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

Question: If g : E → E 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 : C → C 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 : C → C be a mapping defined by

$$\begin{array}{ll}\hfill S_{n,0}& =I,\\ \hfill S_{n,1}& ={\lambda }_{n,1}{T}_1{S}_{n,0}+\left(1-{\lambda }_{n,1}\right)I,\\ \hfill S_{n,2}& ={\lambda }_{n,2}{T}_2{S}_{n,1}+\left(1-{\lambda }_{n,2}\right)I,\\ ⋮\\ \hfill S_{n,N-1}& ={\lambda }_{n,N-1}{T}_{N-1}{S}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right)I,\\ \hfill W_n=S_{n,N}& ={\lambda }_{n,N}{T}_N{S}_{n,N-1}+\left(1-{\lambda }_{n,N}\right)I,\end{array}$$

(1.7)

where I is the identity mapping of C and λ_n,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 λ_n,1, λ_n,2,..., λ_n,N. Many researchers have studied and applied this mapping for finding a common fixed point of nonexpansive mappings, for instance, see [8–23].

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

$$\begin{array}{ll}\hfill U_{n,0}& =I,\\ \hfill U_{n,1}& ={\lambda }_{n,1}{T}_1{U}_{n,0}+\left(1-{\lambda }_{n,1}\right)U_{n,0},\\ \hfill U_{n,2}& ={\lambda }_{n,2}{T}_2{U}_{n,1}+\left(1-{\lambda }_{n,2}\right)U_{n,1},\\ ⋮\\ \hfill U_{n,N-1}& ={\lambda }_{n,N-1}{T}_{N-1}{U}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right)U_{n,N-2},\\ \hfill K_n=U_{n,N}& ={\lambda }_{n,N}{T}_N{U}_{n,N-1}+\left(1-{\lambda }_{n,N}\right)U_{n,N-1},\end{array}$$

(1.8)

where I is the identity mapping of C and λ_n,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 λ_n,1, λ_n,2,..., λ_n,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 [23–26].

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

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

$$u_{n+1}=W_n^{\left(T,N\right)}{u}_n\forall n\ge 1,$$

(1.9)

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

$$\begin{array}{ll}\hfill S_{n,0}& =I,\\ \hfill S_{n,1}& ={\lambda }_{n,1}T{S}_{n,0}+\left(1-{\lambda }_{n,1}\right)I,\\ \hfill S_{n,2}& ={\lambda }_{n,2}T{S}_{n,1}+\left(1-{\lambda }_{n,2}\right)I,\\ ⋮\\ \hfill S_{n,N-1}& ={\lambda }_{n,N-1}T{S}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right)I,\\ \hfill W_n^{\left(T,N\right)}=S_{n,N}& ={\lambda }_{n,N}T{S}_{n,N-1}+\left(1-{\lambda }_{n,N}\right)I,\end{array}$$

(1.10)

where I is the identity mapping of E and λ_n,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 λ_n,1, λ_n,2,..., λ_n,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

$$x_{n+1}=K_n^{\left(T,N\right)}{x}_n\forall n\ge 1,$$

(1.11)

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

$$\begin{array}{ll}\hfill U_{n,0}& =I,\\ \hfill U_{n,1}& ={\lambda }_{n,1}T{U}_{n,0}+\left(1-{\lambda }_{n,1}\right)U_{n,0},\\ \hfill U_{n,2}& ={\lambda }_{n,2}T{U}_{n,1}+\left(1-{\lambda }_{n,2}\right)U_{n,1},\\ ⋮\\ \hfill U_{n,N-1}& ={\lambda }_{n,N-1}T{U}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right)U_{n,N-2},\\ \hfill K_n^{\left(T,N\right)}=U_{n,N}& ={\lambda }_{n,N}T{U}_{n,N-1}+\left(1-{\lambda }_{n,N}\right)U_{n,N-1},\end{array}$$

(1.12)

where I is the identity mapping of E and λ_n,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 λ_n,1,λ_n,2, ..., λ_n,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 : E → E 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) lim_n→∞λ_n,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 ∈ ℝ, a = lim inf_n→∞x _n , b = lim sup_n→∞x _n and a < b.

Next, we show that

$$\text{if}m\in \left(a,b\right),\text{then}Tm=m.$$

(2.1)

To show this, suppose that Tm ≠ m 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

$$Tx-x>0\text{for}\left|x-m\right|\le \delta .$$

(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 U_n,1x _n = λ_n,1Tx _n + (1- λ_n,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 U_n,ix _n - U_n,i-1x _n = λ_n,i(TU_n,i-1x _n - U_n,i-1x _n ) for all i = 1,2,..., N. By condition (C 1) and (C 2), we get lim_n→∞|U_n,ix _n - U_n,i-1x _n |=0 for all i = 1, 2,..., N.

Since

$$\begin{array}{ll}\hfill \left|x_{n+1}-x_n\right|& =\left|U_{n,N}{x}_n-U_{n,0}{x}_n\right|\\ \le \left|x_{n+1}-U_{n,N-1}{x}_n\right|+\left|U_{n,N-1}{x}_n-U_{n,N-2}{x}_n\right|+\cdots +\left|U_{n,1}{x}_n-U_{n,0}{x}_n\right|,\end{array}$$

it implies that lim_n→∞|x_n+1- x _n | = 0. Thus, there exists M₀ such that

$$\left|x_{n+1}-x_n\right|<\frac{\delta }{N}\text{and}\left|U_{n,i}{x}_n-U_{n,i-1}{x}_n\right|<\frac{\delta }{N}\left(i=1,2,...,N-1\right),$$

(2.3)

for all n > M₀. Since b = lim sup_n→∞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 x_k+1> m. If $x_k\in \left(m,m+\frac{\delta }{N}\right)$, then by (2.3), we have

$$m-\frac{\delta }{N}i<U_{k,i}{x}_k<m+\frac{\delta }{N}\left(i+1\right)\text{for}\text{all}i=1,2,...,N-1.$$

So we have

$$\left|x_k-m\right|<\delta \text{and}\left|U_{k,i}{x}_k-m\right|<\delta \text{for}\text{all}i=1,2,...,N-1.$$

This implies by (2.2) that

$$T{x}_k-x_k>0\text{and}T{U}_{k,i}{x}_k-U_{k,i}{x}_k>0\text{for}\text{all}i=1,2,...,N-1.$$

(2.4)

Using (1.11), we obtain

$$\begin{array}{ll}\hfill x_{k+1}& ={\lambda }_{k,N}T{U}_{k,N-1}{x}_k+\left(1-{\lambda }_{k,N}\right)U_{k,N-1}{x}_k\\ =U_{k,N-1}{x}_k+{\lambda }_{k,N}\left(T{U}_{k,N-1}{x}_k-U_{k,N-1}{x}_k\right)\\ =U_{k,N-2}{x}_k+{\lambda }_{k,N-1}\left(T{U}_{k,N-2}{x}_k-U_{k,N-2}{x}_k\right)+{\lambda }_{k,N}\left(T{U}_{k,N-1}{x}_k-U_{k,N-1}{x}_k\right)\\ ⋮\\ =x_k+\sum _{i=1}^N{\lambda }_{k,i}\left(T{U}_{k,i-1}{x}_k-U_{k,i-1}{x}_k\right).\end{array}$$

(2.5)

By (2.4), we have x_k+1> x _k . Thus, x_k+1> m.

By using the above argument, we obtain x_k+j> m for all j ≥ 2. Thus we get x _n > m for all n > k. So a = lim inf_n→∞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 $x_{\bar{M}}$ such that $a<x_{\bar{M}}<b$. Then $T{x}_{\bar{M}}=x_{\bar{M}}$. By using (1.11), we obtain that $U_{\bar{M},i}{x}_{\bar{M}}=x_{\bar{M}}$ for all i = 1, 2,..., N. Thus, we have $x_{\bar{M}+1}=x_{\bar{M}}$. By induction, we obtain $x_{\bar{M}}=x_{\bar{M}+1}=x_{\bar{M}+2}=...$, so $x_n\to x_{\bar{M}}$. This implies that $x_{\bar{M}}=a$ 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 lim_n→∞|x_n+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 sup_n→∞x _n ≤ a, which is a contradiction with a < b. If x _n ≥ b for n > M₁, so we have a = lim inf_n→∞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 lim_n→∞, x _n = p and suppose Tp ≠ p. 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 lim_n→∞U_n,ix _n = p for all i = 1, 2,..., N - 1. Let h_k,i= TU_k,i-1x _k - U_k,i-1x _k for all i = 1, 2,..., N. Continuity of T implies that lim_k→∞h_k,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 lim_k→∞h_k,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 : E → E 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) lim_n→∞λ_n,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 : E → E 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) lim_n→∞λ_n,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 : E → E 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 lim_n→∞, λ_n,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 : E → E 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) lim_n→∞λ_n,1= 0, lim_n→∞λ_n,2= 0 and lim_n→∞λ_n,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 : E → E 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) lim_n→∞λ_n,1= 0 and lim_n→∞λ_n,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 [1–7]. 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 : E → E 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

$$\left|x_n-p\right|\le \left|u_n-p\right|\text{for}\text{all}n\ge 1.$$

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 : E → E 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

$$U_{n,1}{x}_n={\lambda }_{n,1}T{x}_n+\left(1-{\lambda }_{n,1}\right)x_n\ge x_n\text{for}\text{all}n\ge 1.$$

Since T is nondecreasing, we have TU_n,1x _n ≥ Tx _n ≥ x _n . Using (1.11) again, we have

$$U_{n,2}{x}_n={\lambda }_{n,2}T{U}_{n,1}{x}_n+\left(1-{\lambda }_{n,2}\right)U_{n,1}{x}_n\ge x_n\text{for}\text{all}n\ge 1.$$

This implies that TU_n,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 : E → E 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 : E → E 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 ≤ U_k,1x _k ≤ x _k . Since T is nondecreasing, TU_k,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 U_k,2x _k ≤ x _k . By (3.1) and T is nondecreasing, we have TU_k,2x _k ≤ TU_k,1x _k ≤ U_k,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 U_k,3x _k ≤ x _k . By continuity in this way, we can show that

$$U_{k,i}{x}_k\le x_k\text{for}\text{all}i=1,2,...,N.$$

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 TU_k,1x _k ≤ TS_k,1u _k . It follows that

$$\begin{array}{ll}\hfill U_{k,2}{x}_k-S_{k,2}{u}_k& ={\lambda }_{k,2}\left(U_{k,1}{x}_k-u_k\right)+\left(1-{\lambda }_{k,2}\right)\left(T{U}_{k,1}{x}_k-T{S}_{k,1}{u}_k\right)\\ \le {\lambda }_{k,2}\left(U_{k,1}{x}_k-x_k\right)+\left(1-{\lambda }_{k,2}\right)\left(T{U}_{k,1}{x}_k-T{S}_{k,1}{u}_k\right)\\ \le 0.\end{array}$$

That is U_k,2x _k ≤ S_k,2u _k . Since T is nondecreasing, we have TU_k,2x _k ≤ TS_k,2u _k . This implies that

$$\begin{array}{ll}\hfill U_{k,3}{x}_k-S_{k,3}{u}_k& ={\lambda }_{k,3}\left(U_{k,2}{x}_k-u_k\right)+\left(1-{\lambda }_{k,3}\right)\left(T{U}_{k,2}{x}_k-T{S}_{k,2}{u}_k\right)\\ \le {\lambda }_{k,3}\left(U_{k,2}{x}_k-x_k\right)+\left(1-{\lambda }_{k,3}\right)\left(T{U}_{k,2}{x}_k-T{S}_{k,2}{u}_k\right)\\ \le 0.\end{array}$$

That is U_k,3x _k ≤ S_k,3u _k . By continuity in this way we can show that U_k,Nx _k ≤ S_k,Nu _k . Thus, x_k+1≤ u_k+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 - p ≤ u _n - p, so

$$\left|x_n-p\right|\le \left|u_n-p\right|\text{for}\text{all}n\ge 1.$$

(3.2)

Since lim_n→∞, u _n = p, it implies that lim_n→∞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 lim_n→∞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: L ≤ u₁ = x₁ ≤ U. Suppose that Tu₁ ≠ u₁. Without loss of generality, we suppose Tu₁ < u₁. It follows by (1.9) that u _n ≤ u₁ for all n ≥ 1. Since lim_n→∞u _n = p, we must get p < u₁ = x₁. By the same argument as in Case 1, we have p ≤ x _n ≤ u _n for all n ≥ 1. It follows that |x _n - p| ≤ |u _n - p| for all n ≥ 1. Hence, lim_n→∞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 λ_n,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

$$x_{n+1}={\lambda }_{n,N}T{x}_n+\left(1-{\lambda }_{n,N}\right)x_n\text{for}\text{all}n\ge 1$$

(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 ≤ S_k,1u _k ≤ u _k . Since T is nondecreasing, TS_k,1u _k ≤ Tu _k ≤ S_k,1u _k . Thus, TS_k,1u _k ≤ u _k ≤ x _k . It follows that TS_k,1u _k ≤ S_k,2u _k ≤ u _k . Since T is nondecreasing, TS_k,2u _k ≤ Tu _k ≤ S_k,1u _k . Thus, TS_k,2u _k ≤ u _k ≤ x _k . By continuity in this way, we have TS_k,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

$$T{S}_{k,i}{u}_k\le T{x}_k\text{for}\text{all}i=2,3,...,N-1.$$

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 - p ≤ x _n - p for all n ≥ 1. Since lim_n→∞x _n = p, it follows that lim_n→∞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 : E → E 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

$$x_{n+1}=K_n^{\left(T,N\right)}{x}_n\forall n\ge 1,$$

(3.4)

where$K_n^{\left(T,N\right)}$is the K-mapping generated by T and λ_n,1, λ_n,2,..., λ_n,N, and${\left\{x_n^{*}\right\}}_{n=1}^{\infty }$be a sequence defined by$x_1^{*}=x_1\in E$and

$$x_{n+1}^{*}={\bar{K}}_n^{\left(T,N\right)}{x}_n^{*}\forall n\ge 1,$$

(3.5)

where${\bar{K}}_n^{\left(T,N\right)}$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

$$T{U}_{k,i}{x}_k^{*}\le U_{k,i}{x}_k^{*}\text{for}\text{all}i=0,1,...,N.$$

(3.6)

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

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

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

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

By continuity in this way, we can show that

$${\bar{K}}_k^{\left(T,N\right)}{x}_k^{*}-K_k^{\left(T,N\right)}{x}_k=U_{k,N}{x}_k^{*}-U_{k,N}{x}_K\le 0.$$

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: $x_1^{*}=x_1<L$. 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 $x_1^{*}<L$ 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 x_n+1≤ x _n for all n ≥ 1. Since lim_n→∞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.

Table 1 Comparison of the rate of convergence between W^(T,10)-iteration and the K^(T,10)-iteration for the mapping given in Example 3.6, for ${\lambda }_{n,i}=\frac{1}{n^{1.5}+1}$ (i = 1,2,...,9) and ${\lambda }_{n,10}=\frac{1}{n^{0.5}+1}$

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

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

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.

Table 2 Comparison of the rate of convergence between the W^(T,12)-iteration and K^(T,12)-iteration for the mapping given in Example 3.7, for ${\lambda }_{n,i}=\frac{1}{n^2+1}$ (i = 1, 2,..., 11) and ${\lambda }_{n,12}=\frac{1}{n^{0.5}+1}$, with the initial point u₁ = x₁ = 5

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?