941775 3.000000001 15 3.00000331 3.000000012 3.017425149 3.010019562 3.000000000 16 3.000001263 3.000000004 3.016057811 3.009225316 3.000000000 17 3.000000468 3.000000001 3.014867814 3.00853509 3.000000000 18 3.000000172 3.000000000 3.013824028 3.007930452 3.000000000 19 3.000000063 3.000000000 3.012902073 3.007397005 3.000000000 20 3.000000023 3.000000000 3.012082600 3.006923067 3.000000000 21 3.000000008 3.000000000 3.011350076 3.006500078 3.000000000 22 3.000000003 3.000000000 3.010691894 3.006120341 3.000000000 23 3.000000001 3.000000000 3.010097723 3.005777802 3.000000000 24 3.000000000 3.000000000 3.009559027 3.005467464 3.000000000 25 3.000000000 3.000000000 3.009068691 3.00518517 3.000000000 26 3.000000000 3.000000000 3.008620741 3.004927435 3.000000000 27 3.000000000 3.000000000 3.008210132 3.004691315 3.000000000 28 3.000000000 3.000000000 3.007832568 3.004474311 3.000000000 29 3.000000000 3.000000000 3.007484378 3.004274286 3.000000000 30 3.000000000 3.000000000 3.007162403 3.004089403 3.000000000 31 3.000000000 3.000000000 3.00686391 3.003918076 3.000000000The error estimation(en) in Table 1 choosing n at an interval of 5 iterations is presented in Table 3, where en=q-xn.Table 3
The error estimation for various iterations of Example9.
n Error estimation (en) Mann Picard-Mann Ishikawa Thianwan S-iteration 5 0.002702457 0.000008368 0.044414079 0.041829435 0.000021034 10 0.000015892 0.0 0.004248503 0.003761774 0.000000001 15 0.00000093 0.0 0.000406771 0.000338525 0.0 20 0.000000001 0.0 0.000038949 0.000030463 0.0 25 0.0 0.0 0.000003729 0.000002742 0.0 30 0.0 0.0 0.000000357 0.000000247 0.0The calculated error in Table2 choosing n at an interval of 5 iterations is presented in Table 4.Table 4
The error estimation for various iterations of Example10.
n Error estimation (en) Mann Picard-Mann Ishikawa Thianwan S-iteration 5 0.036738167 0.001287744 0.066152455 0.03905625 0.002026325 10 0.000420156 0.000003568 0.028917572 0.016742078 0.000007924 15 0.000003391 0.000000012 0.017425149 0.010019562 0.0 20 0.000000023 0.0 0.012902073 0.006923067 0.0 25 0.0 0.0 0.009559027 0.00518517 0.0 30 0.0 0.0 0.007162403 0.004089403 0.0


## 5. Conclusion

This paper proved the convergence and convergence rate of some two-step iterative schemes with errors using Zamfirescu operator in Banach spaces. It was observed from Example9 that the Picard-Mann iteration converges faster than the S-iteration, Ishikawa iteration, and Thianwan iteration, while, in Example 10, it is seen that the S-iteration has better convergence rate than other iterations. This is so because the speed of the aforementioned iterations depends on the choices of the parametric constants αn and βn.



---

*Source: 9641706-2016-10-10.xml*