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# On rate of convergence of various iterative schemes

Fixed Point Theory and Applications20112011:45

https://doi.org/10.1186/1687-1812-2011-45

• Accepted: 5 September 2011
• Published:

## Abstract

In this note, by taking an counter example, we prove that the iteration process due to Agarwal et al. (J. Nonlinear Convex. Anal. 8 (1), 61-79, 2007) is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

## Keywords

• iteration processes
• Zamfirescu operator

## 1 Introduction

For a nonempty convex subset C of a normed space E and T : CC, (a) the Mann iteration process  is defined by the following sequence{x n }:
$\left\{\begin{array}{c}\hfill {x}_{0}\in C,\hfill \\ \hfill {x}_{n+1}=\left(1-{b}_{n}\right)\phantom{\rule{0.3em}{0ex}}{x}_{n}+{b}_{n}T{x}_{n},\phantom{\rule{2.77695pt}{0ex}}n\ge 0,\hfill \end{array}\right\\phantom{\rule{1em}{0ex}}\left({M}_{n},\right)$
where {b n } is a sequence in [0, 1].
1. (b)
the sequence {x n } defined by
$\left\{\begin{array}{c}\hfill {x}_{0}\in C,\hfill \\ \hfill {y}_{n}=\left(1-{{b}^{\prime }}_{n}\right){x}_{n}+{{b}^{\prime }}_{n}T{x}_{n},\hfill \\ \hfill {x}_{n+1}=\left(1-{b}_{n}\right){x}_{n}+{b}_{n}T{y}_{n},\phantom{\rule{2.77695pt}{0ex}}n\ge 0,\hfill \end{array}\right\\phantom{\rule{1em}{0ex}}\left({I}_{n},\right)$

where {b n }, $\left\{{b}_{n}^{\prime }\right\}$ are sequences in [0, 1] is known as the Ishikawa  iteration process.
1. (c)
the sequence {x n } defined by
$\left\{\begin{array}{c}\hfill {x}_{0}\in C,\hfill \\ \hfill {y}_{n}=\left(1-{{b}^{\prime }}_{n}\right){x}_{n}+{{b}^{\prime }}_{n}T{x}_{n},\hfill \\ \hfill {x}_{n+1}=\left(1-{b}_{n}\right)T{x}_{n}+{b}_{n}T{y}_{n},\phantom{\rule{2.77695pt}{0ex}}n\ge 0,\hfill \end{array}\right\\phantom{\rule{1em}{0ex}}\left(AR{S}_{n},\right)$

where {b n }, $\left\{{b}_{n}^{\prime }\right\}$ are sequences in [0, 1] is known as the Agarwal et al.  iteration process.

Definition 1.  Suppose that {a n } and {b n } are two real convergent sequences with limits a and b, respectively. Then, {a n } is said to converge faster than {b n } if
$\underset{n\to \infty }{lim}\left|\frac{{a}_{n}-a}{{b}_{n}-b}\right|=0.$

Theorem 2.  Let (X, d) be a complete metric space, and T : XX 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 X, at least one of the following is true:

(z 1) d(Tx, Ty) ≤ ad(x, y),

(z 2) d(Tx, Ty) ≤ b [d(x, Tx) + d(y, Ty)],

(z 3) d(Tx, Ty) ≤ c [d(x, Ty) + d(y, Tx)].

Then, T has a unique fixed point p and the Picard iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ defined by
${x}_{n+1}=T{x}_{n},\phantom{\rule{2.77695pt}{0ex}}n=0,1,2,\phantom{\rule{2.77695pt}{0ex}}\dots ,$

converges to p, for any x0 X.

Remark 3. An operator T that satisfies the contraction conditions (z 1) - (z 3) of Theorem 2 will be called a Zamfirescu operator [[4, 6, 7]] and is denoted by Z.

In [6, 7], Berinde introduced a new class of operators on a normed space E satisfying
$||Tx-Ty||\phantom{\rule{0.3em}{0ex}}\le \delta ||x-y||+L||Tx-x||\phantom{\rule{1em}{0ex}}\left(B\right)$

for any x, y E, 0 ≤ δ < 1 and L ≥ 0. He proved that this class is wider than the class of Zamfiresu operators.

The following results are proved in [6, 7].

Theorem 4.  Let C be a nonempty closed convex subset of a normed space E. Let T : CC be an operator satisfying (B). Let {x n } be defined through the iterative process (M n ). If F (T) ≠ Ø and $\sum {b}_{n}=\infty$, then {x n } converges strongly to the unique fixed point of T.

Theorem 5.  Let C be a nonempty closed convex subset of an arbitrary Banach space E and T : CC be an operator satisfying (B). Let {x n } be defined through the iterative process I n and x0 C, where {b n } and $\left\{{b}_{n}^{\prime }\right\}$ are sequences of positive numbers in [0, 1] with {b n } satisfying $\sum {b}_{n}=\infty$. Then {x n } converges strongly to the fixed point of T.

The following theorem was presented in .

Theorem 6. Let C be a closed convex subset of an arbitrary Banach space E. Let the Mann and Ishikawa iteration processes denoted by M n and I n , respectively, with {b n } and $\left\{{b}_{n}^{\prime }\right\}$ be real sequences satisfying (i) 0 ≤ b n , ${b}_{n}^{\prime }\le 1$, and (ii) $\sum {b}_{n}=\infty$. Then, M n and I n converge strongly to the unique fixed point of a Zamfirescu operator T : CC, and moreover, the Mann iteration process converges faster than the Ishikawa iteration process to the fixed point of T.

Remark 7. In , Qing and Rhoades, by taking a counter example, showed that the Ishikawa iteration process is faster than the Mann iteration process for Zamfirescu operators. Thus, Theorem in  and the presentation in  contradict to each other (see also ).

In this note, we establish a general theorem to approximate fixed points of quasi-contractive

operators in a Banach space through the iteration process ARS n , due to Agarwal et al. . Our result generalizes and improves upon, among others, the corresponding results of Babu and Prasad  and Berinde [4, 6, 7].

We also prove that the iteration process ARS n is faster than the Mann iteration process M n and the Ishikawa iteration process I n for Zamfirescu operators.

## 2 Main results

We now prove our main results.

Theorem 8. Let C be a nonempty closed convex subset of an arbitrary Banach space E and T : CC be an operator satisfying (B). Let {x n } be defined through the iterative process ARS n and x0 C, where {b n }, $\left\{{b}_{n}^{\prime }\right\}$ are sequences in [0, 1] satisfying $\sum {b}_{n}=\infty$. Then, {x n } converges strongly to the fixed point of T.

Proof. Assume that F(T) ≠ Ø and w F(T), then using (ARS n ), we have
$\begin{array}{lll}\hfill ||{x}_{n+1}-w||\phantom{\rule{2.77695pt}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}||\left(1-{b}_{n}\right)T{x}_{n}+{b}_{n}T{y}_{n}\phantom{\rule{2.77695pt}{0ex}}-w||\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\phantom{\rule{2.77695pt}{0ex}}||\left(1-{b}_{n}\right)\left(T{x}_{n}\phantom{\rule{2.77695pt}{0ex}}-w\right)+{b}_{n}\left(T{y}_{n}-w\right)||\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \le \left(1-{b}_{n}\right)||T{x}_{n}-w||+{b}_{n}||T{y}_{n}-w||.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$
(2.1)
Now using (B) with x = w, y = x n , and then with x = w, y = y n , we obtain the following two inequalities,
$||T{x}_{n}-w||\phantom{\rule{0.3em}{0ex}}\le \delta ||{x}_{n}-w||,$
(2.2)
and
$||T{y}_{n}-w||\phantom{\rule{0.3em}{0ex}}\le \delta ||{y}_{n}-w||.$
(2.3)
By substituting (2.2) and (2.3) in (2.1), we obtain
$||{x}_{n+1}-w||\phantom{\rule{0.3em}{0ex}}\le \left(1-{b}_{n}\right)\delta ||{x}_{n}-w||+{b}_{n}\delta ||{y}_{n}-w||.$
(2.4)
In a similar fashion, again by using (ARS n ), we can get
$||{y}_{n}-w||\phantom{\rule{0.3em}{0ex}}\le \left(1-\left(1-\delta \right){b}_{n}^{\prime }\right)||{x}_{n}-w||.$
(2.5)
From (2.4) and (2.5), we have
$||{x}_{n+1}-w||\phantom{\rule{0.3em}{0ex}}\le \left[1-\left(1-\delta \right){b}_{n}\left(1+\delta {b}_{n}^{\prime }\right)\right]||{x}_{n}-w||.$
(2.6)
It may be noted that for δ [0, 1) and {η n } [0, 1], the following inequality holds:
$1\le 1+\delta {\eta }_{n}\le 1+\delta .$
(2.7)
From (2.6) and (2.7), we get
$||{x}_{n+1}-w||\phantom{\rule{0.3em}{0ex}}\le \left(1-\left(1-\delta \right){b}_{n}\right)||{x}_{n}-w||.$
(2.8)
By (2.8) we inductively obtain
$||{x}_{n+1}-w||\phantom{\rule{0.3em}{0ex}}\le \prod _{k=0}^{n}\left[1-\delta \left(1-\delta \right){b}_{k}\right]||{x}_{0}-w||,\phantom{\rule{2.77695pt}{0ex}}n=0,1,2,\phantom{\rule{2.77695pt}{0ex}}\dots$
(2.9)
Using the fact that 0 ≤ δ < 1, 0 ≤ b n ≤ 1, and $\sum {b}_{n}=\infty$, it results that
$\underset{n\to \infty }{lim}\prod _{k=0}^{n}\left[1-\delta \left(1-\delta \right){b}_{k}\right]=0,$
which by (2.9) implies
$\underset{n\to \infty }{lim}||{x}_{n+1}-w||\phantom{\rule{0.3em}{0ex}}=0.$

Consequently x n w F and this completes the proof. □

Now by an counter example, we prove that the iteration process ARS n due to Agarwal et al.  is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

Example 9.  Suppose $T:\left[0,1\right]\to \left[0,1\right]:=\frac{1}{2}x$, ${b}_{n}=0={b}_{n}^{\prime }$, n = 1, 2,..., 15. ${b}_{n}=\frac{4}{\sqrt{n}}={b}_{n}^{\prime }$, n ≥ 16.

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.

Proof. Since ${b}_{n}=0={b}_{n}^{\prime }$, n = 1, 2,..., 15, so M n = x0 = I n = ARS n , n = 1, 2,..., 16. Suppose x0 ≠ 0. For M n , I n and ARS n iteration processes, we have
$\begin{array}{lll}\hfill {M}_{n}\phantom{\rule{2.77695pt}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}\left(1-{b}_{n}\right){x}_{n}+{b}_{n}T{x}_{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left(1-\frac{4}{\sqrt{n}}\right){x}_{n}+\frac{4}{\sqrt{n}}\frac{1}{2}{x}_{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left(1-\frac{2}{\sqrt{n}}\right){x}_{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\cdot \cdot \cdot \phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ =\phantom{\rule{2.77695pt}{0ex}}\prod _{i=16}^{n}\left(1-\frac{2}{\sqrt{i}}\right){x}_{0},\phantom{\rule{2em}{0ex}}& \hfill \text{(5)}\\ \hfill \text{(6)}\end{array}$
$\begin{array}{lll}\hfill {I}_{n}\phantom{\rule{2.77695pt}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}\left(1-{b}_{n}\right){x}_{n}+{b}_{n}T\left(\left(1-{{b}^{\prime }}_{n}\right){x}_{n}+{{b}^{\prime }}_{n}T{x}_{n}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left(1-\frac{4}{\sqrt{n}}\right){x}_{n}+\frac{4}{\sqrt{n}}\frac{1}{2}\left(1-\frac{2}{\sqrt{n}}\right){x}_{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left(1-\frac{2}{\sqrt{n}}-\frac{4}{n}\right){x}_{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\cdot \cdot \cdot \phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ =\phantom{\rule{2.77695pt}{0ex}}\prod _{i=16}^{n}\left(1-\frac{2}{\sqrt{i}}-\frac{4}{i}\right){x}_{0},\phantom{\rule{2em}{0ex}}& \hfill \text{(5)}\\ \hfill \text{(6)}\end{array}$
and
$\begin{array}{lll}\hfill AR{S}_{n}\phantom{\rule{2.77695pt}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}\left(1-{b}_{n}\right)T{x}_{n}+{b}_{n}T\left(\left(1-{{b}^{\prime }}_{n}\right){x}_{n}+{{b}^{\prime }}_{n}T{x}_{n}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left(1-\frac{4}{\sqrt{n}}\right)\frac{{x}_{n}}{2}+\frac{4}{\sqrt{n}}\frac{1}{2}\left(1-\frac{2}{\sqrt{n}}\right){x}_{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left(\frac{1}{2}-\frac{4}{n}\right){x}_{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\cdot \cdot \cdot \phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ =\phantom{\rule{2.77695pt}{0ex}}\prod _{i=16}^{n}\left(\frac{1}{2}-\frac{4}{i}\right){x}_{0}.\phantom{\rule{2em}{0ex}}& \hfill \text{(5)}\\ \hfill \text{(6)}\end{array}$
Now consider
$\begin{array}{lll}\hfill \left|\frac{AR{S}_{n}-0}{{M}_{n}-0}\right|& =\left|\frac{\prod _{i=16}^{n}\left(\frac{1}{2}-\frac{4}{i}\right){x}_{0}}{\prod _{i=16}^{n}\left(1-\frac{2}{\sqrt{i}}\right){x}_{0}}\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\left|\frac{\prod _{i=16}^{n}\left(\frac{1}{2}-\frac{4}{i}\right)}{\prod _{i=16}^{n}\left(1-\frac{2}{\sqrt{i}}\right)}\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left|\prod _{i=16}^{n}\left(1-\frac{\frac{1}{2}-\frac{2}{\sqrt{i}}+\frac{4}{i}}{\left(1-\frac{2}{\sqrt{i}}\right)}\right)\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left|\prod _{i=16}^{n}\left(1-\frac{1}{2\sqrt{i}}\frac{i-4\sqrt{i}+8}{\sqrt{i}-2}\right)\right|.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
It is easy to see that
$\begin{array}{lll}\hfill 0\phantom{\rule{2.77695pt}{0ex}}& \le \phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{lim}\prod _{i=16}^{n}\left(1-\frac{1}{2\sqrt{i}}\frac{i-4\sqrt{i}+8}{\sqrt{i}-2}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le \phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{lim}\prod _{i=16}^{n}\left(1-\frac{1}{i}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{lim}\frac{15}{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\phantom{\rule{2.77695pt}{0ex}}0.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
Hence
$\underset{n\to \infty }{lim}\left|\frac{AR{S}_{n}-0}{{M}_{n}-0}\right|=0.$

Thus, the iteration process due to Agarwal et al.  converges faster than the Mann iteration process to the fixed point 0 of T.

Similarly
$\begin{array}{lll}\hfill \left|\frac{AR{S}_{n}-0}{{I}_{n}-0}\right|& =\left|\frac{\prod _{i=16}^{n}\left(\frac{1}{2}-\frac{4}{i}\right){x}_{0}}{\prod _{i=16}^{n}\left(1-\frac{2}{\sqrt{i}}-\frac{4}{i}\right){x}_{0}}\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\left|\frac{\prod _{i=16}^{n}\left(\frac{1}{2}-\frac{4}{i}\right)}{\prod _{i=16}^{n}\left(1-\frac{2}{\sqrt{i}}-\frac{4}{i}\right)}\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left|\prod _{i=16}^{n}\left(1-\frac{\frac{1}{2}-\frac{2}{\sqrt{i}}}{\left(1-\frac{2}{\sqrt{i}}-\frac{4}{i}\right)}\right)\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\phantom{\rule{2.77695pt}{0ex}}\left|\prod _{i=16}^{n}\left(1-\frac{\sqrt{i}}{2}\frac{i-4}{i-2\sqrt{i}-4}\right)\right|,\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
with
$\begin{array}{lll}\hfill 0\phantom{\rule{2.77695pt}{0ex}}& \le \phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{lim}\prod _{i=16}^{n}\left(1-\frac{\sqrt{i}}{2}\frac{\sqrt{i}-4}{i-2\sqrt{i}-4}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le \phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{lim}\prod _{i=16}^{n}\left(1-\frac{1}{i}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty }{lim}\frac{15}{n}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =0,\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
implies
$\underset{n\to \infty }{lim}\left|\frac{AR{S}_{n}-0}{{I}_{n}-0}\right|=0.$

Thus, the iteration process due to Agarwal et al.  converges faster than the Ishikawa iteration process to the fixed point 0 of T. □

## Declarations

### Acknowledgements

Nawab Hussain gratefully acknowledges the support provided by King Abdulaziz University during this research. Boško Damjanović and Rade Lazović are thankful to the Ministry of Science, Technology and Development, Republic of Serbia.

## Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan
(3)
Faculty of Agriculture, Nemanjina 6, Belgrade, Serbia
(4)
Faculty of Organizational Science, Jove Ili ća 154, Belgrade, Serbia

## References 