S-iteration process for quasi-contractive mappings
- Vivek Kumar^{1},
- Abdul Latif^{2}Email author,
- Arif Rafiq^{3} and
- Nawab Hussain^{2}
https://doi.org/10.1186/1029-242X-2013-206
© Kumar et al.; licensee Springer 2013
Received: 19 November 2012
Accepted: 11 April 2013
Published: 25 April 2013
Abstract
In this note, we show that the S-iteration process due to Sahu and Petrusel (Nonlinear Anal. TMA 74(17):6012-6023, 2011) is faster than the Picard, Mann, Ishikawa and Noor iteration processes for Zamfirescu operators. Also, using computer programs in $\mathrm{C}++$, we present some examples to compare the convergence rate of iterative processes due to Picard, Mann, Ishikawa, Noor, Agarwal et al. and Sahu and Petrusel.
Keywords
iteration processes quasi-contractive operator Zamfirescu operator1 Introduction
During the last many years, much attention has been given to the following iteration processes (see, for example, [1–7]).
- (a)
- (c)
- (d)
where $\{{b}_{n}\}$, $\{{b}_{n}^{\prime}\}$ and $\{{b}_{n}^{\mathrm{\prime}\mathrm{\prime}}\}$ are sequences of positive numbers in $[0,1]$ and denoted by ${N}_{n}$, is known as the Noor multi-step iteration process [12].
Definition 1 [13]
Theorem 2 [14]
Let $(X,d)$ be a complete metric space, and let $T:X\to X$ be a mapping for which there exist real numbers a, b and c satisfying $0<a<1$, $0<b,c<\frac{1}{2}$ such that for each pair $x,y\in X$ at least one of the following is true:
(z1) $d(Tx,Ty)\le ad(x,y)$,
(z2) $d(Tx,Ty)\le b[d(x,Tx)+d(y,Ty)]$,
(z3) $d(Tx,Ty)\le c[d(x,Ty)+d(y,Tx)]$.
converges to p for any ${x}_{0}\in X$.
Remark 3 An operator T which satisfies the contraction conditions (z1)-(z3) of Theorem 2 is called a Zamfirescu operator [13, 15, 16] and is denoted by Z.
for any $x,y\in E$ and some $\delta \in [0,1)$, $L\ge 0$.
He proved that this class is wider than the class of Zamfirescu operators. The following results are proved in [15, 16].
Theorem 4 [16]
Let C be a nonempty closed convex subset of a normed space E. Let $T:C\to C$ be an operator satisfying (B). Let $\{{x}_{n}\}$ be defined through the iterative process (${M}_{n}$). If $F(T)\ne \mathrm{\varnothing}$ and $\sum {b}_{n}=\mathrm{\infty}$, then $\{{x}_{n}\}$ converges strongly to the unique fixed point of T.
Theorem 5 [16]
Let C be a nonempty closed convex subset of an arbitrary Banach space E, and let $T:C\to C$ be an operator satisfying (B). Let $\{{x}_{n}\}$ be defined through the Ishikawa iterative process (${I}_{n}$) and ${x}_{0}\in C$, where $\{{b}_{n}\}$ and $\{{b}_{n}^{\prime}\}$ are sequences of positive real numbers in $[0,1]$ with $\{{b}_{n}\}$ satisfying $\sum {b}_{n}=\mathrm{\infty}$. Then $\{{x}_{n}\}$ converges strongly to the fixed point of T.
The following theorem was presented in [17].
- (i)
$0\le {b}_{n},{b}_{n}^{\prime}\le 1$,
- (ii)
$\sum {b}_{n}=\mathrm{\infty}$.
Then ${M}_{n}$ and ${I}_{n}$ converge strongly to the unique fixed point of a Zamfirescu operator $T:C\to C$, and, moreover, the Mann iteration process converges faster than the Ishikawa iteration process to the fixed point of T.
Remark 7 In [18], Qing and Rhoades, by taking a counter example, showed that the Ishikawa iteration process is faster than the Mann iteration process for Zamfirescu operators.
In this note, we establish a general theorem to approximate the fixed points of quasi-contractive operators in a Banach space through the S-iteration process due to Sahu and Petrusel [11]. Our result generalizes and improves upon, among others, the corresponding results of Babu and Prasad [17] and Berinde [13, 15, 16]. We also prove that the S-iteration process is faster than the Mann, Ishikawa, Picard and Noor iteration processes, respectively, for Zamfirescu operators.
2 Main results
We now prove our main results.
Theorem 8 Let C be a nonempty closed convex subset of a normed space E. Let $T:C\to C$ be an operator satisfying (B). Let $\{{x}_{n}\}$ be defined through the iterative process (${S}_{n}$) and ${x}_{0}\in C$, where $\{{b}_{n}^{\prime}\}$ is a sequence in $[0,1]$ satisfying $\sum {b}_{n}^{\prime}=\mathrm{\infty}$. Then $\{{x}_{n}\}$ converges strongly to the fixed point of T.
which by (2.8) implies ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}-w\parallel =0$.
Consequently, ${x}_{n}\to w\in F$ and this completes the proof. □
Now we present an example to show that the S-iteration process is faster than the Mann, Ishikawa, Picard and Noor iteration processes, respectively, for Zamfirescu operators.
Example 9 Let $T:[0,1]\to [0,1]:=\frac{x}{2}$. Let ${b}_{n}=\frac{4}{\sqrt{n}}={b}_{n}^{\prime}={b}_{n}^{\mathrm{\prime}\mathrm{\prime}}$.
It is clear that T is a Zamfirescu operator with a unique fixed point 0. Also, it is easy to see that Example 9 satisfies all the conditions of Theorem 8.
Hence, ${lim}_{n\to \mathrm{\infty}}|\frac{{S}_{n}-0}{{M}_{n}-0}|=0$.
3 Applications
Deceasing cum sublinear functions
Number of iterations | Noor iteration | Picard iteration | Mann iteration | Ishikawa iteration | Agarwal et al. iteration | S-iteration | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
n | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ |
0 | 2.56e−06 | 3.094628e−38 | 2.56e−06 | 2.56e−06 | 2.56e−06 | 2.56e−06 | 1.103589 | 1.137884 | 2.56e−06 | 0.99998 | 2.56e−06 | 0.99998 |
1 | 1 | 0.840894 | 0.99998 | 0.99998 | 0.99998 | 0.84088 | 1.150601 | 1.149315 | 3.094628e−38 | 0.210231 | 3.094628e−38 | 0.250008 |
2 | 06676e−07 | 0.233944 | 3.094628e−38 | 3.094628e−38 | 4.109669e−07 | 0.20195 | – | – | 0.151357 | 0.215059 | 0.100105 | 0.31023 |
3 | 0.1186 | 0.134251 | 1 | 1 | 0.164529 | 0.175489 | 1.157988 | 1.157426 | 0.14411 | 0.209484 | 0.051243 | 0.337071 |
4 | 0.315599 | 0.313519 | 0 | 0 | 0.213585 | 0.200965 | 1.158157 | 1.15772 | 0.152505 | 0.199147 | 0.037302 | 0.308803 |
5 | 0.049321 | 0.151679 | 1 | 1 | 0.166159 | 0.178726 | 1.158273 | 1.157929 | 0.169209 | 0.19168 | 0.052097 | 0.286162 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
8 | 0.240452 | 0.214091 | 0 | 0 | 0.198079 | 0.191799 | 1.158458 | 1.158278 | 0.188331 | 0.188347 | 0.096417 | 0.241306 |
9 | 0.145539 | 0.175829 | 1 | 1 | 0.182035 | 0.186308 | 1.158492 | 1.158344 | 0.188349 | 0.188348 | 0.109782 | 0.231058 |
10 | 0.212882 | 0.197132 | 0 | 0 | 0.192167 | 0.189525 | 1.158518 | 1.158395 | 0.188347 | 0.188348 | 0.122222 | 0.222431 |
11 | 0.172645 | 0.18423 | 1 | 1 | 0.186172 | 0.187724 | 1.158538 | 1.158436 | 0.188348 | 0.188348 | 0.133632 | 0.215183 |
12 | 0.196128 | 0.190408 | 0 | 0 | 0.189509 | 0.188664 | 1.158554 | 1.158468 | 0.188348 | 0.188348 | 0.143929 | 0.209132 |
13 | 0.184556 | 0.187511 | 1 | 1 | 0.187761 | 0.188197 | 1.158567 | 1.158494 | 0.188348 | 0.188348 | 0.153049 | 0.204138 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
18 | 0.188363 | 0.18835 | 0 | 0 | 0.188356 | 0.188349 | 0.165756 | 0.198778 | 0.188348 | 0.188348 | 0.180912 | 0.191033 |
19 | 0.188344 | 0.188347 | 1 | 1 | 0.188345 | 0.188347 | 0.169834 | 0.196745 | 0.188348 | 0.188348 | 0.183419 | 0.190053 |
20 | 0.188348 | 0.188348 | 0 | 0 | 0.188349 | 0.188348 | 0.173312 | 0.195058 | 0.188348 | 0.188348 | 0.185206 | 0.189388 |
21 | 0.188348 | 0.188348 | 1 | 1 | 0.188347 | 0.188348 | 0.176246 | 0.193668 | 0.188348 | 0.188348 | 0.186424 | 0.188957 |
22 | 0.188348 | 0.188348 | 0 | 0 | 0.188348 | 0.188348 | 0.178695 | 0.192533 | 0.188348 | 0.188348 | 0.187219 | 0.18869 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
34 | – | – | – | – | – | – | – | – | – | – | 0.188348 | 0.188348 |
35 | – | – | – | – | – | – | – | – | – | – | 0.188348 | 0.188348 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
53 | 0.188348 | 0.188348 | 1 | 1 | 0.188348 | 0.188348 | 0.188347 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 |
54 | 0.188348 | 0.188348 | 0 | 0 | 0.188348 | 0.188348 | 0.188347 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 |
55 | 0.188348 | 0.188348 | 1 | 1 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 |
56 | 0.188348 | 0.188348 | 0 | 0 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 | 0.188348 |
Increasing functions
Number of iterations | Noor iteration | Picard iteration | Mann iteration | Ishikawa iteration | Agarwal et al. iteration | S-iteration | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
n | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ |
0 | 0.964 | 0.998591 | 0.9 | 0.9 | 0.964 | 0.964 | 0.99293 | 0.99293 | 0.964 | 0.99293 | 0.964 | 0.99293 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
4 | 0.999974 | 0.999922 | 0.99985 | 0.99985 | 0.994714 | 0.999996 | 0.999997 | 0.999937 | 0.999996 | 0.999997 | 0.999998 | 0.999999 |
5 | 0.999984 | 0.99995 | 0.99997 | 0.99997 | 0.996595 | 0.999999 | 0.999999 | 0.999973 | 0.999999 | 0.999999 | 1 | 1 |
6 | 0.99999 | 0.999966 | 0.999994 | 0.999994 | 0.997703 | 1 | 1 | 0.999937 | 1 | 1 | 1 | 1 |
7 | 0.999993 | 0.999976 | 0.999999 | 0.999999 | 0.998396 | 1 | 1 | 0.999988 | 1 | 1 | 1 | 1 |
8 | 0.999995 | 0.999983 | 1 | 1 | 0.998849 | 1 | 1 | 0.999994 | 1 | 1 | 1 | 1 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
23 | 1 | 1 | 1 | 1 | 0.999964 | 1 | 1 | 0.999997 | 1 | 1 | 1 | 1 |
24 | 1 | 1 | 1 | 1 | 0.99997 | 1 | 1 | 0.999998 | 1 | 1 | 1 | 1 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
33 | 1 | 1 | 1 | 1 | 0.999994 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
68 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
69 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Superlinear functions with multiple roots
Number of iterations | Noor iteration | Picard iteration | Mann iteration | Ishikawa iteration | Agarwal et al. iteration | S-iteration | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
n | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ |
0 | 0.944 | 0.999988 | 0.944 | 0.944 | 0.944 | 0.944 | 0.944 | 0.996513 | 0.944 | 0.996513 | 0.944 | 0.996513 |
1 | 1 | 0.999996 | 0.996513 | 0.996513 | 0.996513 | 0.981132 | 0.999988 | 0.998978 | 0.999988 | 0.999996 | 0.999988 | 0.999999 |
2 | 1 | 0.999998 | 0.999988 | 0.999988 | 0.999631 | 0.991812 | 0.999999 | 0.999568 | 1 | 1 | 1 | 1 |
3 | 1 | 0.999999 | 1 | 1 | 0.999932 | 0.995872 | 1 | 0.999784 | 1 | 1 | 1 | 1 |
4 | 1 | 1 | 1 | 1 | 0.999983 | 0.99771 | 1 | 0.999881 | 1 | 1 | 1 | 1 |
5 | 1 | 1 | 1 | 1 | 0.999995 | 0.998643 | 1 | 0.999929 | 1 | 1 | 1 | 1 |
6 | 1 | 1 | 1 | 1 | 0.999998 | 0.999155 | 1 | 0.999956 | 1 | 1 | 1 | 1 |
7 | 1 | 1 | 1 | 1 | 0.999999 | 0.999454 | 1 | 0.999972 | 1 | 1 | 1 | 1 |
8 | 1 | 1 | 1 | 1 | 1 | 0.999636 | 1 | 0.999981 | 1 | 1 | 1 | 1 |
9 | 1 | 1 | 1 | 1 | 1 | 0.999751 | 1 | 0.999987 | 1 | 1 | 1 | 1 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
19 | 1 | 1 | 1 | 1 | 1 | 0.999987 | 1 | 0.999999 | 1 | 1 | 1 | 1 |
20 | 1 | 1 | 1 | 1 | 1 | 0.99999 | 1 | 0.999999 | 1 | 1 | 1 | 1 |
21 | 1 | 1 | 1 | 1 | 1 | 0.999992 | 1 | 1 | 1 | 1 | 1 | 1 |
22 | 1 | 1 | 1 | 1 | 1 | 0.999994 | 1 | 1 | 1 | 1 | 1 | 1 |
23 | 1 | 1 | 1 | 1 | 1 | 0.999995 | 1 | 1 | 1 | 1 | 1 | 1 |
– | – | – | – | – | – | – | – | – | – | – | – | – |
32 | 1 | 1 | 1 | 1 | 1 | 0.999999 | 1 | 1 | 1 | 1 | 1 | 1 |
33 | 1 | 1 | 1 | 1 | 1 | 0.999999 | 1 | 1 | 1 | 1 | 1 | 1 |
34 | 1 | 1 | 1 | 1 | 1 | 0.999999 | 1 | 1 | 1 | 1 | 1 | 1 |
35 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
36 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Oscillatory functions
Number of iterations | Noor iteration | Picard iteration | Mann iteration | Ishikawa iteration | Agarwal et al. iteration | S-iteration | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
n | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ | ${\mathit{f}\mathit{x}}_{\mathit{n}}$ | ${\mathit{x}}_{\mathit{n}\mathbf{+}\mathbf{1}}$ |
0 | 1.25 | 1.25 | 1.25 | 1.25 | 1.25 | 1.25 | 1.25 | 0.8 | 1.25 | 0.8 | 1.25 | 0.8 |
1 | 0.8 | 0.921512 | 0.8 | 0.8 | 0.8 | 0.871597 | 1.25 | 0.840872 | 1.25 | 0.912469 | 1.25 | 0.848606 |
2 | 1.085173 | 1.022975 | 1.25 | 1.25 | 1.14732 | 1.081101 | 1.189241 | 0.889224 | 1.095928 | 0.985571 | 1.178403 | 0.909754 |
3 | 0.977541 | 0.997476 | 0.8 | 0.8 | 0.924983 | 0.970709 | 1.124576 | 0.930283 | 1.01464 | 0.999983 | 1.099198 | 0.95812 |
4 | 1.002531 | 1.000107 | 1.25 | 1.25 | 1.030175 | 1.010476 | 1.074942 | 0.95931 | 1.000017 | 1.000002 | 1.043711 | 0.984874 |
5 | 0.999893 | 1.000002 | 0.8 | 0.8 | 0.989632 | 0.997158 | 1.042416 | 0.977477 | 0.999998 | 1 | 1.015358 | 0.995667 |
6 | 0.999998 | 1 | 1.25 | 1.25 | 1.00285 | 1.000657 | 1.023042 | 0.987967 | 1 | 1 | 1.004352 | 0.998995 |
7 | 1 | 1 | 0.8 | 0.8 | 0.999343 | 0.999876 | 1.01218 | 0.99372 | 1 | 1 | 1.001006 | 0.999809 |
8 | 1 | 1 | 0.8 | 0.8 | 1.000124 | 1.000019 | 1.00632 | 0.996772 | 1 | 1 | 1.000191 | 0.99997 |
9 | 1 | 1 | 1.25 | 1.25 | 0.999981 | 0.999998 | 1.003238 | 0.998358 | 1 | 1 | 1.00003 | 0.999996 |
10 | 1 | 1 | 0.8 | 0.8 | 1.000002 | 1 | 1.001645 | 0.99917 | 1 | 1 | 1.000004 | 1 |
11 | 1 | 1 | 0.8 | 0.8 | 1 | 1 | 1.000831 | 0.999583 | 1 | 1 | 1 | 1 |
12 | 1 | 1 | 1.25 | 1.25 | 1 | 1 | 1.000418 | 0.999791 | 1 | 1 | 1 | 1 |
13 | 1 | 1 | 0.8 | 0.8 | 1 | 1 | 1.000209 | 0.999895 | 1 | 1 | 1 | 1 |
14 | 1 | 1 | 0.8 | 0.8 | 1 | 1 | 1.000105 | 0.999948 | 1 | 1 | 1 | 1 |
15 | 1 | 1 | 1.25 | 1.25 | 1 | 1 | 1.000052 | 0.999974 | 1 | 1 | 1 | 1 |
16 | 1 | 1 | 0.8 | 0.8 | 1 | 1 | 1.000026 | 0.999987 | 1 | 1 | 1 | 1 |
17 | 1 | 1 | 1.25 | 1.25 | 1 | 1 | 1.000013 | 0.999993 | 1 | 1 | 1 | 1 |
18 | 1 | 1 | 0.8 | 0.8 | 1 | 1 | 1.000007 | 0.999997 | 1 | 1 | 1 | 1 |
19 | 1 | 1 | 1.25 | 1.25 | 1 | 1 | 1.000003 | 0.999998 | 1 | 1 | 1 | 1 |
20 | 1 | 1 | 0.8 | 0.8 | 1 | 1 | 1.000002 | 0.999999 | 1 | 1 | 1 | 1 |
21 | 1 | 1 | 1.25 | 1.25 | 1 | 1 | 1.000001 | 1 | 1 | 1 | 1 | 1 |
22 | 1 | 1 | 0.8 | 0.8 | 1 | 1 | 1 | 1 | 1 | 1 |
Decreasing cum sublinear functions
Let $f:[0,1]\to [0,1]$ be defined by $f(x)={(1-x)}^{m}$, $m=7,8\dots $ . Then f is a decreasing function. By taking $m=8$ and ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{4}}}$, the comparison of convergence of the above-mentioned iterative processes to the exact fixed point $p=0.188348$ is listed in Table 1.
Increasing functions
Let $f:[0,8]\to [0,8]$ be defined as $f(x)=\frac{{x}^{2}+9}{10}$. Then f is an increasing function. By taking ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{2}}}$, the comparison of convergence of the above-mentioned iterative processes to the exact fixed point $p=1$ is listed in Table 2.
Superlinear functions with multiple roots
The function defined by $f(x)=2{x}^{3}-7{x}^{2}+8x-2$ is a superlinear function with multiple real roots. By taking ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{2}}}$, the comparison of convergence of the above-mentioned iterative processes to the exact fixed point $p=1$ is listed in Table 3.
Oscillatory functions
The function defined by $f(x)=\frac{1}{x}$ is an oscillatory function. By taking ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{4}}}$, the comparison of convergence of the above-mentioned iterative processes to the exact fixed point $p=1$ is listed in Table 4.
For detailed study, these programs are again executed after changing the parameters, and the readings are recorded as follows.
4 Observations
Decreasing cum sublinear functions
- 1.
For $m=8$ and ${x}_{0}=0.8$, the Picard process never converges (oscillates between 0 and 1), the Mann process converges in 23 iterations, the Ishikawa process converges in 56 iterations, the Noor process converges in 21 iterations, the Agarwal et al. process converges in 12 iterations and the S-iterative process converges in 34 iterations.
- 2.
For $m=30$ and ${x}_{0}=0.8$, the Picard process never converges (oscillates between 0 and 1), the Mann process converges in 37 iterations, the Ishikawa process converges in 118 iterations, the Noor process converges in 47 iterations, the Agarwal et al. process converges in 15 iterations while the S-iterative process never converges.
- 3.
Taking initial guess ${x}_{0}=0.2$ (nearer to the fixed point), the Picard process never converges (oscillates between 0 and 1), the Mann process converges in 24 iterations, the Ishikawa process converges in 56 iterations, the Noor process converges in 22 iterations, the Agarwal et al. process converges in 12 iterations and the S-iterative process converges in 34 iterations.
- 4.
Taking ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{6}}}$ and ${x}_{0}=0.8$, we find that the Mann process converges in 72 iterations, the Ishikawa process converges in 289 iterations, the Noor process converges in 105 iterations, the Agarwal et al. process converges in 36 iterations, while the S-iterative process converges in 240 iterations.
Increasing functions
- 1.
For ${x}_{0}=0.8$, the Picard process converges to a fixed point in 8 iterations, the Mann process converges in 69 iterations, the Ishikawa process converges in 34 iterations, the Noor process converges in 24 iterations, the Agarwal et al. process converges in 7 iterations and the S-iterative process converges in 6 iterations.
- 2.
Taking initial guess ${x}_{0}=0.6$ (away from the fixed point), the Picard process converges to a fixed point in 8 iterations, the Mann process converges in 75 iterations, the Ishikawa process converges in 38 iterations, the Noor process converges in 27 iterations, the Agarwal et al. process converges in 6 iterations and the S-iterative process converges in 6 iterations.
- 3.
Taking ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{4}}}$ and ${x}_{0}=0.8$, we obtain that the Mann process converges in 23 iterations, the Ishikawa process converges in 12 iterations, the Noor process converges in 9 iterations, the Agarwal et al. process converges in 6 iterations and the S-iterative process converges in 6 iterations.
Superlinear functions with multiple roots
- 1.
For ${x}_{0}=0.8$, the Picard process converges to a fixed point in 4 iterations, the Mann process converges in 36 iterations, the Ishikawa process converges in 22 iterations, the Noor process converges in 5 iterations and the Agarwal et al. as well as the S-iterative processes converge in 3 iterations.
- 2.
Taking initial guess ${x}_{0}=0.6$ (away from the fixed point), the Picard process converges to a fixed point in 5 iterations, the Mann process converges in 52 iterations, the Ishikawa process converges in 40 iterations, the Noor process converges in 57 iterations and the Agarwal et al. as well as the S-iterative processes converge in 4 iterations.
- 3.
Taking ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{4}}}$ and ${x}_{0}=0.8$, we obtain that the Mann process converges in 13 iterations, the Ishikawa process converges in 9 iterations, the Noor process converges in 14 iterations and the Agarwal et al. as well as the S-iterative processes converge in 3 iterations.
Oscillatory functions
- 1.
For ${x}_{0}=0.8$, the Picard process never converges to a fixed point, the Mann process converges in 12 iterations, the Ishikawa process converges in 23 iterations, the Noor process converges in 8 iterations, the Agarwal et al. process converges in 7 iterations and the S-iterative process converges in 12 iterations.
- 2.
Taking initial guess ${x}_{0}=0.6$ (away from the fixed point), the Picard process converges to a fixed point in 5 iterations, the Mann process converges in 12 iterations, the Ishikawa process converges in 25 iterations, the Noor process converges in 8 iterations, the Agarwal et al. process converges in 8 iterations and the S-iterative process converges in 13 iterations.
- 3.
Taking ${\alpha}_{n}={\beta}_{n}={\gamma}_{n}=\frac{1}{{(1+n)}^{\frac{1}{6}}}$ and ${x}_{0}=0.8$, we obtain that the Mann process converges in 21 iterations, the Ishikawa process converges in 30 iterations, the Noor process converges in 14 iterations, the Agarwal et al. process converges in 12 iterations and the S-iterative process converges in 22 iterations.
5 Conclusions
- 1.
The Picard process does not converge while the decreasing order of convergence rate of other iterative processes is Agarwal et al., Noor, Mann, S and Ishikawa processes.
- 2.
On increasing the value of m, all the above-mentioned processes require more number of iterations to converge except Picard and S-iterative processes which do not converge.
- 3.
For initial guess nearer to the fixed point, Mann and Noor processes show an increase, while Ishikawa, S and Agarwal et al. processes show no change in the number of iterations to converge.
- 4.
The speed of iterative processes depends on ${\alpha}_{n}$ and ${\beta}_{n}$. If we increase the values of ${\alpha}_{n}$ and ${\beta}_{n}$, the fixed point is obtained in more number of iterations for all processes.
Increasing functions
- 1.
The decreasing order of rate of convergence for iterative processes is S, Agarwal et al., Picard, Noor, Ishikawa and Mann processes.
- 2.
For initial guess away from the fixed point, the number of iterations increases in each iterative process except the S-iterative process which shows no change. Hence, the closer the initial guess to the fixed point, the quicker the result is achieved.
- 3.
If we increase the values of ${\alpha}_{n}$ and ${\beta}_{n}$, the fixed point is obtained in less number of iterations for all processes except the S-iterative process which shows no change.
Superlinear functions with multiple roots
- 1.
The decreasing order of rate of convergence for iterative processes is Agarwal et al., Picard, Noor, Ishikawa and Mann processes.
- 2.
For initial guess away from the fixed point, the number of iterations increases in each iterative process. Hence, the closer the initial guess to the fixed point, the quicker the result is achieved.
- 3.
If we increase the values of ${\alpha}_{n}$ and ${\beta}_{n}$, the fixed point is obtained in less number of iterations for Noor, Ishikawa and Mann processes, while Agarwal et al. and S-iterative processes show no change.
Oscillatory functions
- 1.
The Picard process does not converge, Mann and S-iterative processes show equivalence, while the decreasing order of convergence rate of other iterative processes is Agarwal et al., Noor, Mann and Ishikawa processes.
- 2.
For initial guess away from the fixed point, Ishikawa, Agarwal et al. and S-iterative processes show an increase, while Mann and Noor processes show no change in the number of iterations to converge.
- 3.
The speed of iterative processes depends on ${\alpha}_{n}$ and ${\beta}_{n}$. If we increase the values of ${\alpha}_{n}$ and ${\beta}_{n}$, the fixed point is obtained in more number of iterations for all processes.
Declarations
Acknowledgements
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second and fourth authors, therefore, acknowledge with thanks DSR for financial support. The authors thank the referees for their valuable comments and suggestions for the improvement of the manuscript.
Authors’ Affiliations
References
- Hussain N, Rafiq A, Damjanović B, Lazovic R: On rate of convergence of various iterative processes. Fixed Point Theory Appl. 2011., 2011: Article ID 45Google Scholar
- Hussain N, Chugh R, Kumar V, Rafiq A: On the rate of convergence of Kirk type iterative processes. J. Appl. Math. 2012., 2012: Article ID 526503Google Scholar
- Hussain N, Rafiq A, Ciric LB, Al-Mezel S: Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 207Google Scholar
- Hussain N, Rafiq A, Ciric LB: Stability of the Ishikawa iteration process with errors for two strictly hemicontractive operators in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 160Google Scholar
- Khan SH, Rafiq A, Hussain N: A three-step iterative process for solving nonlinear phi-strongly accretive operator equations in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 149Google Scholar
- Khan AR: On modified Noor iterations for asymptotically nonexpansive mappings. Bull. Belg. Math. Soc. Simon Stevin 2010, 17: 127–140.MathSciNetGoogle Scholar
- Khan AR, Khamsi MA, Fukhar-ud-din H: Strong convergence of a general iteration process in CAT(0) spaces. Nonlinear Anal., Theory Methods Appl. 2011, 74: 783–791. 10.1016/j.na.2010.09.029MathSciNetView ArticleGoogle Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–610. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
- Ishikawa S: Fixed point by a new iteration method. Proc. Am. Math. Soc. 1974, 4(1):147–150.MathSciNetView ArticleGoogle Scholar
- Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 2007, 8(1):61–79.MathSciNetGoogle Scholar
- Sahu DR, Petrusel A: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces. Nonlinear Anal. TMA 2011, 74(17):6012–6023. 10.1016/j.na.2011.05.078MathSciNetView ArticleGoogle Scholar
- Noor MA: Three-step iterative algorithms for multivalued quasi variational inclusions. J. Math. Anal. Appl. 2001, 258(2):589–604.View ArticleGoogle Scholar
- Berinde V: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory Appl. 2004, 2: 97–105.MathSciNetView ArticleGoogle Scholar
- Zamfirescu T: Fixed point theorems in metric spaces. Arch. Math. 1992, 23: 292–298.MathSciNetView ArticleGoogle Scholar
- Berinde V: On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Math. Univ. Comen. 2004, LXXIII(1):119–126.MathSciNetGoogle Scholar
- Berinde V: A convergence theorem for some mean value fixed point iteration procedures. Demonstr. Math. 2005, 38(1):177–184.MathSciNetGoogle Scholar
- Babu GVR, Prasad KNVVV: Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory Appl. 2006., 2006: Article ID 49615Google Scholar
- Qing Y, Rhoades BE: Comments on the rate of convergence between Mann and Ishikawa iterations applied to Zamfirescu operators. Fixed Point Theory Appl. 2008., 2008: Article ID 387504. doi:10.1155/2008/387504Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.