# Strong convergence theorems and rate of convergence of multi-step iterative methods for continuous mappings on an arbitrary interval

## Abstract

In this article, by using the concept of W-mapping introduced by Atsushiba and Takahashi and K-mapping introduced by Kangtunyakarn and Suantai, we define W(T,N)-iteration and K(T,N)-iteration for finding a fixed point of continuous mappings on an arbitrary interval. Then, a necessary and sufficient condition for the strong convergence of the proposed iterative methods for continuous mappings on an arbitrary interval is given. We also compare the rate of convergence of those iterations. It is proved that the W(T,N)-iteration and K(T,N)-iteration are equivalent and the K(T,N)-iteration converges faster than the W(T,N)-iteration. Moreover, we also present numerical examples for comparing the rate of convergence between W(T,N)-iteration and K(T,N)-iteration.

MSC: 26A18; 47H10; 54C05.

## 1 Introduction

There are several classical methods for approximation of solutions of nonlinear equation of one variable

$f\left(x\right)=0$
(1.1)

where f : EE is a continuous function and E is a closed interval on the real line. Classical fixed point iteration method is one of the methods used for this problem. To use this method, we have to transform (1.1) to the following equation:

$g\left(x\right)=x$
(1.2)

where g : EE is a contraction. Then, Picard's iteration can be applied for finding a solution of (1.2).

Question: If g : EE is continuous but not contraction, what iteration methods can be used for finding a solution of (1.2) (that is a fixed point of g) and how about the rate of convergence of those methods.

There are many iterative methods for finding a fixed point of g. For example, the Mann iteration (see [1]) is defined by x1 E and

${x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}g\left({x}_{n}\right)$
(1.3)

for all n ≥ 1, where ${\left\{{\alpha }_{n}\right\}}_{n=1}^{\infty }$ is a sequence in [0,1]. The Ishikawa iteration (see [2]) is

defined by x1 E and

$\left\{\begin{array}{c}{y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}g\left({x}_{n}\right)\hfill \\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}g\left({y}_{n}\right)\hfill \end{array}\right\$
(1.4)

for all n ≥ 1, where ${\left\{{\alpha }_{n}\right\}}_{n=1}^{\infty }$, ${\left\{{\beta }_{n}\right\}}_{n=1}^{\infty }$ are sequences in [0,1]. The Noor iteration (see [3]) is defined by x1 E and

$\left\{\begin{array}{c}{z}_{n}=\left(1-{\gamma }_{n}\right){x}_{n}+{\gamma }_{n}g\left({x}_{n}\right)\hfill \\ {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}g\left({z}_{n}\right)\hfill \\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}g\left({y}_{n}\right)\hfill \end{array}\right\$
(1.5)

for all n ≥ 1, where ${\left\{{\alpha }_{n}\right\}}_{n=1}^{\infty }$, ${\left\{{\beta }_{n}\right\}}_{n=1}^{\infty }$, and ${\left\{{\gamma }_{n}\right\}}_{n=1}^{\infty }$ are sequences in [0,1]. Clearly Mann and Ishikawa iterations are special cases of Noor iteration. The SP-iteration (see [4]) is defined by x1 E and

$\left\{\begin{array}{c}{z}_{n}=\left(1-{\gamma }_{n}\right){x}_{n}+{\gamma }_{n}g\left({x}_{n}\right)\hfill \\ {y}_{n}=\left(1-{\beta }_{n}\right){z}_{n}+{\beta }_{n}g\left({z}_{n}\right)\hfill \\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){y}_{n}+{\alpha }_{n}g\left({y}_{n}\right)\hfill \end{array}\right\$
(1.6)

for all n ≥ 1, where ${\left\{{\alpha }_{n}\right\}}_{n=1}^{\infty }$, ${\left\{{\beta }_{n}\right\}}_{n=1}^{\infty }$, and ${\left\{{\gamma }_{n}\right\}}_{n=1}^{\infty }$ are sequences in [0,1]. Clearly Mann iteration is special cases of SP-iteration.

In 1976, Rhoades [5] proved the convergence of the Mann and Ishikawa iterations to a solution of (1.2) when E = [0,1]. He also proved the Ishikawa iteration converges faster than the Mann iteration for the class of continuous and nondecreasing functions. Later in 1991, Borwein and Borwein [6] proved the convergence of the Mann iteration of continuous functions on a bounded closed interval. In 2006, Qing and Qihou [7] extended their results to an arbitrary interval and to the Ishikawa iteration and gave some control conditions for the convergence of Ishikawa iteration on an arbitrary interval. Recently, Phuengrattana and Suantai [4] obtained a similar result for the new iteration, called the SP-iteration, and they proved the Mann, Ishikawa, Noor and SP-iterations are equivalent and the SP-iteration converges faster than the others for the class of continuous and nondecreasing functions.

In this article, we are interested to employ the concept of W-mappings and K-mappings for approximation of a solution of (1.2) for a continuous function on an arbitrary interval and compare which one converges faster. The concept of W-mapping was first introduced by Atsushiba and Takahashi [8]. They defined W-mapping as follows. Let C be a subset of a Banach space X and T : CC be a mapping. A point x C is a fixed point of T if Tx = x. The set of all fixed points of T is denoted by F(T). Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of mappings of C into itself. Let W n : CC be a mapping defined by

$\begin{array}{ll}\hfill {S}_{n,0}& =I,\phantom{\rule{2em}{0ex}}\\ \hfill {S}_{n,1}& ={\lambda }_{n,1}{T}_{1}{S}_{n,0}+\left(1-{\lambda }_{n,1}\right)I,\phantom{\rule{2em}{0ex}}\\ \hfill {S}_{n,2}& ={\lambda }_{n,2}{T}_{2}{S}_{n,1}+\left(1-{\lambda }_{n,2}\right)I,\phantom{\rule{2em}{0ex}}\\ ⋮\phantom{\rule{2em}{0ex}}\\ \hfill {S}_{n,N-1}& ={\lambda }_{n,N-1}{T}_{N-1}{S}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right)I,\phantom{\rule{2em}{0ex}}\\ \hfill {W}_{n}={S}_{n,N}& ={\lambda }_{n,N}{T}_{N}{S}_{n,N-1}+\left(1-{\lambda }_{n,N}\right)I,\phantom{\rule{2em}{0ex}}\end{array}$
(1.7)

where I is the identity mapping of C and λn,i [0,1] for all i = 1, 2,..., N. Such a mapping W n is called the W-mapping generated by T1, T2,..., T n and λn,1, λn,2,..., λn,N. Many researchers have studied and applied this mapping for finding a common fixed point of nonexpansive mappings, for instance, see [823].

In 2009, Kangtunyakarn and Suantai [24] introduced a new concept of the K-mapping in a Banach space as follows. Let K n : CC be a mapping defined by

$\begin{array}{ll}\hfill {U}_{n,0}& =I,\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{n,1}& ={\lambda }_{n,1}{T}_{1}{U}_{n,0}+\left(1-{\lambda }_{n,1}\right){U}_{n,0},\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{n,2}& ={\lambda }_{n,2}{T}_{2}{U}_{n,1}+\left(1-{\lambda }_{n,2}\right){U}_{n,1},\phantom{\rule{2em}{0ex}}\\ ⋮\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{n,N-1}& ={\lambda }_{n,N-1}{T}_{N-1}{U}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right){U}_{n,N-2},\phantom{\rule{2em}{0ex}}\\ \hfill {K}_{n}={U}_{n,N}& ={\lambda }_{n,N}{T}_{N}{U}_{n,N-1}+\left(1-{\lambda }_{n,N}\right){U}_{n,N-1},\phantom{\rule{2em}{0ex}}\end{array}$
(1.8)

where I is the identity mapping of C and λn,i [0,1] for all i = 1, 2,..., N. Such a mapping K n is called the K-mapping generated by T1,T2,..., T n and λn,1, λn,2,..., λn,N. They showed that if C is a nonempty closed convex subset of a strictly convex Banach space X and ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ is a finite family of nonexpansive mappings of C into itself, then $F\left({K}_{n}\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$ and they also introduced an iterative method by using the concept of K-mapping for finding a common fixed point of a finite family of nonexpansive mappings and a solution of an equilibrium problem. Applications of K-mappings for fixed point problems and equilibrium problems can be found in [2326].

By using the concept of W-mappings and K-mappings, we introduce two new iterations for finding a fixed point of a mapping T : EE on an arbitrary interval E as follows.

The W(T,N)-iteration is defined by u1 E and

${u}_{n+1}={W}_{n}^{\left(T,N\right)}{u}_{n}\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 1,$
(1.9)

where N ≥ 1 and ${W}_{n}^{\left(T,N\right)}$ is a mapping of E into itself generated by

$\begin{array}{ll}\hfill {S}_{n,0}& =I,\phantom{\rule{2em}{0ex}}\\ \hfill {S}_{n,1}& ={\lambda }_{n,1}T{S}_{n,0}+\left(1-{\lambda }_{n,1}\right)I,\phantom{\rule{2em}{0ex}}\\ \hfill {S}_{n,2}& ={\lambda }_{n,2}T{S}_{n,1}+\left(1-{\lambda }_{n,2}\right)I,\phantom{\rule{2em}{0ex}}\\ ⋮\phantom{\rule{2em}{0ex}}\\ \hfill {S}_{n,N-1}& ={\lambda }_{n,N-1}T{S}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right)I,\phantom{\rule{2em}{0ex}}\\ \hfill {W}_{n}^{\left(T,N\right)}={S}_{n,N}& ={\lambda }_{n,N}T{S}_{n,N-1}+\left(1-{\lambda }_{n,N}\right)I,\phantom{\rule{2em}{0ex}}\end{array}$
(1.10)

where I is the identity mapping of E and λn,i [0,1] for all i = 1, 2,..., N. We call a mapping ${W}_{n}^{\left(T,N\right)}$ as the W-mapping generated by T and λn,1, λn,2,..., λn,N. Clearly W(T,1)-iteration is Mann iteration, W(T,2)-iteration is Ishikawa iteration and W(T,3)-iteration is Noor iteration.

The K(T,N)-iteration is defined by x1 E and

${x}_{n+1}={K}_{n}^{\left(T,N\right)}{x}_{n}\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 1,$
(1.11)

where N ≥ 1 and ${K}_{n}^{\left(T,N\right)}$ is a mapping of E into itself generated by

$\begin{array}{ll}\hfill {U}_{n,0}& =I,\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{n,1}& ={\lambda }_{n,1}T{U}_{n,0}+\left(1-{\lambda }_{n,1}\right){U}_{n,0},\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{n,2}& ={\lambda }_{n,2}T{U}_{n,1}+\left(1-{\lambda }_{n,2}\right){U}_{n,1},\phantom{\rule{2em}{0ex}}\\ ⋮\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{n,N-1}& ={\lambda }_{n,N-1}T{U}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right){U}_{n,N-2},\phantom{\rule{2em}{0ex}}\\ \hfill {K}_{n}^{\left(T,N\right)}={U}_{n,N}& ={\lambda }_{n,N}T{U}_{n,N-1}+\left(1-{\lambda }_{n,N}\right){U}_{n,N-1},\phantom{\rule{2em}{0ex}}\end{array}$
(1.12)

where I is the identity mapping of E and λn,i [0,1] for all i = 1, 2,..., N. We call a mapping ${K}_{n}^{\left(T,N\right)}$ as the K-mapping generated by T and λn,1,λn,2, ..., λn,N. Clearly K(T,1)-iteration is Mann iteration and K(T,3)-iteration is SP-iteration.

Obviously the mappings (1.10) and (1.12) are special cases of the W-mapping and K-mapping, respectively.

The purpose of this article is to give a necessary and sufficient condition for the strong convergence of the W(T,N)-iteration and K(T,N)-iteration of continuous mappings on an arbitrary interval. We also prove that the K(T,N)-iteration and W(T,N)-iteration are equivalent and the K(T,N)-iteration converges faster than the W(T,N)-iteration for the class of continuous and nondecreasing mappings. Moreover, we present numerical examples for the K(T,N)-iteration to compare with the W(T,N)-iteration. Our results extend and improve the corresponding results of Rhoades [5], Borwein and Borwein [6], Qing and Qihou [7], Phuengrattana and Suantai [4], and many others.

## 2 Convergence theorems

We first give a convergence theorem for the K(T,N)-iteration for continuous mappings on an arbitrary interval.

Theorem 2.1 Let E be a closed interval on the real line and T : EE be a continuous mapping. For x1 E, let the K(T,N)-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$defined by (1.11), where${\left\{{\lambda }_{n,i}\right\}}_{n=1}^{\infty }$ (i = 1, 2, ..., N) are sequences in [0,1] satisfying the following conditions:

(C1)${\sum }_{n=1}^{\infty }{\lambda }_{n,i}<\infty$for all i = 1, 2,..., N - 1;

(C2) limn→∞λn,N= 0 and${\sum }_{n=1}^{\infty }{\lambda }_{n,N}=\infty$.

Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to a fixed point of T.

Proof. It is obvious that if ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges to a fixed point of T, then it is bounded. Now, assume that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ is bounded. We will show that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges to a fixed point of T. First, we show that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ is convergent. To show this, we suppose not. Then there exist a, b , a = lim infn→∞x n , b = lim supn→∞x n and a < b.

Next, we show that

$\text{if}\phantom{\rule{2.77695pt}{0ex}}m\in \left(a,b\right),\phantom{\rule{2.77695pt}{0ex}}\text{then}\phantom{\rule{2.77695pt}{0ex}}Tm=m.$
(2.1)

To show this, suppose that Tmm for some m (a,b). Without loss of generality, we may assume that Tm - m > 0. By continuity of T, there exists δ (0, b - a) such that

$Tx-x>0\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\left|x-m\right|\le \delta .$
(2.2)

By boundedness of ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$, we have ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ belongs to a bounded closed interval. Continuity of T implies that ${\left\{T{x}_{n}\right\}}_{n=1}^{\infty }$ belongs to another bounded closed interval, so ${\left\{T{x}_{n}\right\}}_{n=1}^{\infty }$ is bounded. Since Un,1x n = λn,1Tx n + (1- λn,1)x n , we get ${\left\{{U}_{n,1}{x}_{n}\right\}}_{n=1}^{\infty }$ is bounded, and thus ${\left\{T{U}_{n,1}{x}_{n}\right\}}_{n=1}^{\infty }$ is bounded. Similarly, by using (1.11), we have ${\left\{{U}_{n,i}{x}_{n}\right\}}_{n=1}^{\infty }$ and ${\left\{T{U}_{n,i}{x}_{n}\right\}}_{n=1}^{\infty }$ are bounded for all i = 2, 3,..., N - 1. It follows by (1.11) that Un,ix n - Un,i-1x n = λn,i(TUn,i-1x n - Un,i-1x n ) for all i = 1,2,..., N. By condition (C 1) and (C 2), we get limn→∞|Un,ix n - Un,i-1x n |=0 for all i = 1, 2,..., N.

Since

$\begin{array}{ll}\hfill \left|{x}_{n+1}-{x}_{n}\right|& =\left|{U}_{n,N}{x}_{n}-{U}_{n,0}{x}_{n}\right|\phantom{\rule{2em}{0ex}}\\ \le \left|{x}_{n+1}-{U}_{n,N-1}{x}_{n}\right|+\left|{U}_{n,N-1}{x}_{n}-{U}_{n,N-2}{x}_{n}\right|+\cdots +\left|{U}_{n,1}{x}_{n}-{U}_{n,0}{x}_{n}\right|,\phantom{\rule{2em}{0ex}}\end{array}$

it implies that limn→∞|xn+1- x n | = 0. Thus, there exists M0 such that

$\left|{x}_{n+1}-{x}_{n}\right|<\frac{\delta }{N}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\left|{U}_{n,i}{x}_{n}-{U}_{n,i-1}{x}_{n}\right|<\frac{\delta }{N}\left(i=1,2,...,N-1\right),$
(2.3)

for all n > M0. Since b = lim supn→∞x n > m, there exists k1 > M0 such that ${x}_{{k}_{1}}>m$. Let k = k1, then x k > m. If ${x}_{k}\ge m+\frac{\delta }{N}$, then by (2.3), we have ${x}_{k+1}>{x}_{k}-\frac{\delta }{N}\ge m$, so xk+1> m. If ${x}_{k}\in \left(m,m+\frac{\delta }{N}\right)$, then by (2.3), we have

$m-\frac{\delta }{N}i<{U}_{k,i}{x}_{k}

So we have

$\left|{x}_{k}-m\right|<\delta \phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\left|{U}_{k,i}{x}_{k}-m\right|<\delta \phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}i=1,2,...,N-1.$

This implies by (2.2) that

$T{x}_{k}-{x}_{k}>0\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}T{U}_{k,i}{x}_{k}-{U}_{k,i}{x}_{k}>0\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}i=1,2,...,N-1.$
(2.4)

Using (1.11), we obtain

$\begin{array}{ll}\hfill {x}_{k+1}& ={\lambda }_{k,N}T{U}_{k,N-1}{x}_{k}+\left(1-{\lambda }_{k,N}\right){U}_{k,N-1}{x}_{k}\phantom{\rule{2em}{0ex}}\\ ={U}_{k,N-1}{x}_{k}+{\lambda }_{k,N}\left(T{U}_{k,N-1}{x}_{k}-{U}_{k,N-1}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ ={U}_{k,N-2}{x}_{k}+{\lambda }_{k,N-1}\left(T{U}_{k,N-2}{x}_{k}-{U}_{k,N-2}{x}_{k}\right)+{\lambda }_{k,N}\left(T{U}_{k,N-1}{x}_{k}-{U}_{k,N-1}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ ⋮\phantom{\rule{2em}{0ex}}\\ ={x}_{k}+\sum _{i=1}^{N}{\lambda }_{k,i}\left(T{U}_{k,i-1}{x}_{k}-{U}_{k,i-1}{x}_{k}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(2.5)

By (2.4), we have xk+1> x k . Thus, xk+1> m.

By using the above argument, we obtain xk+j> m for all j ≥ 2. Thus we get x n > m for all n > k. So a = lim infn→∞x n m, which is a contradiction with a < m. Thus Tm = m. Therefore, we obtain (2.1).

For the sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$, we consider the following two cases:

Case 1: There exists ${x}_{\stackrel{̄}{M}}$ such that $a<{x}_{\stackrel{̄}{M}}. Then $T{x}_{\stackrel{̄}{M}}={x}_{\stackrel{̄}{M}}$. By using (1.11), we obtain that ${U}_{\stackrel{̄}{M},i}{x}_{\stackrel{̄}{M}}={x}_{\stackrel{̄}{M}}$ for all i = 1, 2,..., N. Thus, we have ${x}_{\stackrel{̄}{M}+1}={x}_{\stackrel{̄}{M}}$. By induction, we obtain ${x}_{\stackrel{̄}{M}}={x}_{\stackrel{̄}{M}+1}={x}_{\stackrel{̄}{M}+2}=...$, so ${x}_{n}\to {x}_{\stackrel{̄}{M}}$. This implies that ${x}_{\stackrel{̄}{M}}=a$ and x n a, which contradicts with our assumption.

Case 2: For all n, x n a or x n b. Because b - a > 0 and limn→∞|xn+1- x n | = 0, there exists M1 such that $\left|{x}_{n+1}-{x}_{n}\right|<\frac{b-a}{N}$ for all n > M1. It implies that either x n a for all n > M1 or x n b for all n > M1. If x n a for n > M1, then b = lim supn→∞x n a, which is a contradiction with a < b. If x n b for n > M1, so we have a = lim infn→∞x n b, which is a contradiction with a < b.

Hence, we have ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ is convergent.

Finally, we show that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges to a fixed point of T. Let limn→∞, x n = p and suppose Tpp. Since ${\left\{{U}_{n,i}{x}_{n}\right\}}_{n=1}^{\infty }$ is bounded for all i = 1, 2,..., N - 1, it implies by (1.11), condition (C 1) and (C 2) that limn→∞Un,ix n = p for all i = 1, 2,..., N - 1. Let hk,i= TUk,i-1x k - Uk,i-1x k for all i = 1, 2,..., N. Continuity of T implies that limk→∞hk,i= Tp - p ≠ 0 for all i = 1, 2,..., N. Put w = Tp - p. Then w ≠ 0. By (2.5), we have

$\sum _{k=1}^{n-1}\left({x}_{k+1}-{x}_{k}\right)=\sum _{k=1}^{n-1}\left({\lambda }_{k,1}{h}_{k,1}+{\lambda }_{k,2}{h}_{k,2}+\cdots +{\lambda }_{k,N}{h}_{k,N}\right).$

This implies that

${x}_{n}={x}_{1}+\sum _{k=1}^{n-1}\left({\lambda }_{k,1}{h}_{k,1}+{\lambda }_{k,2}{h}_{k,2}+\cdots +{\lambda }_{k,N}{h}_{k,N}\right).$
(2.6)

By condition (C 1), (C 2), and limk→∞hk,i= w ≠ 0 for all i = 1, 2,..., N, we get that ${\sum }_{k=1}^{\infty }{\lambda }_{k,i}{h}_{k,i}$ is convergent for all i = 1, 2,..., N - 1 and ${\sum }_{k=1}^{\infty }{\lambda }_{k,N}{h}_{k,N}$ is divergent. It follows by (2.6) that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ is divergent, which is a contradiction. Hence, ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges to a fixed point of T.

We now obtain the convergence theorem of W(T,N)-iteration. The proof is omitted because it is similar as above theorem and Theorem 2.2 of [4].

Theorem 2.2 Let E be a closed interval on the real line and T : EE be a continuous mapping. For x1 E, let the W(T,N)-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$defined by (1.9), where${\left\{{\lambda }_{n,1}\right\}}_{n=1}^{\infty },{\left\{{\lambda }_{n,2}\right\}}_{n=1}^{\infty }$ (i = 1,2,...,N) are sequences in [0,1] satisfying the following conditions:

(C1) limn→∞λn,i= 0 for all i = 1,2,..., N;

(C2)${\sum }_{n=1}^{\infty }{\lambda }_{n,N}=\infty$.

Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to a fixed point of T.

The following results are obtained direclty from Theorem 2.1.

Corollary 2.3 ([4, Theorem 2.1]) Let E be a closed interval on the real line and T : EE be a continuous mapping. For x1 E, let the SP-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$defined by (1.6), where${\left\{{\lambda }_{n,1}\right\}}_{n=1}^{\infty }$, ${\left\{{\lambda }_{n,2}\right\}}_{n=1}^{\infty }$, and${\left\{{\lambda }_{n,3}\right\}}_{n=1}^{\infty }$are sequences in [0,1] satisfying the following conditions:

(C1)${\sum }_{n=1}^{\infty }{\lambda }_{n,1}<\infty$and${\sum }_{n=1}^{\infty }{\lambda }_{n,2}<\infty$;

(C2) limn→∞λn,3= 0 and${\sum }_{n=1}^{\infty }{\lambda }_{n,3}=\infty$.

Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to a fixed point of T.

Corollary 2.4 ([7, Theorem 3]) Let E be a closed interval on the real line and T : EE be a continuous mapping. For x1 E, let the Mann iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$defined by (1.3), where${\left\{{\lambda }_{n,1}\right\}}_{n=1}^{\infty }$is a sequence in [0,1] satisfying limn→∞, λn,1= 0 and${\sum }_{n=1}^{\infty }{\lambda }_{n,1}=\infty$. Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to a fixed point of T.

The following results are obtained directly from Theorem 2.2.

Corollary 2.5 ([4, Theorem 2.2]) Let E be a closed interval on the real line and T : EE be a continuous mapping. For x1 E, let the Noor iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$defined by (1.5), where${\left\{{\lambda }_{n,1}\right\}}_{n=1}^{\infty }$, ${\left\{{\lambda }_{n,2}\right\}}_{n=1}^{\infty }$, ${\left\{{\lambda }_{n,3}\right\}}_{n=1}^{\infty }$are sequences in [0,1] satisfying the following conditions:

(C1) limn→∞λn,1= 0, limn→∞λn,2= 0 and limn→∞λn,3= 0;

(C2)${\sum }_{n=1}^{\infty }{\lambda }_{n,3}=\infty$.

Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to a fixed point of T.

Corollary 2.6 ([7]) Let E be a closed interval on the real line and T : EE be a continuous mapping. For x1 E, let the Ishikawa iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$defined by (1.4), where${\left\{{\lambda }_{n,1}\right\}}_{n=1}^{\infty }$are sequences in [0,1] satisfying the following conditions:

(C1) limn→∞λn,1= 0 and limn→∞λn,2= 0;

(C2)${\sum }_{n=1}^{\infty }{\lambda }_{n,2}=\infty$.

Then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$is bounded if and only if${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to a fixed point of T.

## 3 Rate of convergence and numerical examples

There are many articles have been published on the iterative methods using for approximation of fixed points of nonlinear mappings, see for instance [17]. However, there are only a few articles concerning comparison of those iterative methods in order to establish which one converges faster. As far as we know, there are two ways for comparison of the rate of convergence. The first one was introduced by Berinde [27]. He used this idea to compare the rate of convergence of Picard and Mann iterations for a class of Zamfirescu operators in arbitrary Banach spaces. Popescu [28] also used this concept to compare the rate of convergence of Picard and Mann iterations for a class of quasi-contractive operators. It was shown in [29] that the Mann and Ishikawa iterations are equivalent for the class of Zamfirescu operators. In 2006, Babu and Prasad [30] showed that the Mann iteration converges faster than the Ishikawa iteration for this class of operators. Two years later, Qing and Rhoades [31] provided an example to show that the claim of Babu and Prasad [30] is false.

However, this concept is not suitable or cannot be applied to a class of continuous self-mappings defined on a closed interval. In order to compare the rate of convergence of continuous self-mappings defined on a closed interval, Rhoades [5] introduced the other concept which is slightly different from that of Berinde to compare iterative methods which one converges faster as follows.

Definition 3.1 Let E be a closed interval on the real line and T : EE be a continuous mapping. Suppose that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ and ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$ are two iterations which converge to the fixed point p of T. We say that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges faster than ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$ if

$\left|{x}_{n}-p\right|\le \left|{u}_{n}-p\right|\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}n\ge 1.$

In this section, we study the rate of convergence of W(T,N)-iteration and K(T,N)-iteration for continuous and nondecreasing mappings on an arbitrary interval in the sense of Rhoades. The following lemmas are useful and crucial for our following results.

Lemma 3.2 Let E be a closed interval on the real line and T : EE be a continuous and nondecreasing mapping such that F(T) is nonempty and bounded with x1 > sup{p E : p = Tp}. Let${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$be defined by W(T,N)-iteration or K(T,N)-iteration. If Tx1 > x1, then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$does not converge to a fixed point of T.

Proof. We prove only the case that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ is defined by K(T,N)-iteration because the other case can be proved similarly.

Let Tx1 > x1. Since x1 > sup{p E : p = Tp} and by using (1.11) and mathematical induction, we can show that x n ≥ sup{p E : p = Tp} for all n ≥ 1. It is clear that Tx n x n for all n ≥ 1. Using (1.11), we have

${U}_{n,1}{x}_{n}={\lambda }_{n,1}T{x}_{n}+\left(1-{\lambda }_{n,1}\right){x}_{n}\ge {x}_{n}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}n\ge 1.$

Since T is nondecreasing, we have TUn,1x n Tx n x n . Using (1.11) again, we have

${U}_{n,2}{x}_{n}={\lambda }_{n,2}T{U}_{n,1}{x}_{n}+\left(1-{\lambda }_{n,2}\right){U}_{n,1}{x}_{n}\ge {x}_{n}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}n\ge 1.$

This implies that TUn,2x n Tx n x n . By continuity in this way, we can show that ${x}_{n+1}={K}_{n}^{\left(T,N\right)}{x}_{n}={U}_{n,N}{x}_{n}\ge {x}_{n}$ for all n ≥ 1. Thus ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ is nondecreasing. But x1 > sup{p E : p = Tp}, it implies that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ does not converges to a fixed point of T.

By using the same argument of proof as in above lemma, we get the following result.

Lemma 3.3 Let E be a closed interval on the real line and T : EE be a continuous and nondecreasing mapping such that F(T) is nonempty and bounded with x1 < inf{p E : p = Tp}. Let${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$be defined by W(T,N)-iteration or K(T,N)-iteration. If Tx1 < x1, then${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$does not converge to a fixed point of T.

We now get the following theorem for compare rate of convergence between W(T,N)-iteration and K(T,N)-iteration.

Theorem 3.4 Let E be a closed interval on the real line and T : EE be a continuous and nondecreasing mapping such that F(T) is nonempty and bounded. For u1 = x1 E, let${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$and${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$are the sequences defined by (1.9) and (1.11), respectively. Let${\left\{{\lambda }_{n,i}\right\}}_{n=1}^{\infty }$be sequences in [0,1) for all i = 1,2,..., N. Then, the W(T,N)-iteration${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$converges to the fixed point p of T if and only if the K(T,N)-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to p . Moreover, the K(T,N)-iteration converges faster than the W(T,N)-iteration.

Proof. Put L = inf{p E : p = Tp} and U = sup{p E : p = Tp}.

() Suppose that the W(T,N)-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$ converges to the fixed point p of T.

We divide our proof into the following three cases:

Case 1: u1 = x1 > U. By Lemma 3.2, we have Tu1 < u1 and Tx1 < x1. We now show that x n u n for all n ≥ 1. Assume that x k u k . Thus, Tx k Tu k . Since x1 > U and by using (1.11) and mathematical induction, we can show that x n U for all n ≥ 1. It is clear that Tx k x k . This implies that Tx k Uk,1x k x k . Since T is nondecreasing, TUk,1x k Tx k . Thus, we have

$T{U}_{k,1}{x}_{k}\le {U}_{k,2}{x}_{k}\le {U}_{k,1}{x}_{k}.$
(3.1)

It follows that Uk,2x k x k . By (3.1) and T is nondecreasing, we have TUk,2x k TUk,1x k Uk,2x k . This implies that

$T{U}_{k,2}{x}_{k}\le {U}_{k,3}{x}_{k}\le {U}_{k,2}{x}_{k}.$

Thus, we have Uk,3x k x k . By continuity in this way, we can show that

${U}_{k,i}{x}_{k}\le {x}_{k}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}i=1,2,...,N.$

Using (1.9) and (1.11), we get

${U}_{k,1}{x}_{k}-{S}_{k,1}{u}_{k}={\lambda }_{k,1}\left({x}_{k}-{u}_{k}\right)+\left(1-{\lambda }_{k,1}\right)\left(T{x}_{k}-T{u}_{k}\right)\le 0.$

Since T is nondecreasing, we have TUk,1x k TSk,1u k . It follows that

$\begin{array}{ll}\hfill {U}_{k,2}{x}_{k}-{S}_{k,2}{u}_{k}& ={\lambda }_{k,2}\left({U}_{k,1}{x}_{k}-{u}_{k}\right)+\left(1-{\lambda }_{k,2}\right)\left(T{U}_{k,1}{x}_{k}-T{S}_{k,1}{u}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le {\lambda }_{k,2}\left({U}_{k,1}{x}_{k}-{x}_{k}\right)+\left(1-{\lambda }_{k,2}\right)\left(T{U}_{k,1}{x}_{k}-T{S}_{k,1}{u}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le 0.\phantom{\rule{2em}{0ex}}\end{array}$

That is Uk,2x k Sk,2u k . Since T is nondecreasing, we have TUk,2x k TSk,2u k . This implies that

$\begin{array}{ll}\hfill {U}_{k,3}{x}_{k}-{S}_{k,3}{u}_{k}& ={\lambda }_{k,3}\left({U}_{k,2}{x}_{k}-{u}_{k}\right)+\left(1-{\lambda }_{k,3}\right)\left(T{U}_{k,2}{x}_{k}-T{S}_{k,2}{u}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le {\lambda }_{k,3}\left({U}_{k,2}{x}_{k}-{x}_{k}\right)+\left(1-{\lambda }_{k,3}\right)\left(T{U}_{k,2}{x}_{k}-T{S}_{k,2}{u}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le 0.\phantom{\rule{2em}{0ex}}\end{array}$

That is Uk,3x k Sk,3u k . By continuity in this way we can show that Uk,Nx k Sk,Nu k . Thus, xk+1uk+1. Hence, by mathematical induction, we obtain x n u n for all n ≥ 1. By x n U for all n ≥ 1, we get 0 ≤ x n - pu n - p, so

$\left|{x}_{n}-p\right|\le \left|{u}_{n}-p\right|\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}n\ge 1.$
(3.2)

Since limn→∞, u n = p, it implies that limn→∞x n = p. That is, the K(T,N)-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges to the same fixed point p. Moreover, by (3.2), we see that the K(T,N)-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges faster than the W(T,N)-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$.

Case 2: u1 = x1 < L. By Lemma 3.3, we have Tu1 > u1 and Tx1 > x1. By using (1.9), (1.11) and the same argument as in Case 1, we can show that x n u n for all n ≥ 1. We note that x1 < L and by using (1.11) and mathematical induction, we can show that x n L for all n ≥ 1. Thus, we have |x n - p| ≤ |u n - p| for all n ≥ 1. It follows that limn→∞x n = p and the K(T,N)-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges faster than the W(T,N)-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$.

Case 3: Lu1 = x1U. Suppose that Tu1u1. Without loss of generality, we suppose Tu1 < u1. It follows by (1.9) that u n u1 for all n ≥ 1. Since limn→∞u n = p, we must get p < u1 = x1. By the same argument as in Case 1, we have px n u n for all n ≥ 1. It follows that |x n - p| ≤ |u n - p| for all n ≥ 1. Hence, limn→∞x n = p and the K(T,N)-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges faster than the W(T,N)-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$.

() Suppose that the K(T,N)-iteration ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges to the fixed point p of T. Put λn,i= 0 for all i = 1, 2,..., N - 1 and n ≥ 1, we get the sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ generated by

${x}_{n+1}={\lambda }_{n,N}T{x}_{n}+\left(1-{\lambda }_{n,N}\right){x}_{n}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}n\ge 1$
(3.3)

that converges to p. We will show that W(T,N)-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$ converges to p. We shall prove only the case x1 = u1 > U, because other cases can be proved similarly as the first part. By Proposition 3.5 in [4], we get Tx1 < x1 and Tu1 < u1. Assume that u k x k . Thus Tu k Tx k . Since u1 > U and by using (1.9) and mathematical induction, we can show that u n U for all n ≥ 1. It is clear that Tu k u k . This implies that Tu k Sk,1u k u k . Since T is nondecreasing, TSk,1u k Tu k Sk,1u k . Thus, TSk,1u k u k x k . It follows that TSk,1u k Sk,2u k u k . Since T is nondecreasing, TSk,2u k Tu k Sk,1u k . Thus, TSk,2u k u k x k . By continuity in this way, we have TSk,iu k x k for all i = 1, 2,..., N. By (1.9) and (3.3), we obtain

${S}_{k,i}{u}_{k}-{x}_{k}={\lambda }_{k,i}\left({u}_{k}-{x}_{k}\right)+\left(1-{\lambda }_{k,i}\right)\left(T{S}_{k,i-1}{u}_{k}-{x}_{k}\right)\le 0,$

for all i = 2, 3,..., N - 1. Since T is nondecreasing, we have

$T{S}_{k,i}{u}_{k}\le T{x}_{k}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}i=2,3,...,N-1.$

It follows by (1.9) and (3.3) that

${u}_{k+1}-{x}_{k+1}={\lambda }_{k,N}\left({u}_{k}-{x}_{k}\right)+\left(1-{\lambda }_{k,N}\right)\left(T{S}_{k,N-1}{u}_{k}-T{x}_{k}\right)\le 0.$

By mathematical induction, we have u n x n for all n ≥ 1. We note that x1 > U and by using (3.3) and mathematical induction, we can show that x n U for all n ≥ 1. Thus, we have 0 ≤ u n - px n - p for all n ≥ 1. Since limn→∞x n = p, it follows that limn→∞u n = p That is, the W(T,N)-iteration ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$ converges to the same fixed point p.

We also consider the speed of convergence of the K(T,N)-iteration which depends on the choice of control sequences ${\left\{{\lambda }_{n,i}\right\}}_{n=1}^{\infty }$ (i = 1,2,..., N) as the following theorem.

Theorem 3.5 Let E be a closed interval on the real line and T : EE be a continuous and nondecreasing mapping such that F(T) is nonempty and bounded. Let${\left\{{\lambda }_{n,i}\right\}}_{n=1}^{\infty }$, ${\left\{{\lambda }_{n,i}^{*}\right\}}_{n=1}^{\infty }$are the sequences in [0,1) such that${\lambda }_{n,i}\le {\lambda }_{n,i}^{*}$for all i = 1,2,..., N. Let${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$be a sequence defined by x1 E and

${x}_{n+1}={K}_{n}^{\left(T,N\right)}{x}_{n}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 1,$
(3.4)

where${K}_{n}^{\left(T,N\right)}$is the K-mapping generated by T and λn,1, λn,2,..., λn,N, and${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty }$be a sequence defined by${x}_{1}^{*}={x}_{1}\in E$and

${x}_{n+1}^{*}={\stackrel{̄}{K}}_{n}^{\left(T,N\right)}{x}_{n}^{*}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 1,$
(3.5)

where${\stackrel{̄}{K}}_{n}^{\left(T,N\right)}$is the K-mapping generated by T and${\lambda }_{n,1}^{*},{\lambda }_{n,2}^{*},...,{\lambda }_{n,N}^{*}$.

If${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$converges to the fixed point p of T, then${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty }$converges to p. Moreover, ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty }$converges faster than${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$.

Proof. Put L = inf{p E : p = Tp} and U = sup{p E : p = Tp}. Suppose that ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges to a fixed point p of T. We divide our proof into the following three cases:

Case 1: ${x}_{1}^{*}={x}_{1}>U$. By Lemma 3.2, we have $T{x}_{1}^{*}<{x}_{1}^{*}$ and Tx1 < x1. Assume that ${x}_{k}^{*}\le {x}_{k}$. Thus, $T{x}_{k}^{*}\le T{x}_{k}^{*}$. Since ${x}_{1}^{*}>U$ and by using (3.5) and mathematical induction, we can show that ${x}_{n}^{*}\ge U$ for all n ≥ 1. It is clear that $T{x}_{k}^{*}\le {x}_{k}^{*}$. This implies that $T{x}_{k}^{*}\le {U}_{k,1}{x}_{k}^{*}\le {x}_{k}^{*}$. Since T is nondecreasing, $T{U}_{k,1}{x}_{k}^{*}\le T{x}_{k}^{*}$. Thus, we have

$T{U}_{k,1}{x}_{k}^{*}\le {U}_{k,2}{x}_{k}^{*}\le {U}_{k,1}{x}_{k}^{*}.$

It follows that $T{U}_{k,2}{x}_{k}^{*}\le T{U}_{k,1}{x}_{k}^{*}\le {U}_{k,2}{x}_{k}^{*}$. This implies that

$T{U}_{k,2}{x}_{k}^{*}\le {U}_{k,3}{x}_{k}^{*}\le {U}_{k,2}{x}_{k}^{*}.$

By continuity in this way, we can show that

$T{U}_{k,i}{x}_{k}^{*}\le {U}_{k,i}{x}_{k}^{*}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}i=0,1,...,N.$
(3.6)

Using (3.4), (3.5), and (3.6), we have

$\begin{array}{ll}\hfill {U}_{k,1}{x}_{k}^{*}-{U}_{k,1}{x}_{k}& =\left({U}_{k,0}{x}_{k}^{*}-{U}_{k,0}{x}_{k}\right)+{\lambda }_{k,1}^{*}\left(T{U}_{k,0}{x}_{k}^{*}-{U}_{k,0}{x}_{k}^{*}\right)+{\lambda }_{k,1}\left({U}_{k,0}{x}_{k}-T{U}_{k,0}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le \left({U}_{k,0}{x}_{k}^{*}-{U}_{k,0}{x}_{k}\right)+{\lambda }_{k,1}^{*}\left(T{U}_{k,0}{x}_{k}^{*}-{U}_{k,0}{x}_{k}^{*}\right)+{\lambda }_{k,1}^{*}\left({U}_{k,0}{x}_{k}-T{U}_{k,0}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =\left(1-{\lambda }_{k,1}^{*}\right)\left({U}_{k,0}{x}_{k}^{*}-{U}_{k,0}{x}_{k}\right)+{\lambda }_{k,1}^{*}\left(T{U}_{k,0}{x}_{k}^{*}-T{U}_{k,0}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le 0.\phantom{\rule{2em}{0ex}}\end{array}$

This implies $T{U}_{k,1}{x}_{k}^{*}\le T{U}_{k,1}{x}_{k}$. It follows that

$\begin{array}{ll}\hfill {U}_{k,2}{x}_{k}^{*}-{U}_{k,2}{x}_{k}& =\left({U}_{k,1}{x}_{k}^{*}-{U}_{k,1}{x}_{k}\right)+{\lambda }_{k,2}^{*}\left(T{U}_{k,1}{x}_{k}^{*}-{U}_{k,1}{x}_{k}^{*}\right)+{\lambda }_{k,2}\left({U}_{k,1}{x}_{k}-T{U}_{k,1}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le \left({U}_{k,1}{x}_{k}^{*}-{U}_{k,1}{x}_{k}\right)+{\lambda }_{k,2}^{*}\left(T{U}_{k,1}{x}_{k}^{*}-{U}_{k,1}{x}_{k}^{*}\right)+{\lambda }_{k,2}^{*}\left({U}_{k,1}{x}_{k}-T{U}_{k,1}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =\left(1-{\lambda }_{k,2}^{*}\right)\left({U}_{k,1}{x}_{k}^{*}-{U}_{k,1}{x}_{k}\right)+{\lambda }_{k,2}^{*}\left(T{U}_{k,1}{x}_{k}^{*}-T{U}_{k,1}{x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le 0.\phantom{\rule{2em}{0ex}}\end{array}$

By continuity in this way, we can show that

${\stackrel{̄}{K}}_{k}^{\left(T,N\right)}{x}_{k}^{*}-{K}_{k}^{\left(T,N\right)}{x}_{k}={U}_{k,N}{x}_{k}^{*}-{U}_{k,N}{x}_{K}\le 0.$

That is, ${x}_{k+1}^{*}\le {x}_{k+1}$. By mathematical induction, we obtain ${x}_{n}^{*}\le {x}_{n}$ for all n ≥ 1. Since ${x}_{n}^{*}\ge U$ for all n ≥ 1, we get $0\le {x}_{n}^{*}-p\le {x}_{n}-p$, so $\left|{x}_{n}^{*}-p\right|\le \left|{x}_{n}-p\right|$ for all n ≥ 1. It follows that ${\text{lim}}_{n\to \infty }{x}_{n}^{*}=p$ and ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty }$ converges faster than ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$.

Case 2: ${x}_{1}^{*}={x}_{1}. By Lemma 3.3, we have $T{x}_{1}^{*}>{x}_{1}^{*}$ and Tx1 > x1. By using (3.4), (3.5) and the same argument as in Case 1, we can show that ${x}_{n}^{*}\ge {x}_{n}$ for all n ≥ 1. We note that ${x}_{1}^{*} and by using (3.5) and mathematical induction, we can show that ${x}_{n}^{*}\le L$ for all n ≥ 1. Thus, we have $\left|{x}_{n}^{*}-p\right|\le \left|{x}_{n}-p\right|$ for all n ≥ 1. It follows that ${\text{lim}}_{n\to \infty }{x}_{n}^{*}=p$ and ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty }$ converges faster than ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$.

Case 3: $L\le {x}_{1}^{*}={x}_{1}\le U$. Suppose that $T{x}_{1}^{*}\ne {x}_{1}^{*}$. Without loss of generality, we suppose $T{x}_{1}^{*}<{x}_{1}^{*}$. It follows by (3.5) that xn+1x n for all n ≥ 1. Since limn→∞x n = p, we must get $p<{x}_{1}^{*}={x}_{1}$. By the same argument as in Case 1, we have $p\le {x}_{n}^{*}\le {x}_{n}$ for all n ≥ 1. It follows that $\left|{x}_{n}^{*}-p\right|\le \left|{x}_{n}-p\right|$ for all n ≥ 1. Hence, ${\text{lim}}_{n\to \infty }{x}_{n}^{*}=p$ and ${\left\{{x}_{n}^{*}\right\}}_{n=1}^{\infty }$ converges faster than ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$.

Finally, we present two numerical examples for comparing rate of convergence between W(T,N)-iteration and K(T,N)-iteration.

Example 3.6 Let T : [0,8] → [0,8] be defined by$Tx=-\text{sin}\left(\frac{x-3}{2}\right)+x+\frac{1}{2}$. Then T is a continuous and nondecreasing mapping. The comparison of the rate of convergence of the W(T,N)-iteration${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$and K(T,N)-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$to a fixed point of T are given in Table1, with the initial point u1 = x1 = 1 when N = 10.

From Table 1, we see that the K(T,10)-iteration converges faster than the W(T,10)-iteration under the same control conditions. We also observe that x45 = 4.047155172 is an approximation of the fixed point of T with accuracy at 6 significant digits.

Example 3.7 Let T : [-7, 7] → [-7, 7] be defined by

$Tx=\left\{\begin{array}{cc}0.7x+{e}^{-0.8}+0.8,\hfill & if\phantom{\rule{2.77695pt}{0ex}}x\in \left[-7,-4\right)\hfill \\ {e}^{\frac{x}{5}}-2,\hfill & if\phantom{\rule{2.77695pt}{0ex}}x\in \left[-4,5\right)\hfill \\ {\left(x-5\right)}^{2}+e-2,\hfill & if\phantom{\rule{2.77695pt}{0ex}}x\in \left[5,7\right].\hfill \end{array}\right\$

Then T is a continuous and nondecreasing mapping. The comparison of the rate of convergence of the W(T,N)-iteration${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$and K(T,N)-iteration${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$to a fixed point of T are given in Table2, when N = 12.

In Example 3.7, the mapping T is continuous on [-7,7] but it not differentiable at x = -4 and x = 5. In Table 2, we observe that the K(T,12)-iteration and W(T,12)-iteration with the initial point is x = 5 converge to a fixed point p ≈ -1.215863862 of T. Moreover, the K(T,12)-iteration converges faster than the W(T,12)-iteration.

Open Problem: Is it possible to prove the convergence theorem of a finite family of continuous mappings on an arbitrary interval by using W-mappings and K-mappings and how about the rate of convergence of those methods?

## References

1. Mann WR: Mean value methods in iteration. Proc Amer Math Soc 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3

2. Ishikawa S: Fixed points by a new iteration method. Proc Amer Math Soc 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5

3. Noor MA: New approximation schemes for general variational inequalities. J Math Anal Appl 2000, 251: 217–229. 10.1006/jmaa.2000.7042

4. Phuengrattana W, Suantai S: On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J Comput Appl Math 2011, 235: 3006–3014. 10.1016/j.cam.2010.12.022

5. Rhoades BE: Comments on two fixed point iteration methods. J Math Anal Appl 1976, 56: 741–750. 10.1016/0022-247X(76)90038-X

6. Borwein D, Borwein J: Fixed point iterations for real functions. J Math Anal Appl 1991, 157: 112–126. 10.1016/0022-247X(91)90139-Q

7. Qing Y, Qihou L: The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval. J Math Anal Appl 2006, 323: 1383–1386. 10.1016/j.jmaa.2005.11.058

8. Atsushiba S, Takahashi W: Strong convergence theorems for a finite family of nonex-pansive mappings and applications, in: B.N. Prasad Birth Centenary Commemoration Volume. Indian J Math 1999, 41: 435–453.

9. Ceng LC, Cubiotti P, Yao JC: Strong convergence theorems for finitely many nonexpansive mappings and applications. Nonlinear Anal 2007, 67: 1464–1473. 10.1016/j.na.2006.06.055

10. Cho YJ, Kang SM, Qin X: Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces. Comput Math Appl 2008, 56: 2058–2064. 10.1016/j.camwa.2008.03.035

11. Cholamjiak P, Suantai S: A new hybrid algorithm for variational inclusions, generalized equilibrium problems and a finite family of quasi-nonexpansive mappings. Fixed Point Theory and Applications 2009, 2009: 20. Article ID 350979

12. Colao V, Marino G, Xu H-K: An iterative method for finding common solutions of equilibrium and fixed point problems. J Math Anal Appl 2008, 344: 340–352. 10.1016/j.jmaa.2008.02.041

13. Kangtunyakarn A, Suantai S: Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal Hybrid Syst 2009, 3: 296–309. 10.1016/j.nahs.2009.01.012

14. Kimura Y, Takahashi W: Weak convergence to common fixed points of countable nonexpansive mappings and its applications. J Korean Math Soc 2001, 38: 1275–1284.

15. Nakajo K, Shimoji K, Takahashi W: Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces. J Nonlinear Convex Anal 2007, 8: 11–34.

16. Nakajo K, Shimoji K, Takahashi W: On strong convergence by the hybrid method for families of mappings in Hilbert spaces. Nonlinear Anal 2009, 71: 112–119. 10.1016/j.na.2008.10.034

17. Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal 2010, 72: 99–112. 10.1016/j.na.2009.06.042

18. Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J Math 2001, 5: 387–404.

19. Singthong U, Suantai S: A new general iterative method for a finite family of nonexpansive mappings in Hilbert spaces. Fixed Point Theory and Appl 2010, 2010: 12. Article ID 262691

20. Takahashi W: Weak and strong convergence theorems for families of nonexpansive mappings and their applications. Ann Univ Mariae Curie-Sklodowska 1997, 51: 277–292.

21. Takahashi W, Shimoji K: Convergence theorems for nonexpansive mappings and feasibility problems. Math Comput Model 2000, 32: 1463–1471. 10.1016/S0895-7177(00)00218-1

22. Yao Y: A general iterative method for a finite family of nonexpansive mappings. Nonlinear Anal 2007, 66: 2676–2687. 10.1016/j.na.2006.03.047

23. Yao Y, Noor MA, Liou Y-C: On iterative methods for equilibrium problems. Nonlinear Anal 2009, 70: 497–509. 10.1016/j.na.2007.12.021

24. Kangtunyakarn A, Suantai S: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal 2009, 71: 4448–4460. 10.1016/j.na.2009.03.003

25. Onjai-uea N, Jaiboon C, Kumam P: A strong convergence theorem based on a relaxed extragradient method for generalized equilibrium problems. Appl Math Sci 2010, 4: 691–708.

26. Peathanom S, Phuengrattana W: A hybrid method for generalized equilibrium, variational inequality and fixed point problems of finite family of nonexpansive mappings. Thai J Math 2011, 9: 95–119.

27. Berinde V: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory Appl 2004, 2: 97–105.

28. Popescu O: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Math Commun 2007, 12: 195–202.

29. Soltuz SM: The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators. Math Commun 2005, 10: 81–88.

30. Babu GV, Prasad KN: Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory and Appl 2006, 2006: 6. Article ID49615

31. Qing Y, Rhoades BE: Comments on the Rate of Convergence between Mann and Ishikawa Iterations Applied to Zamfirescu Operators. Fixed Point Theory and Appl 2008, 2008: 3. Article ID 387504

## Acknowledgements

The authors would like to thank the Centre of Excellence in Mathematics, the Commission on Higher Education for financial support. WP is supported by the Office of the Higher Education Commission and the Graduate School of Chiang Mai University, Thailand.

## Author information

Authors

### Corresponding author

Correspondence to Suthep Suantai.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Phuengrattana, W., Suantai, S. Strong convergence theorems and rate of convergence of multi-step iterative methods for continuous mappings on an arbitrary interval. Fixed Point Theory Appl 2012, 9 (2012). https://doi.org/10.1186/1687-1812-2012-9

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2012-9

### Keywords

• fixed point
• continuous mapping
• W-mapping
• K-mapping
• rate of convergence