Open Access

A comparative study on the convergence rate of some iteration methods involving contractive mappings

Fixed Point Theory and Applications20152015:234

https://doi.org/10.1186/s13663-015-0490-3

Received: 23 October 2015

Accepted: 20 December 2015

Published: 30 December 2015

Abstract

We compare the rate of convergence for some iteration methods for contractions. We conclude that the coefficients involved in these methods have an important role to play in determining the speed of the convergence. By using Matlab software, we provide numerical examples to illustrate the results. Also, we compare mathematical and computer-calculating insights in the examples to explain the reason of the existence of the old difference between the points of view.

Keywords

contractive map fixed point iteration method rate of convergence

MSC

47H09 47H10

1 Introduction

Iteration schemes for numerical reckoning fixed points of various classes of nonlinear operators are available in the literature. The class of contractive mappings via iteration methods is extensively studied in this regard. In 1952, Plunkett published a paper on the rate of convergence for relaxation methods [1]. In 1953, Bowden presented a talk in a symposium on digital computing machines entitled ‘Faster than thought’ [2]. Later, this basic idea has been used in engineering, statistics, numerical analysis, approximation theory, and physics for many years (see, for example, [39] and [10]). In 1991, Argyros published a paper about iterations converging faster than Newton’s method to the solutions of nonlinear equations in Banach spaces [11, 12]. In 1997, Lucet presented a method faster than the fast Legendre transform [13]. In 2004, Berinde used the notion of rate of convergence for iterations method and showed that the Picard iteration converges faster than the Mann iteration for a class of quasi-contractive operators [14]. Later, he provided some results in this area [15, 16]. In 2006, Babu and Vara Prasad showed that the Mann iteration converges faster than the Ishikawa iteration for the class of Zamfirescu operators [17]. In 2007, Popescu showed that the Picard iteration converges faster than the Mann iteration for the class of quasi-contractive operators [18]. Recently, there have been published some papers about introducing some new iterations and comparing of the rates of convergence for some iteration methods (see, for example, [1922] and [23]).

In this paper, we compare the rates of convergence of some iteration methods for contractions and show that the involved coefficients in such methods have an important role to play in determining the rate of convergence. During the preparation of this work, we found that the efficiency of coefficients had been considered in [24] and [25]. But we obtained our results independently, before reading these works, and one can see it by comparing our results and those ones.

2 Preliminaries

As we know, the Picard iteration has been extensively used in many works from different points of view. Let \((X, d)\) be a metric space, \(x_{0}\in X\), and \(T\colon X\to X\) a selfmap. The Picard iteration is defined by
$$ x_{n+1}=Tx_{n} $$
for all \(n\geq0\). Let \(\{\alpha_{n}\}_{n\geq0}\), \(\{\beta_{n}\}_{n\geq 0}\), and \(\{\gamma_{n}\}_{n\geq0}\) be sequences in \([0, 1]\). Then the Mann iteration method is defined by
$$ x_{n+1}=\alpha_{n} x_{n} + (1- \alpha_{n}) Tx_{n} $$
(2.1)
for all \(n\geq0\) (for more information, see [26]). Also, the Ishikawa iteration method is defined by
$$ \begin{aligned} &x_{n+1} =(1- \alpha_{n})x_{n}+\alpha_{n}Ty_{n}, \\ &y_{n} = (1-\beta_{n})x_{n} +\beta_{n}Tx_{n} \end{aligned} $$
(2.2)
for all \(n\geq0\) (for more information, see [27]). The Noor iteration method is defined by
$$\begin{aligned}& x_{n+1} =(1-\alpha_{n})x_{n} + \alpha_{n} Ty_{n}, \\& y_{n} =(1-\beta_{n})x_{n} + \beta_{n} Tz_{n}, \\& z_{n} =(1-\gamma_{n})x_{n} +\gamma_{n} Tx_{n} \end{aligned}$$
(2.3)
for all \(n\geq0\) (for more information, see [28]). In 2007, Agarwal et al. defined their new iteration methods by
$$ \begin{aligned} &x_{n+1} = (1 - \alpha_{n})Tx_{n} + \alpha_{n}Ty_{n}, \\ &y_{n} = (1 - \beta_{n})x_{n} + \beta_{n}Tx_{n} \end{aligned} $$
(2.4)
for all \(n\geq0\) (for more information, see [29]). In 2014, Abbas et al. defined their new iteration methods by
$$\begin{aligned}& x_{n+1} =(1 - \alpha_{n})Ty_{n} + \alpha_{n}Tz_{n} , \\& y_{n} = (1-\beta_{n})Tx_{n} +\beta_{n}Tz_{n}, \\& z_{n} = (1-\gamma_{n})x_{n} + \gamma_{n}Tx_{n} \end{aligned}$$
(2.5)
for all \(n\geq0\) (for more information, see [30]). In 2014, Thakur et al. defined their new iteration methods by
$$\begin{aligned}& x_{n+1} =(1 - \alpha_{n})Tx_{n} + \alpha_{n}Ty_{n}, \\& y_{n} = (1-\beta_{n})z_{n} +\beta_{n}Tz_{n}, \\& z_{n} = (1-\gamma_{n})x_{n} + \gamma_{n}Tx_{n} \end{aligned}$$
(2.6)
for all \(n\geq0\) (for more information, see [23]). Also, the Picard S-iteration was defined by
$$\begin{aligned}& x_{n+1} = Ty_{n}, \\& y_{n} = (1-\beta_{n})Tx_{n} +\beta_{n}Tz_{n}, \\& z_{n} = (1-\gamma_{n})x_{n} + \gamma_{n}Tx_{n} \end{aligned}$$
(2.7)
for all \(n\geq0\) (for more information, see [20] and [22]).

3 Self-comparing of iteration methods

Now, we are ready to provide our main results for contractive maps. In this respect, we assume that \((X, \|\cdot\|)\) is a normed space, \(x_{0}\in X\), \(T\colon X\to X\) is a selfmap and \(\{\alpha_{n}\}_{n\geq 0}\), \(\{\beta_{n}\}_{n\geq0}\) and \(\{\gamma_{n}\}_{n\geq0}\) are sequences in \((0, 1)\).

The Mann iteration is given by \(x_{n+1}= (1-\alpha_{n})x_{n}+\alpha_{n} Tx_{n}\) for all \(n\geq0\).

Note that we can rewrite it as \(x_{n+1}= \alpha_{n} x_{n}+(1-\alpha_{n}) Tx_{n}\) for all \(n\geq0\).

We call these cases the first and second forms of the Mann iteration method.

In the next result we show that choosing a type of sequence \(\{\alpha_{n}\}_{n\geq0}\) in the Mann iteration has a notable role to play in the rate of convergence of the sequence \(\{x_{n}\}_{n\geq0}\).

Let \(\{u_{n}\}_{n\geq0}\) and \(\{v_{n}\}_{n\geq0}\) be two fixed point iteration procedures that converge to the same fixed point p and \(\|u_{n}-p\|\leq a_{n}\) and \(\|v_{n}-p\|\leq b_{n}\) for all \(n\geq0\). If the sequences \(\{a_{n}\}_{n\geq0}\) and \(\{b_{n}\}_{n\geq0}\) converge to a and b, respectively, and \(\lim_{n\to\infty}\frac{\|a_{n}-a\|}{\|b_{n}-b\|}=0\), then we say that \(\{u_{n}\}_{n\geq0}\) converges faster than \(\{v_{n}\}_{n\geq0}\) to p (see [14] and [23]).

Proposition 3.1

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\) and p a fixed point of T. Consider the first case for Mann iteration. If the coefficients of \(Tx_{n}\) are greater than the coefficients of \(x_{n}\), that is, \(1-\alpha_{n} < \alpha_{n}\) for all \(n\geq0\) or equivalently \(\{\alpha_{n}\}_{n\geq0}\) is a sequence in \((\frac{1}{2}, 1)\), then the Mann iteration converges faster than the Mann iteration which the coefficients of \(x_{n}\) are greater than the coefficients of \(Tx_{n}\).

Proof

Let \(\{x_{n}\}\) be the sequence in the Mann iteration which the coefficients of \(Tx_{n}\) are greater than the coefficients of \(x_{n}\), that is,
$$ x_{n+1}=(1-\alpha_{n}) x_{n} + \alpha_{n} Tx_{n} $$
(3.1)
for all n. In this case, we have
$$\begin{aligned} \begin{aligned} \Vert x_{n+1}-p\Vert &=\bigl\Vert (1-\alpha_{n})x_{n}+ \alpha_{n} Tx_{n}-p\bigr\Vert \leq(1-\alpha_{n}) \Vert x_{n}-p\Vert +\alpha_{n}\Vert Tx_{n}-p \Vert \\ &\leq \bigl(1-\alpha_{n}(1-k) \bigr) \Vert x_{n}-p\Vert \end{aligned} \end{aligned}$$
for all n. Since \(\alpha_{n} \in(\frac{1}{2}, 1)\), \(1-\alpha_{n}(1-k) < 1-\frac{1}{2}(1-k)\). Put \(a_{n} = (1-\frac{1}{2}(1-k) )^{n} \Vert x_{1}-p\Vert \) for all n. Now, let \(\{x_{n}\}\) be the sequence in the Mann iteration of which the coefficients of \(x_{n}\) are greater than the coefficients of \(Tx_{n}\). In this case, we have
$$\begin{aligned} \Vert x_{n+1}-p\Vert =&\bigl\Vert \alpha_{n} x_{n}+(1- \alpha _{n})Tx_{n}-p\bigr\Vert \leq \alpha_{n} \Vert x_{n}-p\Vert +(1-\alpha_{n}) \Vert Tx_{n}-p\Vert \\ \leq& \bigl(1-(1-\alpha_{n}) (1-k) \bigr) \Vert x_{n}-p \Vert \end{aligned}$$
for all n. Since \(1-\alpha_{n} < \alpha_{n}\) for all \(n\geq0\), we get \(1-(1-\alpha_{n})(1-k) < 1\) for all \(n\geq0\). Put \(b_{n}=\Vert x_{1}-p\Vert \) for all n. Note that \(\lim\frac{a_{n}}{b_{n}}=\lim\frac{ (1-\frac{1}{2}(1-k) )^{n} \Vert x_{1}-p\Vert }{ \Vert x_{1}-p\Vert }=0\). This completes the proof. □

Note that we can use \(1-\alpha_{n} < \alpha_{n}\), for n large enough, instead of the condition \(1-\alpha_{n} < \alpha_{n}\), for all \(n\geq0\). One can use similar conditions instead of the conditions which we will use in our results.

As we know, we can consider four cases for writing the Ishikawa iteration method. In the next result, we indicate each case by different enumeration. Similar to the last result, we want to compare the Ishikawa iteration method with itself in the four possible cases. Again, we show that the coefficient sequences \(\{\alpha_{n}\}_{n\geq 0}\) and \(\{\beta_{n}\}_{n\geq0}\) have effective roles to play in the rate of convergence of the sequence \(\{x_{n}\}_{n\geq0}\) in the Ishikawa iteration method.

Proposition 3.2

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C\) a contraction with constant \(k\in(0, 1)\), and p a fixed point of T. Consider the following cases of the Ishikawa iteration method:
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} x_{n+1} =(1 - \alpha_{n})x_{n} + \alpha_{n}Ty_{n}, \\ y_{n} = (1-\beta_{n})x_{n} +\beta_{n}Tx_{n}, \end{array}\displaystyle \right . \end{aligned}$$
(3.2)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} x_{n+1} =\alpha_{n} x_{n} + (1-\alpha_{n})Ty_{n} , \\ y_{n} = \beta_{n} x_{n} +(1-\beta_{n})Tx_{n}, \end{array}\displaystyle \right . \end{aligned}$$
(3.3)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} x_{n+1} = \alpha_{n}x_{n} + (1-\alpha_{n})Ty_{n} , \\ y_{n} = (1-\beta_{n})x_{n} +\beta_{n}Tx_{n}, \end{array}\displaystyle \right . \end{aligned}$$
(3.4)
and
$$ \left \{ \textstyle\begin{array}{l} x_{n+1} =(1 - \alpha_{n})x_{n} + \alpha_{n}Ty_{n} , \\ y_{n} = \beta_{n} x_{n} +(1-\beta_{n})Tx_{n} \end{array}\displaystyle \right . $$
(3.5)
for all \(n\geq0\). If \(1-\alpha_{n} < \alpha_{n}\) and \(1-\beta_{n} < \beta_{n}\) for all \(n\geq0\), then the case (3.2) converges faster than the others. In fact, the Ishikawa iteration method is faster whenever the coefficients of \(Ty_{n}\) and \(Tx_{n}\) simultaneously are greater than the related coefficients of \(x_{n}\) for all \(n\geq0\).

Proof

Let \(\{x_{n}\}_{n\geq0}\) be the sequence in the case (3.2). Then we have
$$\begin{aligned} \Vert y_{n}-p\Vert &= \bigl\Vert (1-\beta_{n}) x_{n} + \beta_{n}Tx_{n}-p\bigr\Vert \\ &\leq(1 -\beta_{n})\Vert x_{n}-p\Vert + \beta_{n}\Vert Tx_{n}-p\Vert \\ & \leq \bigl((1-\beta_{n}) + \beta_{n} k \bigr) \Vert x_{n} -p \Vert \end{aligned}$$
and
$$\begin{aligned} \Vert x_{n+1}-p\Vert &= \bigl\Vert (1-\alpha_{n}) x_{n} + \alpha_{n}Ty_{n}-p\bigr\Vert \\ &\leq(1-\alpha_{n})\Vert x_{n}-p\Vert + \alpha_{n} \Vert Ty_{n}-p\Vert \\ &\leq(1-\alpha_{n})\Vert x_{n}-p\Vert + k \alpha_{n} \Vert y_{n}-p\Vert \\ & \leq \bigl(1-\alpha_{n} + k\alpha_{n}\bigl[(1- \beta_{n}) +\beta_{n} k\bigr] \bigr) \Vert x_{n} -p\Vert \\ & \leq \bigl(1-\alpha_{n} + \alpha_{n} k- \alpha_{n}\beta_{n} k +\alpha _{n} \beta_{n} k^{2} \bigr) \Vert x_{n} -p\Vert \\ & \leq \bigl(1 - \alpha_{n}(1-k)-\alpha_{n} \beta_{n} k (1 - k) \bigr) \Vert x_{n} -p\Vert \end{aligned}$$
for all \(n\geq0\). Since \(\alpha_{n}, \beta_{n} \in(\frac{1}{2}, 1)\), \(1-\alpha_{n}(1-k)-\alpha_{n}\beta_{n} k (1-k)< 1-\frac{1}{2}(1-k)-\frac {1}{4}k(1-k)\) for all \(n\geq0\). Put \(a_{n}= (1-\frac {1}{2}(1-k)-\frac{1}{4}k(1-k) ) ^{n} \Vert x_{1}-p\Vert \) for all \(n\geq0\). If \(\{x_{n}\}_{n\geq0}\) is the sequence in the case (3.3), then we get
$$\begin{aligned} \Vert y_{n}-p\Vert &= \bigl\Vert \beta_{n} x_{n} + (1-\beta _{n})Tx_{n}-p\bigr\Vert \\ &\leq\beta_{n}\Vert x_{n}-p\Vert + (1- \beta_{n})\Vert Tx_{n}-p\Vert \\ & \leq \bigl(1-(1-\beta_{n}) (1-k) \bigr) \Vert x_{n} -p \Vert \end{aligned}$$
and
$$\begin{aligned} \Vert x_{n+1}-p\Vert &= \bigl\Vert \alpha_{n} x_{n} + (1-\alpha_{n})Ty_{n}-p\bigr\Vert \\ &\leq\alpha_{n}\Vert x_{n}-p\Vert + (1- \alpha_{n}) \Vert Ty_{n}-p\Vert \\ &\leq\alpha_{n}\Vert x_{n}-p\Vert + k (1- \alpha_{n}) \Vert y_{n}-p\Vert \\ &\leq \bigl(\alpha_{n}+k(1-\alpha_{n}) \bigl(1-(1- \beta_{n}) (1-k)\bigr)\bigr) \Vert x_{n}-p\Vert \\ &= \bigl(\alpha_{n}+(1-\alpha_{n})k-k(1- \alpha_{n}) (1-\beta_{n}) (1-k) \bigr) \Vert x_{n}-p\Vert \\ &= \bigl(1-(1-\alpha_{n}) (1-k)-(1-\alpha_{n}) (1- \beta_{n})k(1-k) \bigr) \Vert x_{n}-p\Vert \end{aligned}$$
for all \(n\geq0\). Since \(\alpha_{n}, \beta_{n} \in(\frac{1}{2}, 1)\), \(1-(1-\alpha_{n}) (1-k)-(1-\alpha_{n})(1-\beta_{n})(1-k)< 1\) for all \(n\geq0\). Put \(b_{n} =\Vert x_{1}-p\Vert \) for all \(n\geq0\). Since
$$1-\frac{1}{2}(1-k)-\frac{1}{4}k(k-1) < 1+\frac{1}{2}k(1-k), $$
we get \(\lim\frac{a_{n}}{b_{n}}=\lim\frac{ (1-\frac{1}{2}(1-k)-\frac {1}{4}k(1-k) )^{n} \Vert x_{1}-p\Vert }{\Vert x_{1}-p\Vert }=0\) and so the iteration (3.2) converges faster than the case (3.3). Now, let \(\{x_{n}\}_{n\geq0}\) be the sequence in the case (3.4). Then
$$\begin{aligned} \Vert y_{n}-p\Vert &= \bigl\Vert \beta x_{n} +(1- \beta _{n})Tx_{n}-p\bigr\Vert \\ &\leq\beta_{n}\Vert x_{n}-p\Vert + (1 - \beta_{n}) \Vert Tx_{n}-p\Vert \\ &\leq \bigl(\beta_{n}+k(1-\beta_{n}) \bigr) \Vert x_{n} -p\Vert \\ &= \bigl(1-(1-\beta_{n}) (1-k) \bigr) \Vert x_{n} -p \Vert \end{aligned}$$
and
$$\begin{aligned} \Vert x_{n+1}-p\Vert &= \bigl\Vert (1-\alpha_{n}) x_{n} + \alpha_{n}Ty_{n}-p\bigr\Vert \\ &\leq(1-\alpha_{n})\Vert x_{n}-p\Vert + \alpha_{n} \Vert Ty_{n}-p\Vert \\ &\leq \bigl(1-\alpha_{n} + k\alpha_{n}\bigl[ \bigl(1-(1- \beta_{n}) (1-k) \bigr)\bigr] \bigr) \Vert x_{n} -p\Vert \\ &= \bigl(1-\alpha_{n} + k\alpha_{n}-\alpha_{n}(1- \beta_{n})k(1-k) \bigr) \Vert x_{n}-p\Vert \\ &= \bigl(1-\alpha_{n}(1-k)-\alpha_{n}(1- \beta_{n})k(1-k) \bigr) \Vert x_{n}-p\Vert \end{aligned}$$
for all \(n\geq0\). Since \(\alpha_{n}, \beta_{n}\in(\frac{1}{2}, 1)\) for all \(n\geq0\), \(-(1-k)<-\alpha_{n}(1-k)< -\frac{1}{2}(1-k)\) and \(\frac{-1}{2}k(1-k)<-\alpha_{n}(1-\beta_{n})k(1-k)<0\) for all n. Hence,
$$1-\alpha_{n}(1-k)-\alpha_{n}(1-\beta_{n})k(1-k)< 1- \frac{1}{2}(1-k) $$
for all \(n\geq0\). Put \(c_{n} = (1-\frac{1}{2}(1-k) )^{n} \Vert x_{1}-p\Vert \) for all \(n\geq0\). Thus, we obtain
$$ \lim\frac{a_{n}}{c_{n}}=\lim\frac{ (1-\frac{1}{2}(1-k)-\frac {1}{4}k(1-k) )^{n} \Vert x_{1} -p\Vert }{ (1-\frac{1}{2}(1-k) )^{n} \Vert x_{1} -p\Vert }=0 $$
and so the iteration (3.2) converges faster than the case (3.4). Now, let \(\{x_{n}\}_{n\geq0}\) be the sequence in the case (3.5). Then we have
$$\begin{aligned} \Vert y_{n}-p\Vert &= \bigl\Vert (1-\beta)x_{n} + \beta _{n} Tx_{n}-p\bigr\Vert \\ &\leq(1-\beta_{n})\Vert x_{n}-p\Vert + \beta_{n}\Vert Tx_{n}-p\Vert \\ & \leq \bigl(1-\beta_{n}(1-k) \bigr) \Vert x_{n} -p\Vert \end{aligned}$$
and
$$\begin{aligned} \Vert x_{n+1}-p\Vert &= \bigl\Vert \alpha_{n} x_{n} + (1-\alpha_{n})Ty_{n}\bigr\Vert \\ &\leq\alpha_{n}\Vert x_{n}-p\Vert + (1- \alpha_{n}) \Vert Ty_{n}-p\Vert \\ &\leq\alpha_{n}\Vert x_{n}-p\Vert + k (1- \alpha_{n}) \Vert y_{n}-p\Vert \\ &\leq \bigl(\alpha_{n} + k(1-\alpha_{n})\bigl[1- \beta_{n}(1-k)\bigr] \bigr) \Vert x_{n} -p\Vert \\ &\leq \bigl(\alpha_{n} + k(1-\alpha_{n})-(1- \alpha_{n})\beta_{n}k(1-k) \bigr) \Vert x_{n} -p \Vert \\ &\leq \bigl(1-(1-\alpha_{n} ) + k(1-\alpha_{n})-(1- \alpha_{n})\beta _{n}k(1-k) \bigr) \Vert x_{n} -p \Vert \\ &\leq \bigl(1 - (1-\alpha_{n}) (1-k)-(1-\alpha_{n}) \beta_{n}k(1-k) \bigr) \Vert x_{n} -p\Vert \end{aligned}$$
for all \(n\geq0\). Since \(\alpha_{n}, \beta_{n}\in(\frac{1}{2}, 1)\) for all n, \(-(1-k^{2})<-\alpha_{n}(1-k^{2})< -\frac{1}{2}(1-k^{2})\), and \(-\frac{1}{2}k(1-k)<-(1-\alpha_{n})\beta_{n}k(1-k)<0\) and so
$$1-\alpha_{n}(1-k)-(1-\alpha_{n})\beta_{n}k(1-k)< 1- \frac{1}{2}(1-k) $$
for all \(n\geq0\). Put \(d_{n} = (1-\frac{1}{2}(1-k) )^{n} \Vert x_{1}-p \Vert \) for all \(n\geq0\). Then we have
$$ \lim\frac{a_{n}}{d_{n}}=\lim\frac{ (1-\frac{1}{2}(1-k)-\frac {1}{4}k(1-k) )^{n} \Vert x_{1}-p\Vert }{ (1-\frac {1}{2}(1-k) )^{n} \Vert x_{1}-p\Vert }=0 $$
and so the iteration (3.2) converges faster than the case (3.5). □

By using a similar condition, one can show that the iteration (3.5) is faster than the case (3.3).

Now consider eight cases for writing the Noor iteration method. By using a condition, we show that the coefficient sequences \(\{\alpha_{n}\}_{n\geq0}\), \(\{\beta_{n}\}_{n\geq0}\), and \(\{\gamma_{n}\}_{n\geq0}\) have effective roles to play in the rate of convergence of the sequence \(\{x_{n}\}_{n\geq0}\) in the Noor iteration method. We enumerate the cases of the Noor iteration method during the proof of our next result.

Theorem 3.1

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\) and p a fixed point of T. Consider the case (2.3) of the Noor iteration method
$$ \left \{ \textstyle\begin{array}{l} x_{n+1} =(1-\alpha_{n})x_{n}+\alpha_{n} Ty_{n}, \\ y_{n} = (1-\beta_{n})x_{n} +\beta_{n}Tz_{n}, \\ z_{n} = (1-\gamma_{n})x_{n} + \gamma_{n}Tx_{n} \end{array}\displaystyle \right . $$
for all \(n\geq0\). If \(1-\alpha_{n} <\alpha_{n}\), \(1-\beta_{n} <\beta_{n}\), and \(1-\gamma_{n}<\gamma_{n}\) for all \(n\geq0\), then the iteration (2.3) is faster than the other possible cases.

Proof

First, we compare the case (2.3) with the following Noor iteration case:
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} = (1-\alpha_{n})u_{n}+\alpha_{n} Tv_{n}, \\ v_{n} = (1-\beta_{n})u_{n} +\beta_{n}Tw_{n}, \\ w_{n} = \gamma_{n}u_{n} +(1- \gamma_{n}) Tu_{n} \end{array}\displaystyle \right . $$
(3.6)
for all \(n\geq0\). Note that
$$\begin{aligned} \Vert z_{n}-p\Vert &= \bigl\Vert (1-\gamma_{n})x_{n} +\gamma _{n}Tx_{n} -p\bigr\Vert \\ &\leq(1-\gamma_{n})\Vert x_{n}-p\Vert + k \gamma_{n} \Vert x_{n}-p\Vert \\ &= \bigl(1-(1-k)\gamma_{n}\bigr) \Vert x_{n}-p\Vert \end{aligned}$$
and
$$\begin{aligned} \Vert y_{n}-p\Vert &= \bigl\Vert (1-\beta_{n})x_{n} +\beta _{n}Tz_{n} -p\bigr\Vert \\ &\leq(1-\beta_{n})\Vert x_{n}-p\Vert + k \beta_{n} \Vert z_{n}-p\Vert \\ &\leq(1-\beta_{n}) +k \beta_{n} \bigl(\bigl(1-(1-k) \gamma_{n}\bigr)\bigr) \Vert x_{n}-p\Vert \\ &\leq\bigl[1-\beta_{n}(1-k)-\beta_{n}\gamma_{n} k(1-k)\bigr]\Vert x_{n}-p \Vert \end{aligned}$$
for all \(n\geq0\). Also, we have
$$\begin{aligned} \Vert x_{n+1}-p\Vert &= \bigl\Vert (1-\alpha _{n})x_{n}+ \alpha_{n}Ty_{n} -p\bigr\Vert \\ &\leq(1-\alpha_{n}) \Vert x_{n}-p\Vert + k \alpha_{n} \Vert y_{n}-p\Vert \\ &\leq(1-\alpha_{n} ) \Vert x_{n}-p\Vert + k \alpha_{n} \bigl[1-\beta_{n}(1-k)-\beta_{n} \gamma_{n} k(1-k)\bigr] \Vert x_{n}-p \Vert \\ &\leq\bigl(1-\alpha_{n} + k\alpha_{n} \bigl(1- \beta_{n}(1-k)-\beta_{n}\gamma_{n} k(1-k)\bigr) \bigr)\Vert x_{n}-p\Vert \\ &\leq\bigl(1-\alpha_{n} + k\alpha_{n} -k(1-k) \beta_{n}\alpha_{n}-\alpha_{n}\beta _{n} \gamma_{n} k^{2} (1-k)\bigr) \Vert x_{n}-p\Vert \\ &\leq\bigl(1 - (1-k)\alpha_{n} -k(1-k)\beta_{n} \alpha_{n}-\alpha_{n}\beta _{n}\gamma_{n} k^{2} (1-k)\bigr) \Vert x_{n}-p\Vert \end{aligned}$$
for all \(n\geq0\). Since \(\alpha_{n}, \beta_{n}, \gamma_{n}\in(\frac {1}{2}, 1)\) for all n, \(-(1-k^{2})< -\alpha_{n}(1-k^{2})< -\frac{1}{2}(1 -k^{2})\), \(-k(1-k)<-\alpha_{n}\beta_{n}k(1-k)< -\frac{1}{4}k(1-k)\), and
$$-k^{2}(1-k) < -\alpha_{n}\beta_{n} \gamma_{n} k^{2} (1-k) < -\frac{1}{8}k^{2}(1-k) $$
for all n. This implies that
$$1 - (1-k)\alpha_{n} -k(1-k)\beta_{n}\alpha_{n}- \alpha_{n}\beta_{n}\gamma_{n} k^{2} (1-k)< 1-\frac{1}{2}(1-k)-\frac{1}{4}k(1-k)-\frac{1}{8}k^{2}(1-k) $$
for all n. Put \(a_{n} = (1-\frac{1}{2}(1-k) -\frac {1}{8}k^{2}(1-k))^{n}\Vert x_{1}-p\Vert \) for all \(n\geq0\). Now for the sequences \(\{u_{n}\}_{n\geq0}\) with \(u_{1}=x_{1}\) and \(\{v_{n}\} _{n\geq0}\) in (3.6), we have
$$\begin{aligned} \Vert w_{n}-p\Vert &= \bigl\Vert \gamma_{n} u_{n} +(1-\gamma _{n})Tu_{n} -p\bigr\Vert \\ &\leq\gamma_{n}\Vert u_{n}-p\Vert + k(1- \gamma_{n} ) \Vert u_{n}-p\Vert \\ &= \bigl(1-(1-\gamma_{n}) (1-k)\bigr) \Vert u_{n}-p\Vert \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} \Vert v_{n}-p\Vert &= \bigl\Vert (1-\beta_{n})u_{n} +\beta _{n}Tw_{n} -p\bigr\Vert \\ &\leq(1-\beta_{n})\Vert u_{n}-p\Vert + k \beta_{n} \Vert w_{n}-p\Vert \\ &\leq(1-\beta_{n}) + k\beta_{n} \bigl(1-(1- \gamma_{n}) (1-k)\bigr) \Vert u_{n}-p\Vert \\ &\leq\bigl(1-\beta_{n} + k\beta_{n} -\beta_{n}(1- \gamma_{n})k(1-k)\bigr) \Vert u_{n}-p\Vert \\ & \leq\bigl(1-\beta_{n} (1-k) -\beta_{n}(1- \gamma_{n})k(1-k)\bigr) \Vert u_{n}-p\Vert \end{aligned} \end{aligned}$$
for all \(n\geq0\). Hence,
$$\begin{aligned} \Vert u_{n+1}-p\Vert &= \bigl\Vert (1-\alpha_{n})u_{n} +\alpha_{n} Tv_{n} -p\bigr\Vert \\ &\leq(1 - \alpha_{n}) \Vert u_{n}-p\Vert + k \alpha_{n} \Vert v_{n}-p\Vert \\ &\leq(1-\alpha_{n}) \Vert u_{n}-p\Vert + k \alpha_{n} \bigl(1-\beta_{n} (1-k) -\beta_{n}(1- \gamma_{n})k(1-k)\bigr)\Vert u_{n}-p \Vert \\ &\leq\bigl( (1-\alpha_{n}) + k\alpha_{n} - \alpha_{n}\beta_{n}k(1-k) - \alpha \beta_{n}(1- \gamma_{n}) K^{2}(1-k) \bigr)\Vert u_{n}-p\Vert \\ &\leq\bigl( 1-\alpha_{n} (1-k) -\alpha_{n} \beta_{n}k(1-k) - \alpha\beta _{n}(1-\gamma_{n}) K^{2}(1-k) \bigr)\Vert u_{n}-p\Vert \end{aligned}$$
for all n. Since \(\alpha_{n}, \beta_{n}, \gamma_{n}\in(\frac{1}{2}, 1)\) for all n, \(-k(1-k)<-\alpha_{n}\beta_{n}k(1-k)< -\frac{1}{4}k (1-k)\) and \(\frac{1}{2}k^{2}(1-k)< -\alpha_{n}\beta_{n}(1-\gamma _{n})k^{2}(1-k) < 0 \) for all n. Hence,
$$1-\alpha_{n} (1-k) -\alpha_{n}\beta_{n}k(1-k) - \alpha\beta_{n}(1-\gamma _{n}) k^{2}(1-k)< 1- \frac{1}{2}(1-k) -\frac{1}{4}k(1-k) $$
for all n. Put \(b_{n}=(1-\frac{1}{2}(1-k) -\frac{1}{4}k(1-k))^{n}\| u_{1}-p\|\) for all \(n\geq0\). Then we have
$$\begin{aligned} \lim_{n\to\infty} \frac{a_{n}}{b_{n}}= \frac{(1-\frac {1}{2}(1-k)-\frac{1}{4}k(1-k)-\frac{1}{8}k^{2}(1-k))^{n}\Vert x_{1}-p\Vert }{ {(1-\frac{1}{2}(1-k) -\frac{1}{4}k(1-k))}^{n} \Vert u_{1} -p\Vert }=0. \end{aligned}$$
Thus, \(\{x_{n}\}_{n\geq0}\) converges faster than the sequence \(\{u_{n}\} _{n\geq0}\). Now, we compare the case (2.3) with the following Noor iteration case:
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n} )u_{n} + \alpha_{n}Tv_{n}, \\ v_{n} = \beta_{n} u_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.7)
for all \(n\geq0\). Note that
$$\begin{aligned} \Vert w_{n}-p\Vert &= \bigl\Vert (1-\gamma_{n}) u_{n} +\gamma _{n}Tu_{n} -p\bigr\Vert \\ &\leq(1-\gamma_{n})\Vert u_{n}-p\Vert + k \gamma_{n} \Vert u_{n}-p\Vert \\ &= \bigl(1-(1 - k)\gamma_{n}\bigr) \Vert u_{n}-p\Vert \end{aligned}$$
and
$$\begin{aligned} \Vert v_{n}-p\Vert &= \bigl\Vert \beta_{n}u_{n} +(1-\beta _{n})Tw_{n} -p\bigr\Vert \\ &\leq\beta_{n}\Vert u_{n}-p\Vert + k(1- \beta_{n}) \Vert w_{n}-p\Vert \\ &\leq\bigl(\beta_{n} + k(1-\beta_{n} )-\beta_{n} \gamma_{n} k(1-k)\bigr) \Vert u_{n}-p\Vert \\ &\leq\bigl(1-(1-k) (1-\beta_{n} )-\beta_{n} \gamma_{n} k(1-k)\bigr) \Vert u_{n}-p\Vert \end{aligned}$$
for all \(n\geq0\). Hence,
$$\begin{aligned} \Vert u_{n+1}-p\Vert &= \bigl\Vert (1-\alpha_{n})u_{n} +\alpha_{n} Tv_{n} -p\bigr\Vert \\ &\leq(1 - \alpha_{n}) \Vert u_{n}-p\Vert + k \alpha_{n} \Vert w_{n}-p\Vert \\ &\leq(1-\alpha_{n}) \Vert u_{n}-p\Vert + k \alpha_{n} \bigl(1-(1-k) (1-\beta_{n} )-\beta_{n} \gamma_{n} k(1-k)\bigr) \Vert u_{n}-p \Vert \\ &\leq\bigl((1-\alpha_{n}) + k\alpha_{n}-k(1-k) \alpha_{n}(1-\beta_{n} )-\alpha _{n} \beta_{n}\gamma_{n} k^{2}(1-k)\bigr) \Vert u_{n}-p\Vert \\ &\leq\bigl(1-(1-k)\alpha_{n} -\alpha_{n}(1- \beta_{n} )k(1-k)-\alpha_{n}\beta _{n} \gamma_{n} k^{2}(1-k) \bigr)\Vert u_{n}-p\Vert \end{aligned}$$
for all \(n\geq0\). Since \(\alpha_{n}, \beta_{n}, \gamma_{n}\in(\frac {1}{2}, 1)\) for all n, \(-\frac{1}{2}k(1-k)<-\alpha_{n}(1-\beta _{n})k(1-k)< 0 \), and \(-k^{2}(1-k)< -\alpha_{n}\beta_{n}(1-\gamma_{n})k^{2}(1-k) < -\frac{1}{8}k^{2}(1-k)\) and so
$$1-(1-k)\alpha_{n} -\alpha_{n}(1-\beta_{n} )k(1-k)- \alpha_{n}\beta_{n}\gamma _{n} k^{2}(1-k)< 1- \frac{1}{2}(1-k) -\frac{1}{8}k^{2}(1-k) $$
for all n. Put \(c_{n}=(1-\frac{1}{2}(1-k) -\frac {1}{8}k^{2}(1-k))^{n}\Vert u_{1}-p\Vert \) for all \(n\geq0\). Then we have
$$ \lim_{n\to\infty}\frac{a_{n}}{c_{n}}= \frac{(1-\frac{1}{2}(1-k)-\frac {1}{4}k(1-k)-\frac{1}{8}k^{2}(1-k))^{n}\Vert x_{1}-p\Vert }{ {(1-\frac{1}{2}(1-k) -\frac{1}{8}k^{2}(1-k))}^{n} \Vert u_{1} -p\Vert }=0. $$
Thus, \(\{x_{n}\}_{n\geq0}\) converges faster than the sequence \(\{u_{n}\} _{n\geq0}\). Now, we compare the case (2.3) with the following Noor iteration case:
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n}) u_{n} + \alpha_{n}Tv_{n}, \\ v_{n} = \beta_{n}u_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = \gamma_{n}u_{n} +(1-\gamma_{n} )Tu_{n} \end{array}\displaystyle \right . $$
(3.8)
for all \(n\geq0\). Note that
$$\begin{aligned} \Vert w_{n}-p\Vert &= \bigl\Vert \gamma_{n} u_{n} +(1-\gamma _{n})Tu_{n} -p\bigr\Vert \\ &\leq\gamma_{n}\Vert u_{n}-p\Vert + k(1- \gamma_{n} ) \Vert u_{n}-p\Vert \\ &= \bigl(1-(1-\gamma_{n}) (1-k)\bigr) \Vert u_{n}-p\Vert \end{aligned}$$
and
$$\begin{aligned} \Vert v_{n}-p\Vert &= \bigl\Vert (1-\beta_{n})u_{n} +\beta _{n}Tw_{n} -p\bigr\Vert \\ &\leq(1-\beta_{n})\Vert u_{n}-p\Vert + k \beta_{n} \Vert w_{n}-p\Vert \\ &\leq\bigl(1-\beta_{n} + k\beta_{n} \bigl(1-(1- \gamma_{n}) (1-k)\bigr)\bigr) \Vert u_{n}-p\Vert \\ &\leq\bigl(1-\beta_{n} + k\beta_{n} -\beta_{n}(1- \gamma_{n})k(1-k)\bigr) \Vert u_{n}-p\Vert \\ &\leq\bigl(1-\beta_{n} (1-k) -\beta_{n}(1- \gamma_{n})k(1-k)\bigr) \Vert u_{n}-p\Vert \end{aligned}$$
and so
$$\begin{aligned} \Vert u_{n+1}-p\Vert &= \bigl\Vert (1-\alpha_{n})u_{n} +\alpha_{n} Tv_{n} -p\bigr\Vert \\ &\leq(1 - \alpha_{n}) \Vert u_{n}-p\Vert + k \alpha_{n} \Vert w_{n}-p\Vert \\ &\leq(1-\alpha_{n}) \Vert u_{n}-p\Vert + k \alpha_{n} \bigl(1-\beta_{n} (1-k) -\beta_{n}(1- \gamma_{n})k(1-k)\bigr) \Vert u_{n}-p \Vert \\ &\leq\bigl(1-\alpha_{n} + k\alpha_{n} -\alpha_{n} \beta_{n} k(1-k) - \alpha _{n}\beta_{n}(1- \gamma_{n} )k^{2}(1-k) \bigr)\Vert u_{n}-p\Vert \\ &\leq\bigl(1-(1-k)\alpha_{n} -\alpha_{n}\beta_{n} k(1-k) - \alpha_{n}\beta _{n}(1-\gamma_{n}) k^{2}(1-k) \bigr)\Vert u_{n}-p\Vert \end{aligned}$$
for all n. Since \(\alpha_{n}, \beta_{n}, \gamma_{n}\in(\frac{1}{2}, 1)\) for all n, \(-k(1-k)<-\alpha_{n}\beta_{n}k(1-k)< -\frac {1}{4}k(1-k)\), and \(-\frac{1}{2}k^{2}(1-k)< -\alpha_{n}\beta_{n}(1-\gamma _{n})k^{2}(1-k) < 0\) for all n. This implies that
$$1-(1-k)\alpha_{n} -\alpha_{n}\beta_{n} k(1-k)- \alpha_{n}\beta_{n}(1-\gamma _{n}) k^{2}(1-k)< 1-\frac{1}{2}(1-k) -\frac{1}{4}k(1-k) $$
for all n. Put \(d_{n}=(1-\frac{1}{2}(1-k) -\frac{1}{4}k(1-k))^{n} \Vert u_{1}-p\Vert \) for all \(n\geq0\). Then we get
$$ \lim_{n\to\infty}\frac{a_{n}}{d_{n}}=\frac{(1-\frac{1}{2}(1-k)-\frac {1}{4}k(1-k)-\frac{1}{8}k^{2}(1-k))^{n}\Vert x_{1}-p\Vert }{( 1-\frac{1}{2}(1-k) -\frac{1}{4}k(1-k))^{n} \Vert u_{1} -p \Vert }=0 $$
and so the sequence \(\{x_{n}\}_{n\geq0}\) converges faster than the sequence \(\{u_{n}\}_{n\geq0}\). By using similar proofs, one can show that the case (2.3) is faster than the following cases of the Noor iteration method:
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} u_{n+1} = \alpha_{n}u_{n} +(1-\alpha_{n}) Tv_{n}, \\ v_{n} = (1-\beta_{n})u_{n} +\beta_{n}Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} +\gamma_{n} Tu_{n} , \end{array}\displaystyle \right . \end{aligned}$$
(3.9)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} u_{n} +(1- \alpha_{n})Tv_{n}, \\ v_{n} = (1-\beta_{n})u_{n} +\beta_{n}Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n} , \end{array}\displaystyle \right . \end{aligned}$$
(3.10)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} u_{n+1} = \alpha_{n}u_{n} +(1-\alpha_{n}) Tv_{n}, \\ v_{n} = \beta_{n}u_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} =(1 - \gamma_{n}) u_{n} +\gamma_{n} Tu_{n}, \end{array}\displaystyle \right . \end{aligned}$$
(3.11)
and
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n}u_{n}+(1-\alpha_{n}) Tv_{n}, \\ v_{n} = \beta_{n} u_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n} \end{array}\displaystyle \right . $$
(3.12)
for all \(n\geq0\). This completes the proof. □

By using similar conditions, one can show that the case (3.7) converges faster than (3.8), (3.9) converges faster than (3.11), (3.11) converges faster than (3.10) and (3.10) converges faster than (3.12).

As we know, the Agarwal iteration method could be written in the following four cases:
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} x_{n+1} =(1 - \alpha_{n})Tx_{n} + \alpha_{n}Ty_{n}, \\ y_{n} = (1-\beta_{n})x_{n} +\beta_{n}Tx_{n}, \end{array}\displaystyle \right . \end{aligned}$$
(3.13)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} x_{n+1} =\alpha_{n} Tx_{n} + (1-\alpha_{n})Ty_{n}, \\ y_{n} = \beta_{n} x_{n} +(1-\beta_{n})Tx_{n}, \end{array}\displaystyle \right . \end{aligned}$$
(3.14)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} x_{n+1} = \alpha_{n}Tx_{n} + (1-\alpha_{n})Ty_{n}, \\ y_{n} = (1-\beta_{n})x_{n} +\beta_{n}Tx_{n}, \end{array}\displaystyle \right . \end{aligned}$$
(3.15)
and
$$ \left \{ \textstyle\begin{array}{l} x_{n+1} =(1 - \alpha_{n})Tx_{n} + \alpha_{n}Ty_{n}, \\ y_{n} = \beta_{n} x_{n} +(1-\beta_{n})Tx_{n} \end{array}\displaystyle \right . $$
(3.16)
for all \(n\geq0\). One can easily show that the case (3.13) converges faster than the other ones for contractive maps. We record it as the next lemma.

Lemma 3.1

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\) and p a fixed point of T. If \(1-\alpha_{n}<\alpha_{n}\) and \(1-\beta_{n} <\beta_{n}\) for all \(n\geq0\), then the case (3.13) converges faster than (3.14), (3.15), and (3.16).

Also by using a similar condition, one can show that the case (3.16) converges faster than (3.14). Similar to Theorem 3.1, we can prove that for contractive maps one case in the Abbas iteration method converges faster than the other possible cases whenever the elements of the sequences \(\{\alpha_{n}\}_{n\geq0}\), \(\{\beta_{n}\}_{n\geq0}\), and \(\{\gamma_{n}\}_{n\geq0}\) are in \((\frac{1}{2}, 1)\) for sufficiently large n. Also, one can show that for contractive maps the case (2.6) of the Thakur-Thakur-Postolache iteration method converges faster than the other possible cases whenever elements of the sequences \(\{\alpha_{n}\}_{n\geq0}\), \(\{\beta_{n}\}_{n\geq0}\), and \(\{\gamma_{n}\}_{n\geq0}\) are in \((\frac{1}{2}, 1)\) for sufficiently large n. We record these results as follows.

Lemma 3.2

Let C be a nonempty, closed, and convex subset of a Banach space X, \(u_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\), and p a fixed point of T. Consider the following case in the Abbas iteration method:
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} Tv_{n} + (1-\alpha_{n})Tw_{n}, \\ v_{n} = (1-\beta_{n})Tu_{n} +\beta_{n}Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.17)
for all n. If \(1-\alpha_{n} <\alpha_{n}\), \(1-\beta_{n} <\beta_{n}\), and \(1-\gamma_{n}< \gamma_{n}\) for sufficiently large n, then the case (3.17) converges faster than the other possible cases.
Also by using similar conditions in the Abbas iteration method, one can show that the cases
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} Tv_{n} + (1-\alpha_{n})Tw_{n}, \\ v_{n} = \beta_{n}Tu_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.18)
and
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} Tv_{n} + (1-\alpha_{n})Tw_{n}, \\ v_{n} = (1-\beta_{n})Tu_{n} +\beta_{n}Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n} \end{array}\displaystyle \right . $$
(3.19)
converge faster than the case
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} Tv_{n} + (1-\alpha_{n})Tw_{n}, \\ v_{n} = \beta_{n}Tu_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = \gamma_{n}u_{n} +(1-\gamma_{n} )Tu_{n}. \end{array}\displaystyle \right . $$
(3.20)
Also the case
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} = (1-\alpha_{n} )Tv_{n}+\alpha_{n}Tw_{n}, \\ v_{n} =(1- \beta_{n})Tu_{n} +\beta_{n}Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.21)
converges faster than the cases
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n}) Tv_{n} + \alpha_{n}Tw_{n}, \\ v_{n} = \beta_{n}Tu_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} =(1 - \gamma_{n}) u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.22)
and
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n}) Tv_{n} + \alpha_{n}Tw_{n}, \\ v_{n} = (1-\beta_{n})Tu_{n} +\beta_{n}Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n}, \end{array}\displaystyle \right . $$
(3.23)
and
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n}) Tv_{n} + \alpha_{n}Tw_{n}, \\ v_{n} = \beta_{n}Tu_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n}. \end{array}\displaystyle \right . $$
(3.24)

Lemma 3.3

Let C be a nonempty, closed, and convex subset of a Banach space X, \(u_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\) and p a fixed point of T. If \(1-\alpha_{n} <\alpha_{n}\), \(1-\beta_{n} <\beta_{n}\), and \(1-\gamma_{n}< \gamma_{n}\) for sufficiently large n, then the case (2.6) in the Thakur-Thakur-Postolache iteration method converges faster than the other possible cases.

Also by using similar conditions, one can show that the cases
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n} )Tu_{n} + \alpha_{n}Tv_{n}, \\ v_{n} = \beta_{n} w_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.25)
and
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} = (1-\alpha_{n})Tu_{n} +\alpha_{n} Tv_{n}, \\ v_{n} = (1-\beta_{n})w_{n} +\beta_{n}Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n} \end{array}\displaystyle \right . $$
(3.26)
converge faster than the case
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n}) Tu_{n} + \alpha_{n}Tv_{n}, \\ v_{n} = \beta_{n}w_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = \gamma_{n}u_{n} +(1-\gamma_{n} )Tu_{n}. \end{array}\displaystyle \right . $$
(3.27)
Also the case
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} Tu_{n} +(1- \alpha_{n})Tv_{n}, \\ v_{n} = (1-\beta_{n} )w_{n} +\beta_{n}Tw_{n}, \\ w_{n} = (1-\gamma_{n}) u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.28)
converges faster than the cases
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} Tu_{n} +(1- \alpha_{n})Tv_{n}, \\ v_{n} = \beta_{n} w_{n} +(1-\beta_{n}) Tw_{n}, \\ w_{n} =(1- \gamma_{n}) u_{n} +\gamma_{n} Tu_{n} \end{array}\displaystyle \right . $$
(3.29)
and
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n} Tu_{n} +(1- \alpha_{n})Tv_{n}, \\ v_{n} = (1-\beta_{n})w_{n} +\beta_{n}Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n}, \end{array}\displaystyle \right . $$
(3.30)
and
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n}Tu_{n}+(1-\alpha_{n}) Tv_{n}, \\ v_{n} = \beta_{n}w_{n} +(1-\beta_{n})Tw_{n}, \\ w_{n} = \gamma_{n} u_{n} +(1-\gamma_{n}) Tu_{n}. \end{array}\displaystyle \right . $$
(3.31)
Finally, we have a similar situation for the Picard S-iteration which we record here.

Lemma 3.4

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\) and p a fixed point of T. If \(1-\alpha_{n} <\alpha_{n}\) and \(1-\beta_{n} <\beta_{n}\) for sufficiently large n, then the case (2.7) in the Picard S-iteration method converges faster than the other possible cases.

4 Comparing different iterations methods

In this section, we compare the rate of convergence of some different iteration methods for contractive maps. Our goal is to show that the rate of convergence relates to the coefficients.

Theorem 4.1

Let C be a nonempty, closed, and convex subset of a Banach space X, \(u_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\) and p a fixed point of T. Consider the case (2.5) in the Abbas iteration method
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n}) Tv_{n} + \alpha_{n}Tw_{n}, \\ v_{n} = (1-\beta_{n})Tu_{n} +\beta_{n}Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} + \gamma_{n}Tu_{n}, \end{array}\displaystyle \right . $$
the case (3.17) in the Abbas iteration method
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =\alpha_{n}Tv_{n} +(1-\alpha_{n})Tw_{n}, \\ v_{n} = (1-\beta_{n})Tu_{n} +\beta_{n}Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} + \gamma_{n}Tu_{n}, \end{array}\displaystyle \right . $$
and the case (2.6) in the Thakur-Thakur-Postolache iteration method
$$ \left \{ \textstyle\begin{array}{l} u_{n+1} =(1-\alpha_{n})Tu_{n}+\alpha_{n} Tv_{n}, \\ v_{n} = (1-\beta_{n})w_{n} +\beta_{n}Tw_{n}, \\ w_{n} = (1-\gamma_{n})u_{n} + \gamma_{n}Tu_{n} \end{array}\displaystyle \right . $$
for all \(n\geq0\). If \(1-\alpha_{n} <\alpha_{n}\), \(1-\beta_{n} <\beta_{n}\), and \(1-\gamma_{n}< \gamma_{n}\) for sufficiently large n, then the case (3.17) in the Abbas iteration method converges faster than the case (2.6) in the Thakur-Thakur-Postolache iteration method. Also, the case (2.6) in the Thakur-Thakur-Postolache iteration method is faster than the case (2.5) in the Abbas iteration method.

Proof

Let \(\{u_{n}\}_{n\geq0}\) be the sequence in the case (3.17). Then we have
$$\begin{aligned}& \Vert w_{n}-p\Vert = \bigl\Vert (1-\gamma_{n})u_{n} +\gamma _{n}Tu_{n} -p\bigr\Vert \\& \hphantom{\Vert w_{n}-p\Vert }\leq(1-\gamma_{n})\Vert u_{n}-p\Vert + k \gamma_{n} \Vert u_{n}-p\Vert \\& \hphantom{\Vert w_{n}-p\Vert }= \bigl(1-(1-k)\gamma_{n}\bigr) \Vert u_{n}-p\Vert , \\& \Vert v_{n}-p\Vert = \bigl\Vert (1-\beta_{n})Tu_{n} +\beta _{n}Tw_{n} -p\bigr\Vert \\& \hphantom{\Vert v_{n}-p\Vert }\leq k(1-\beta_{n})\Vert u_{n}-p\Vert + k \beta_{n} \Vert w_{n}-p\Vert \\& \hphantom{\Vert v_{n}-p\Vert }\leq k \bigl[ (1-\beta_{n}) + \beta_{n} \bigl(1-(1-k)\gamma_{n}\bigr)\bigr] \Vert u_{n}-p\Vert \\& \hphantom{\Vert v_{n}-p\Vert }\leq k\bigl[1 - \beta_{n}\gamma_{n}(1-k) \bigr]\Vert u_{n}-p\Vert , \end{aligned}$$
and
$$\begin{aligned} \Vert u_{n+1}-p\Vert &= \bigl\Vert \alpha_{n}Tv_{n} +(1-\alpha_{n})Tw_{n} -p\bigr\Vert \\ &\leq\alpha_{n} k\Vert v_{n}-p\Vert + k \alpha_{n} \Vert w_{n}-p\Vert \\ &\leq\alpha_{n} k^{2} \bigl(1 - \beta_{n} \gamma_{n} (1-k)\bigr) \Vert u_{n}-p\Vert + k(1 - \alpha_{n}) \bigl(1-(1-k)\gamma_{n}\bigr) \Vert u_{n}-p\Vert \\ &\leq k\bigl[ k\alpha_{n} -\alpha_{n}\beta_{n} \gamma_{n} k(1-k) + (1-\alpha_{n}) \bigl(1-(1-k) \gamma_{n}\bigr)\bigr]\Vert u_{n}-p\Vert \\ &=k\bigl[k\alpha_{n} -\alpha_{n}\beta_{n} \gamma_{n} k(1-k)+1-\alpha_{n} - (1-\alpha_{n}) \gamma_{n}(1-k)\bigr]\Vert u_{n}-p\Vert \\ &=k\bigl[1 -\alpha_{n}(1-k)- (1-\alpha_{n}) \gamma_{n}(1-k)-\alpha_{n}\beta _{n} \gamma_{n} k(1-k) \bigr]\Vert u_{n}-p\Vert \end{aligned}$$
for all n. Since \(\alpha_{n}, \beta_{n}, \gamma_{n}\in(\frac{1}{2}, 1)\) for sufficiently large n, we have
$$-(1-k)< -\alpha_{n}(1-k)< -\frac{1}{2}(1-k), $$
\(-\frac{1}{2}(1-k)<-\alpha_{n}\gamma_{n}(1-k)<0\), and \(-k(1-k)<-\alpha _{n}\beta_{n}\gamma_{n} k (1-k) <-\frac{1}{8}k(1-k)\) for sufficiently large n. Hence,
$$1-\alpha_{n}(1-k)-(1-\alpha_{n})\gamma_{n}(1-k)- \alpha_{n}\beta_{n}\gamma_{n} k(1-k)< 1- \frac{1}{2}(1-k)-\frac{1}{8}k(1-k) $$
for sufficiently large n. Put \(a_{n} =k^{n} (1-\frac{1}{2}(1-k)-\frac {1}{8}k(1-k))^{n}\Vert u_{1}-p\Vert \) for all n. Now, let \(\{ u_{n}\}_{n\geq0}\) be the sequence in the case (2.6). Then we have
$$\begin{aligned}& \Vert w_{n}-p\Vert = \bigl\Vert (1-\gamma_{n})u_{n} +\gamma _{n}Tu_{n} -p\bigr\Vert \\& \hphantom{\Vert w_{n}-p\Vert } \leq(1-\gamma_{n})\Vert u_{n}-p\Vert + k \gamma_{n} \Vert u_{n}-p\Vert \\& \hphantom{\Vert w_{n}-p\Vert } = \bigl(1-(1-k)\gamma_{n}\bigr) \Vert u_{n}-p\Vert , \\& \Vert v_{n}-p\Vert = \bigl\Vert (1-\beta_{n})w_{n} +\beta _{n}Tw_{n} -p\bigr\Vert \\& \hphantom{\Vert v_{n}-p\Vert } \leq(1-\beta_{n})\Vert u_{n}-p\Vert + k \beta_{n} \Vert w_{n}-p\Vert \\& \hphantom{\Vert v_{n}-p\Vert } \leq(1-\beta_{n}) \bigl(1-(1-k)\gamma_{n} \bigr) +k \beta_{n} \bigl(\bigl(1-(1-k)\gamma_{n}\bigr)\bigr) \Vert u_{n}-p\Vert \\& \hphantom{\Vert v_{n}-p\Vert } \leq\bigl[1-\beta_{n}(1-k)\bigr] \bigl[1- \gamma_{n}(1-k)\bigr]\Vert u_{n}-p\Vert , \end{aligned}$$
and
$$\begin{aligned} \Vert u_{n+1}-p\Vert =& \bigl\Vert (1-\alpha _{n})Tu_{n}+\alpha_{n}Tv_{n} -p\bigr\Vert \\ \leq&(1-\alpha_{n}) k\Vert u_{n}-p\Vert + k \alpha_{n} \Vert v_{n}-p\Vert \\ \leq& k(1-\alpha_{n} ) \Vert u_{n}-p\Vert + k \alpha_{n} \bigl[1-\beta_{n}(1-k)\bigr] \bigl[1- \gamma_{n}(1-k)\bigr] \Vert u_{n}-p\Vert \\ \leq& k\bigl[ 1-\alpha_{n} + \alpha_{n} \bigl(1- \beta_{n}(1-k)\bigr) \bigl(1-\gamma _{n}(1-k)\bigr)\bigr] \Vert u_{n}-p\Vert \\ \leq& k\bigl[ 1-\alpha_{n} + \bigl(\alpha_{n} -(1-k) \beta_{n}\alpha_{n}\bigr) \bigl((1-\gamma _{n})+k \gamma_{n}\bigr)\bigr] \Vert u_{n}-p\Vert \\ \leq& k\bigl[ 1-\alpha_{n} + \alpha_{n} (1- \gamma_{n}) + \alpha_{n}\gamma_{n} k - \beta_{n}\alpha_{n}(1-\gamma_{n}) (1-k) \\ &{} -\alpha_{n}\beta_{n}\gamma_{n} k(1-k)\bigr] \Vert u_{n}-p\Vert \\ \leq& k\bigl[ 1 -\alpha_{n}\gamma_{n}(1-k) - \alpha_{n}\beta_{n}(1-\gamma _{n}) (1-k)- \alpha_{n}\beta_{n}\gamma_{n}k(1-k)\bigr] \Vert u_{n}-p \Vert \end{aligned}$$
for all n. Since \(\alpha_{n}, \beta_{n}, \gamma_{n}\in(\frac{1}{2}, 1)\) for sufficiently large n, we have
$$-(1-k)< -\alpha_{n}\gamma_{n}(1-k)< -\frac{1}{4}(1-k), $$
\(-\frac{1}{2}(1-k)<-\alpha_{n}\beta_{n}(1-\gamma_{n})(1-k)<0\), and \(-k(1-k)<-\alpha_{n}\beta_{n}\gamma_{n} k (1-k) <-\frac{1}{8}k(1-k)\) for sufficiently large n. Hence,
$$1-\alpha_{n}\gamma_{n}(1-k)-\alpha_{n} \beta_{n}(1-\gamma_{n}) (1-k)-\alpha _{n} \beta_{n}\gamma_{n}k(1-k)< 1-\frac{1}{4}(1-k) - \frac{1}{8}k(1-k) $$
for sufficiently large n. Put \(b_{n} =k^{n} (1-\frac{1}{4}(1-k) -\frac {1}{8}k(1-k))^{n}\Vert u_{1}-p\Vert \) for all n. Then
$$ \lim_{n\to\infty}\frac{a_{n}}{b_{n}}=\frac{k^{n}(1-\frac{1}{2}(1-k) -\frac{1}{8}k(1-k))^{n}\Vert u_{1}-p\Vert }{k^{n} (1-\frac {1}{4}(1-k) -\frac{1}{8}k(1-k))^{n} \Vert u_{1} -p\Vert }=0. $$
Thus, the case (3.17) in the Abbas iteration method converges faster than the case (2.6) in the Thakur-Thakur-Postolache iteration method.
Now for the case (2.5), we have
$$\begin{aligned}& \Vert w_{n}-p\Vert = \Vert 1-\gamma_{n} u_{n} +\gamma _{n}Tu_{n} -p\Vert \\& \hphantom{\Vert w_{n}-p\Vert } \leq(1-\gamma_{n})\Vert u_{n}-p\Vert + k \gamma_{n} \Vert u_{n}-p\Vert \\& \hphantom{\Vert w_{n}-p\Vert } = \bigl(1-(1-k)\gamma_{n}\bigr) \Vert u_{n}-p\Vert , \\& \Vert v_{n}-p\Vert = \bigl\Vert (1-\beta_{n})Tu_{n} +\beta _{n}Tw_{n} -p\bigr\Vert \\& \hphantom{\Vert v_{n}-p\Vert } \leq k(1-\beta_{n})\Vert u_{n}-p\Vert + k\beta_{n} \Vert w_{n}-p\Vert \\& \hphantom{\Vert v_{n}-p\Vert } \leq k \bigl[ (1-\beta_{n}) + \beta_{n} \bigl(1 -(1-k)\gamma_{n}\bigr)\bigr] \Vert u_{n}-p\Vert \\& \hphantom{\Vert v_{n}-p\Vert } \leq k\bigl[1 - \beta_{n}\gamma_{n}(1-k) \bigr]\Vert u_{n}-p\Vert , \end{aligned}$$
and
$$\begin{aligned} \Vert u_{n+1}-p\Vert &= \bigl\Vert (1-\alpha_{n})Tv_{n} +\alpha_{n}Tw_{n} -p\bigr\Vert \\ &\leq(1-\alpha_{n}) k\Vert v_{n}-p\Vert + k \alpha_{n} \Vert w_{n}-p\Vert \\ &\leq(1 - \alpha_{n}) k^{2} \bigl(1-\beta_{n} \gamma_{n} (1-k)\bigr) \Vert u_{n}-p\Vert + k \alpha_{n} \bigl(1-(1-k)\gamma_{n}\bigr) \Vert u_{n}-p\Vert \\ &\leq k\bigl[(1- \alpha_{n})k -(1-\alpha_{n}) \beta_{n}\gamma_{n} k(1-k) + \alpha _{n} - \alpha_{n}\gamma_{n}(1-k)\bigr]\Vert u_{n}-p \Vert \\ &\leq k\bigl[1-(1-\alpha_{n}) (1-k) - \alpha_{n} \gamma_{n}(1-k)-(1-\alpha _{n})\beta_{n} \gamma_{n} k(1-k)\bigr]\Vert u_{n}-p\Vert \end{aligned}$$
for all n. Since \(\alpha_{n}, \beta_{n}, \gamma_{n}\in(\frac{1}{2}, 1)\) for sufficiently large n, \(-\frac{1}{2}(1-k)<-(1-\alpha _{n})(1-k)< 0\), \(-(1-k)<-\alpha_{n}\gamma_{n}(1-k)<-\frac{1}{4}(1-k)\), and \(-\frac {1}{2}k(1-k)<-(1-\alpha_{n})\beta_{n}\gamma_{n} k (1-k) <0\) for sufficiently large n. Hence,
$$1-(1-\alpha_{n}) (1-k)-\alpha_{n}\gamma_{n}(1-k)-(1- \alpha_{n})\beta _{n}\gamma_{n} k(1-k)< 1- \frac{1}{4}(1-k) $$
for sufficiently large n. Put \(c_{n} =k^{n} (1-\frac{1}{4}(1-k) )^{n} \Vert x_{1}-p\Vert \) for all n. Then we have
$$ \lim_{n\to\infty}\frac{b_{n}}{c_{n}}=\frac{k^{n}(1-\frac {1}{4}(1-k)-\frac{1}{8}k(1-k))^{n}\Vert u_{1}-p\Vert }{k^{n}(1-\frac{1}{4}(1-k))^{n} \Vert u_{1}-p\Vert }=0 $$
and so the case (2.6) in the Thakur-Thakur-Postolache iteration method is faster than the case (2.5) in the Abbas iteration method. □

By using a similar proof, we can compare the Thakur-Thakur-Postolache and the Agarwal iteration methods as follows.

Theorem 4.2

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\) and p a fixed point of T. If \(1-\alpha_{n} <\alpha_{n}\), \(1-\beta_{n} <\beta_{n}\), and \(1-\gamma_{n}< \gamma_{n}\) for sufficiently large n, then the case (2.6) in the Thakur-Thakur-Postolache iteration method converges faster than the case (2.4) in the Agarwal iteration method and the case (2.4) in the Agarwal iteration method is faster than the cases (3.29) and (3.30) in the Thakur-Thakur-Postolache iteration method.

Also by using similar proofs, we can compare some another iteration methods. We record those as follows.

Theorem 4.3

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\), and p a fixed point of T. If \(1-\alpha_{n} <\alpha_{n}\), \(1-\beta_{n} <\beta_{n}\), and \(1-\gamma_{n}< \gamma_{n}\) for sufficiently large n, then the case (2.3) in the Abbas iteration method converges faster than the case (2.2) in the Ishikawa iteration method and the case (2.2) in the Ishikawa iteration method is faster than the cases (3.11) and (3.12) in the Abbas iteration method.

It is notable that there are some cases which the coefficients have no effective roles to play in the rate of convergence. By using similar proofs, one can check the next result. One can obtain some similar cases. This shows us that researchers should stress more the probability of the efficiency of coefficients in the rate of convergence for iteration methods.

Theorem 4.4

Let C be a nonempty, closed, and convex subset of a Banach space X, \(x_{1}\in C\), \(T\colon C \to C \) a contraction with constant \(k\in(0, 1)\), p a fixed point of T, and \(\alpha_{n}, \beta_{n}, \gamma_{n} \in(0, 1)\) for all \(n\geq0\). Then the case (2.4) in the Agarwal iteration method is faster than the case (2.1) in the Mann iteration method, the case (2.5) in the Abbas iteration method is faster than the case (2.1) in the Mann iteration method, the case (2.6) in the Thakur-Thakur-Postolache iteration method is faster than the case (2.1) in the Mann iteration method, the case (2.4) in the Agarwal iteration method is faster than the case (2.2) in the Ishikawa iteration method, the case (2.5) in the Abbas iteration method is faster than the case (2.2) in the Ishikawa iteration method and the case (2.6) in the Thakur-Thakur-Postolache iteration method is faster than the case (2.2) in the Ishikawa iteration method.

5 Examples and figures

In this section, we provide some examples to illustrate our results.

Example 1

Let \(X = \mathbb{R}\), \(C=[1, 60]\), \(x_{0}=20\), \(\alpha_{n}=0.7\), and \(\beta_{n}=0.85\) for all \(n\geq0\). Define the map \(T\colon C \to C\) by the formula \(T(x)=(3x+18)^{\frac{1}{3}}\) for all \(x\in C\). It is easy to see that T is a contraction. In Tables 1-3, we first compare two cases of the Mann iteration method and also four cases of the Ishikawa and Agarwal iteration methods separately. From a mathematical point of view, one can see that the Mann iteration (3.1) is more than 2.82 times faster than the Mann iteration (2.1), the Ishikawa iteration (3.2) is more than 1.07 times faster than the Ishikawa iteration (3.4), the Ishikawa iteration (3.2) is more than 11.33 times faster than the Ishikawa iteration (3.3), the Ishikawa iteration (3.2) is more than 11 times faster than the Ishikawa iteration (3.5), the Ishikawa iteration (3.4) is more than 8.75 times faster than the Ishikawa iteration (3.5), the Agarwal iteration (3.13) is 1.22 times faster than the Agarwal iteration (3.14), the Agarwal iteration (3.13) is 1.11 times faster than the Agarwal iteration (3.15), the Agarwal iteration (3.13) is 1.22 times faster than the Agarwal iteration (3.16) and so on. We first add our CPU time in Tables 1-3 for each iteration method. Also, we provide Figure 1 by using at least 30 times calculating of CPU times for our faster cases in the methods. From a computer-calculation point of view, we get a different answer. As one can see in the CPU time table, we found that the Agarwal iteration (3.13) and the Mann iteration (3.1) are faster than the Ishikawa iteration (3.2). This note emphasizes the difference of the mathematical results and computer-calculation results which have appeared many times in the literature.
Figure 1

CPU time.

Table 1

Cases of Mann iteration

Step

Mann ( 2.1 )

Mann ( 3.1 )

1

15.2817976045

8.9908610772

2

11.8962912491

5.186577882

3

9.4591508761

3.8138707904

4

7.6992520365

3.305644632

5

6.4247631019

3.1152016077

6

5.4994648986

3.0434826465

7

4.8262347919

3.0164213456

8

4.3355308466

3.0062028434

9

3.977352589

3.0023431856

10

3.7156123245

3.0008851876

11

3.5241766763

3.000334402

12

3.3840675849

3.0001263293

13

3.2814716521

3.0000477244

14

3.2063163994

3.0000180292

15

3.1512468009

3.000006811

16

3.11088634

3.0000025731

17

3.0813015724

3.000000972

18

3.0596130334

3.0000003672

19

3.0437118532

3.0000001387

20

3.0320530065

3.0000000524

21

3.0235042722

3.0000000198

22

3.0172357852

3.0000000075

23

3.0126392095

3.0000000028

24

3.0092685565

 

25

3.0067968355

 

26

3.004984289

 

 

63

3.0000000517

 

64

3.0000000379

 

65

3.0000000278

 

66

3.0000000204

 

CPU time

0.0010

0.0007

Table 2

Cases of Ishikawa iteration

Step

Ishikawa ( 3.2 )

Ishikawa ( 3.3 )

Ishikawa ( 3.5 )

Ishikawa ( 3.4 )

1

6.022745179

17.599516463

6.397259957

17.53342562

2

3.55504988

15.542488073

3.725044385

15.426710149

3

3.102829451

13.778956254

3.157958555

13.626959863

4

3.019085154

12.266356345

3.034584416

12.089125019

5

3.003543432

10.968408676

3.007580568

10.774826445

6

3.000657931

9.854176549

3.001661995

9.651358665

7

3.000122164

8.89726621

3.000364402

8.690843013

8

3.000022683

8.07514758

3.000079898

7.86950815

9

3.000004212

7.368577613

3.000017518

7.167078769

10

3.000000782

6.76111087

3.000003841

6.566256169

11

3.000000145

6.23868412

3.000000842

6.052276815

12

3.000000027

5.789263769

3.000000185

5.612537089

13

3.000000005

5.402546543

3.00000004

5.23627424

14

3.000000001

5.069705312

3.000000009

4.91429501

15

 

4.783173147

3.000000002

4.638744748

16

 

4.536459758

 

4.402910896

17

 

4.323995342

 

4.201055645

18

 

4.14099766

 

4.028273397

19

 

3.983358785

 

3.880369278

20

 

3.847548529

 

3.753755571

21

 

3.730532022

 

3.645363373

22

 

3.629699305

 

3.55256722

23

 

3.542805134

 

3.473120743

24

 

3.467917475

 

3.405101727

25

 

3.403373393

 

3.346865184

26

 

3.347741258

 

3.297003256

27

 

3.299788327

 

3.254310946

28

 

3.258452935

 

3.217756821

29

 

3.222820611

 

3.18645797

30

 

3.192103569

 

3.159658578

31

 

3.165623078

 

3.136711605

32

 

3.142794307

 

3.117063114

33

 

3.123113286

 

3.100238856

34

 

3.1061457

 

3.0858328

35

 

3.09151723

 

3.07349731

CPU time

0.00086

0.0035

0.0016

0.0085

Table 3

Cases of Agarwal iteration

Step

Agarwal ( 3.13 )

Agarwal ( 3.14 )

Agarwal ( 3.16 )

Agarwal ( 3.15 )

1

3.663643981

4.231276342

4.038158759

4.165185499

2

3.034148064

3.125898552

3.08652991

3.112771857

3

3.001785887

3.013368608

3.007415671

3.011314821

4

3.000093479

3.001425297

3.000637055

3.001139398

5

3.000004893

3.000152024

3.000054738

3.000114779

6

3.000000256

3.000016216

3.000004703

3.000011563

7

3.000000013

3.00000173

3.000000404

3.000001165

8

3.000000001

3.000000184

3.000000035

3.000000117

9

3

3.00000002

3.000000003

3.000000012

10

 

3.000000002

3

3.000000001

11

 

3

 

3

CPU time

0.00095

0.0034

0.0011

0.0011

The next example illustrates Lemma 3.2.

Example 2

Let \(X = \mathbb{R}\), \(C = [0, 2000]\), \(x_{0}=1000\), \(\alpha_{n}=0. 85\), \(\beta_{n}=0. 65\), and \(\gamma_{n}=0. 75\) for all \(n\geq0\). Define the map \(T\colon C\to C\) by the formula \(T(x) =\sqrt[3]{x^{2}}\) for all \(x\in C\). Table 4 shows us that the Abbas iteration (3.17) converges faster than the other cases, the Abbas iteration (3.18) is 1.1 times faster than the Abbas iteration (3.20), the Abbas iteration (3.19) is 1.05 times faster than the Abbas iteration (3.20), the Abbas iteration (3.21) is 1.04 times faster than the Abbas iteration (3.22) and 1.3 times faster than the Abbas iteration (3.23) and the Abbas iteration (3.24). One can get similar results about difference of the mathematical and computer-calculating points of views for this example.
Table 4

Cases of Abbas iteration

Step

Abbas ( 3.17 )

Abbas ( 3.18 )

Abbas ( 3.19 )

Abbas ( 3.20 )

Abbas ( 3.21 )

Abbas ( 3.22 )

Abbas ( 3.23 )

Abbas ( 3.24 )

1

20.933947

23.074444

29.706456

30.294581

42.622758

43.000492

74.725586

74.829373

2

3.501533

3.915728

4.912771

5.052334

6.872931

6.975246

14.057893

14.097781

3

1.650123

1.789347

2.07514

2.127569

2.605814

2.644699

4.919453

4.938021

4

1.218545

1.278689

1.392374

1.417334

1.596596

1.615195

2.581994

2.592232

5

1.080705

1.109014

1.161005

1.174049

1.254442

1.264388

1.750749

1.757015

6

1.030883

1.044439

1.069469

1.076461

1.115609

1.121158

1.389425

1.39348

7

1.011982

1.018426

1.030642

1.034379

1.054109

1.057231

1.212285

1.214975

8

1.004673

1.007695

1.013649

1.015622

1.025684

1.02743

1.119022

1.120821

9

1.001827

1.003223

1.006106

1.007132

1.012273

1.013239

1.067815

1.069015

10

1.000715

1.001351

1.002737

1.003264

1.005884

1.006411

1.038999

1.039794

11

1.00028

1.000567

1.001228

1.001495

1.002825

1.00311

1.022548

1.02307

12

1.000109

1.000238

1.000551

1.000685

1.001357

1.00151

1.013078

1.013417

13

1.000043

1.0001

1.000247

1.000314

1.000653

1.000733

1.007599

1.007818

14

1.000017

1.000042

1.000111

1.000144

1.000314

1.000356

1.00442

1.00456

15

1.000007

1.000018

1.00005

1.000066

1.000151

1.000173

1.002572

1.002661

16

1.000003

1.000007

1.000022

1.00003

1.000073

1.000084

1.001498

1.001554

17

1.000001

1.000003

1.00001

1.000014

1.000035

1.000041

1.000872

1.000907

18

 

1.000001

1.000005

1.000006

1.000017

1.00002

1.000508

1.00053

19

 

1.000001

1.000002

1.000003

1.000008

1.00001

1.000296

1.00031

20

 

1

1.000001

1.000001

1.000004

1.000005

1.000172

1.000181

21

 

1

1

1.000001

1.000002

1.000002

1.0001

1.000106

22

 

1

1

1

1.000001

1.000001

1.000058

1.000062

23

 

1

1

1

1

1.000001

1.000034

1.000036

24

  

1

1

1

1

1.00002

1.000021

25

   

1

1

1

1.000012

1.000012

26

   

1

1

1

1.000007

1.000007

27

    

1

1

1.000004

1.000004

28

    

1

1

1.000002

1.000002

29

    

1

1

1.000001

1.000001

30

      

1.000001

1.000001

31

      

1

1

32

      

1

1

33

      

1

1

34

      

1

1

35

      

1

1

36

      

1

1

37

      

1

1

The next example illustrates Theorem 3.1.

Example 3

Let \(X = \mathbb{R}\), \(C = [1, 60]\), \(x_{0}=40\), \(\alpha_{n}=0.9\), \(\beta _{n}=0.6\), and \(\gamma_{n}=0.8\) for all \(n\geq0\). Define the map \(T\colon C\to C\) by \(T(x) =\sqrt{x^{2}-8x+40}\) for all \(x\in C\) (see [23]). Table 5 shows the Abbas iteration (3.17) converges 1.09 times faster than the Thakur-Thakur-Postolache iteration (2.6) and the Thakur-Thakur-Postolache iteration (2.6) is 1.16 times faster than the Abbas iteration (2.5) from the mathematical point of view. Again, we get different results from the computer-calculating point of view by checking Table 5 and Figures 2 and 3.
Figure 2

Convergence behavior of the iteration methods of Thakur equation ( 2.6 ), Abbas equation ( 2.5 ), and Abbas equation ( 3.17 ).

Figure 3

CPU time.

Table 5

Comparison between Thakur iteration and Abbas iteration

Step

Thakur ( 2.6 )

Abbas ( 2.5 )

Abbas ( 3.17 )

1

31.77453587

33.18158852

31.22317681

2

23.81196041

26.52340588

22.75386567

3

16.33019829

20.11920431

14.88031305

4

9.89958703

14.1634562

8.4317634

5

5.97706669

9.11456867

5.36305686

6

5.07407177

5.96019967

5.01260299

7

5.00409402

5.0925653

5.00037245

8

5.00022019

5.00645474

5.00001094

9

5.00001182

5.00043527

5.00000032

10

5.00000063

5.00002928

5.00000001

11

5.00000003

5.00000197

5

12

5

5.00000013

 

13

5

5.00000001

 

14

 

5

 

CPU time

0.0012

0.0012

0.0009

The next example shows that choosing the coefficients is very important in the rate of convergence of an iteration method.

Example 4

Let \(X=\mathbb{R}\), \(C=[0, 30]\), and \(x_{0}=20\). Define the map \(T\colon \mathbb{R}\to\mathbb{R}\) by \(T(x) = \frac{x}{2}+1\) for all \(x\in C\). Consider the following coefficients separately in the Thakur-Thakur-Postolache iteration (2.6):
  1. (a)

    \(\alpha_{n}=\beta_{n}=\gamma_{n}= 1-\frac{1}{(n+1)^{10}}\),

     
  2. (b)

    \(\alpha_{n}=\beta_{n}=\gamma_{n}= 1-\frac{1}{n+1}\),

     
  3. (c)

    \(\alpha_{n}=\beta_{n}=\gamma_{n}= 1-\frac{1}{(n+1)^{\frac{1}{2}}}\),

     
  4. (d)

    \(\alpha_{n}=\beta_{n}=\gamma_{n}=1 -\frac{1}{(n+1)^{\frac{1}{5}}}\)

     
for all \(n\geq0\). Table 6 shows that the Thakur-Thakur-Postolache iteration (2.6) with coefficients (a) is 1.25 times faster than the Thakur-Thakur-Postolache iteration (2.6) with coefficients (b), the Thakur-Thakur-Postolache iteration (2.6) with coefficients (a) is 1.6 times faster than the Thakur-Thakur-Postolache iteration (2.6) with coefficients (c) and the Thakur-Thakur-Postolache iteration (2.6) with coefficients (a) is 2.16 times faster than the Thakur-Thakur-Postolache iteration (2.6) with coefficients (d). This note satisfies other iteration methods of course from the mathematical point of view. Here, we find a little different computer-calculating result for the CPU time table of this example, which one can check in Figure 4.
Figure 4

CPU time.

Table 6

Cases of Thakur iteration

Step

(a)

(b)

(c)

(d)

1

4.2609841803

9.03125

10.2844561595

10.8540663001

2

2.2826469537

4.2135416667

5.4804739263

6.2632682688

3

2.0353310377

2.6009419759

3.3595601275

4.0142167756

4

2.004416382

2.1466298421

2.5007642765

2.9360724936

5

2.0005520478

2.0330086855

2.1756764587

2.4287794141

6

2.000069006

2.0069770545

2.0591364356

2.1939030837

7

2.0000086257

2.0014018838

2.0192087915

2.0866824323

8

2.0000010782

2.0002701847

2.0060472121

2.0383477219

9

2.0000001348

2.0000502881

2.0018515929

2.0168034488

10

2.0000000168

2.0000090866

2.0005529869

2.0072985299

11

2.0000000021

2.0000016005

2.0001614712

2.0031443476

12

2.0000000003

2.0000002757

2.0000461907

2.0013443922

13

 

2.0000000466

2.0000129668

2.0005707329

14

 

2.0000000077

2.0000035774

2.0002406784

15

 

2.0000000013

2.0000009712

2.0001008564

16

  

2.0000002597

2.0000420126

17

  

2.0000000685

2.0000174019

18

  

2.0000000178

2.0000071693

19

  

2.0000000046

2.0000029385

20

  

2.0000000012

2.0000011985

21

   

2.0000004865

22

   

2.0000001966

23

   

2.0000000791

24

   

2.0000000317

25

   

2.0000000127

26

   

2.000000005

CPU time

0.0013

0.0014

0.0015

0.0017

Declarations

Acknowledgements

The basic idea of this work has been given to the fourth author by Professor Mihai Postolache during his visit to University Politehnica of Bucharest in September 2014. The first and fourth authors was supported by Azarbaijan Shahid Madani University.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Azarbaijan Shahid Madani University
(2)
Department of Mathematics and Informatics, University Politehnica of Bucharest

References

  1. Plunkett, R: On the rate of convergence of relaxation methods. Q. Appl. Math. 10, 263-266 (1952) MATHMathSciNetGoogle Scholar
  2. Bowden, BV: Faster than Thought: A Symposium on Digital Computing Machines. Pitman, London (1953) MATHGoogle Scholar
  3. Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20(1), 103-120 (2004) MATHMathSciNetView ArticleGoogle Scholar
  4. Dykstra, R, Kochar, S, Robertson, T: Testing whether one risk progresses faster than the other in a competing risks problem. Stat. Decis. 14(3), 209-222 (1996) MATHMathSciNetGoogle Scholar
  5. Hajela, D: On faster than Nyquist signaling: computing the minimum distance. J. Approx. Theory 63(1), 108-120 (1990) MATHMathSciNetView ArticleGoogle Scholar
  6. Hajela, D: On faster than Nyquist signaling: further estimations on the minimum distance. SIAM J. Appl. Math. 52(3), 900-907 (1992) MATHMathSciNetView ArticleGoogle Scholar
  7. Longpre, L, Young, P: Cook reducibility is faster than Karp reducibility in NP. J. Comput. Syst. Sci. 41(3), 389-401 (1990) MATHMathSciNetView ArticleGoogle Scholar
  8. Shore, GM: Faster than light: photons in gravitational fields - causality, anomalies and horizons. Nucl. Phys. B 460(2), 379-394 (1996) MATHMathSciNetView ArticleGoogle Scholar
  9. Shore, GM: Faster than light: photons in gravitational fields. II. Dispersion and vacuum polarisation. Nucl. Phys. B 633(1-2), 271-294 (2002) MATHMathSciNetView ArticleGoogle Scholar
  10. Stark, RH: Rates of convergence in numerical solution of the diffusion equation. J. Assoc. Comput. Mach. 3, 29-40 (1956) MathSciNetView ArticleGoogle Scholar
  11. Argyros, IK: Iterations converging faster than Newton’s method to the solutions of nonlinear equations in Banach space. Ann. Univ. Sci. Bp. Rolando Eötvös Nomin., Sect. Comput. 11, 97-104 (1991) MATHGoogle Scholar
  12. Argyros, IK: Sufficient conditions for constructing methods faster than Newton’s. Appl. Math. Comput. 93, 169-181 (1998) MATHMathSciNetView ArticleGoogle Scholar
  13. Lucet, Y: Faster than the fast Legendre transform: the linear-time Legendre transform. Numer. Algorithms 16(2), 171-185 (1997) MATHMathSciNetView ArticleGoogle Scholar
  14. Berinde, V: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory Appl. 2004(2), 97-105 (2004) MATHMathSciNetView ArticleGoogle Scholar
  15. Berinde, V, Berinde, M: The fastest Krasnoselskij iteration for approximating fixed points of strictly pseudo-contractive mappings. Carpath. J. Math. 21(1-2), 13-20 (2005) MathSciNetGoogle Scholar
  16. Berinde, V: A convergence theorem for Mann iteration in the class of Zamfirescu operators. An. Univ. Vest. Timiş., Ser. Mat.-Inform. 45(1), 33-41 (2007) MATHMathSciNetGoogle Scholar
  17. Babu, GVR, Vara Prasad, KNVV: Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory Appl. 2006, Article ID 49615 (2006) MathSciNetGoogle Scholar
  18. Popescu, O: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Math. Commun. 12(2), 195-202 (2007) MATHMathSciNetGoogle Scholar
  19. Akbulut, S, Ozdemir, M: Picard iteration converges faster than Noor iteration for a class of quasi-contractive operators. Chiang Mai J. Sci. 39(4), 688-692 (2012) MATHMathSciNetGoogle Scholar
  20. Gorsoy, F, Karakaya, V: A Picard S-hybrid type iteration method for solving a differential equation with retarded argument (2014). arXiv:1403.2546v2 [math.FA]
  21. Hussain, N, Chugh, R, Kumar, V, Rafiq, A: On the rate of convergence of Kirk-type iterative schemes. J. Appl. Math. 2012, Article ID 526503 (2012) MathSciNetView ArticleGoogle Scholar
  22. Ozturk Celikler, F: Convergence analysis for a modified SP iterative method. Sci. World J. 2014, Article ID 840504 (2014) Google Scholar
  23. Thakur, D, Thakur, BS, Postolache, M: New iteration scheme for numerical reckoning fixed points of nonexpansive mappings. J. Inequal. Appl. 2014, 328 (2014) View ArticleGoogle Scholar
  24. Berinde, V: Iterative Approximation of Fixed Points. Springer, Berlin (2007) MATHGoogle Scholar
  25. Chugh, R, Kumar, S: On the rate of convergence of some new modified iterative schemes. Am. J. Comput. Math. 3, 270-290 (2013) View ArticleGoogle Scholar
  26. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953) MATHView ArticleGoogle Scholar
  27. Ishikawa, S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44, 147-150 (1974) MATHMathSciNetView ArticleGoogle Scholar
  28. Noor, MA: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217-229 (2000) MATHMathSciNetView ArticleGoogle Scholar
  29. Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8(1), 61-79 (2007) MATHMathSciNetGoogle Scholar
  30. Abbas, M, Nazir, T: A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesn. 66(2), 223-234 (2014) MathSciNetGoogle Scholar

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© Fathollahi et al. 2015