Journal of Mathematics
Volume 2016 (2016), Article ID 9641706, 8 pages
http://dx.doi.org/10.1155/2016/9641706

Research Article

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

O. T. Wahab,¹ R. O. Olawuyi,¹ K. Rauf,² and I. F. Usamot²

¹Department of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria
²Department of Mathematics, University of Ilorin, Ilorin, Nigeria

Received 11 July 2016; Accepted 6 September 2016

Academic Editor: Ji Gao

Copyright © 2016 O. T. Wahab et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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

and the concept of Picard iteration process with error is defined as follows:

where satisfy .

In [2], Banach proved the convergence of (1) with the aid of the following contractive mapping:

where , is a metric space, , and . Banach’s theorem is given as follows.

Theorem 1. Let be a metric space and be a contraction map on . Then has a unique fixed point .

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 , of a closed subset of normed space:

where is a sequence in .

The concept of Mann iteration with error was discussed in [4] and it is given by the sequence

where is a sequence in and satisfy .

Ishikawa [5] defined another iteration process given by the sequence

where and are sequences in . The Ishikawa iteration is a double Mann iteration and has better approximation than Mann iteration (5). The Ishikawa iteration with errors is given as

where and satisfy and . 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

where and are sequences in .

The S-iteration with errors can be given as

where and satisfy and .

In [7], Thianwan defined a new iteration process given by the sequence

where and are sequences in .

The Thianwan iteration with errors is given by

where and satisfy and .

Khan [8] gave a different perspective to iteration procedure. He introduced the following hybrid Picard-Mann iterative scheme for a single nonexpansive mapping which is defined as

where is a sequence in with . While the Picard-Mann iteration process with errors is given as

where and satisfy and .

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 is called a Kannan mapping if there exists such that

Another similar definition due to Chatterjea mapping is as follows: there exists such that

By combining (3), (14), and (15) conditions, we have the Zamfirescu operator given, for , by

The equivalence of (16) is given as follows:

where and .

The following Lemma is useful in the proof of our results.

Lemma 2 (see [12]). Let be a real number such that and is a sequence of nonnegative numbers such that ; then for any sequence of positive numbers satisfying

we have

The Zamfirescu operator was used to prove the strong convergence of (1) as follows.

Theorem 3. Let be a nonempty subset of a normed space . Let be z-operator. If ,  , . Then converges strongly to a fixed point of .

Proof. By Lemma 2, has a unique fixed point in , say . Let ; since is a -operator, at least one of each conditions is satisfied. If holds, then

is equivalent to

This implies

Therefore,

Similarly, if holds, we obtain

Let

Then we have and in view of (25) it results in that inequalities (23) and (24) become

Now, for and in (26), we obtain

which implies

By Lemma 2, we conclude that converges strongly to .

3. Analytical Results

We present our main results using z-operators (16) on normed spaces for the two-step iterative schemes with errors defined in the last section.

Throughout, denotes the set of all positive integers, is nonempty convex subset of a complete normed space , and is a self-map.

Theorem 4. Let be a nonempty subset of a normed space . Let be -operator (16). Let be defined with iterative process (5). If , , . Then converges strongly to a fixed point of .

Proof. Let ; by using (5) and (16), we have

Substituting (28) in (29),

Hence, by Lemma 2 and using the fact that , , and , it results in that

Therefore,

Theorem 5. Let be a nonempty subset of a normed . Let be -operator (16). Let be defined with iterative process (13). If , , . Then converges strongly to a fixed point of .

Proof. From (26), we have

Also from (28), we have

Now, using (13) and (28), we have

This implies

Also,

Substituting (37) in (36), we have

Using (28) in (38), we have

where .
By Lemma 2 and using the fact that ,   and , this becomes

Hence,

Theorem 6. Let be a nonempty subset of a normed space . Let be -operator. Let be defined with iterative process (7). If , , . Then, converges strongly to a fixed point of .

Proof. From (26), we have

Also from (28), we have

Now, by combining (7) and (28), we have

This gives

Also, from (7), we have

This also gives

By combining (47) and (45), we have

Using (28) in (48), we have

By letting and by Lemma 2, using the fact that , , and , we get

Therefore,

Theorem 7. Let be a nonempty subset of a normed space . Let be -operator (16). Let be defined with iterative process (9). If , , . Then converges strongly to a fixed point of .

Proof. From (26), we have

Also from (28), we have

Applying (28) to (9), we obtain

Also, from (9), we have

Substituting (55) in (54), we have

Inequality (56) becomes

By letting and by Lemma 2, using the condition of the theorem, we have

Hence,

Theorem 8. Let be a nonempty subset of a normed space . Let be -operator (16). Let be defined with iterative process (11). If , , . Then converges strongly to a fixed point of .

Proof. From (26), we have

Also from (28), we have

Now, applying (28) to in (11), we get

This implies

Also, applying (28) to in (11), we get

This gives

By combining (63) and (65), we have

Thus, applying Lemma 2 and the condition of the theorem, we obtain

Therefore,

4. Numerical Results

In this section, we support our analytical results with the following two numerical examples.

Example 9. Let the function be defined by with fixed point . Initial guess is , .

Example 10. Let the function be defined by with fixed point . Initial guess is , , .

The results for Examples 9 and 10 with various iterations are presented in Tables 1 and 2, respectively.

Table 1: The results for various iterations of Example 9.

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
1.521379707

Table 2: The results for various iterations of Example 10.

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
3.000000000

The error estimation in Table 1 choosing at an interval of 5 iterations is presented in Table 3, where .

Table 3: The error estimation for various iterations of Example 9.

Error estimation ()

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
0.0

The calculated error in Table 2 choosing at an interval of 5 iterations is presented in Table 4.

Table 4: The error estimation for various iterations of Example 10.

Error estimation ()

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 Example 9 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 and .

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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