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.
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 [
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.
The Picard iteration process is defined by the sequence
In [
Let
When condition (
Mann [
The concept of Mann iteration with error was discussed in [
Ishikawa [
In an attempt to reduce computational cost, Agarwal et al. [
The S-iteration with errors can be given as
In [
The Thianwan iteration with errors is given by
Khan [
The most generalized operator used to approximate fixed point is the one proved by Zamfirescu [
The following Lemma is useful in the proof of our results.
Let
The Zamfirescu operator was used to prove the strong convergence of (
Let
By Lemma
We present our main results using
Throughout,
Let
Let
Let
From (
By Lemma
Let
From (
Let
From (
Let
From (
In this section, we support our analytical results with the following two numerical examples.
Let the function
Let the function
The results for Examples
The results for various iterations of Example
| 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 |
The results for various iterations of Example
| 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
The error estimation for various iterations of Example
| 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
The error estimation for various iterations of Example
| 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 |
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
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.
The authors hereby declare that there are no competing interests.