Data dependence results of new multi-step and S-iterative schemes for contractive-like operators

Abstract

In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature.

MSC:47H10.

1 Introduction

Contractive mappings and iteration procedures are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration schemes that have been introduced and developed by several authors to serve various purposes in the literature of this highly active research area, viz., [1–12] among others.

Whether an iteration method used in any investigation converges to a fixed point of a contractive type mapping corresponding to a particular iteration process is of utmost importance. Therefore it is natural to see many works related to the convergence of iteration methods such as [13–22].

Fixed point theory is concerned with investigating a wide variety of issues such as the existence (and uniqueness) of fixed points, the construction of fixed points, etc. One of these themes is data dependency of fixed points. Data dependency of fixed points has been the subject of research in fixed point theory for some time now, and data dependence research is an important theme in its own right.

Several authors who have made contributions to the study of data dependence of fixed points are Rus and Muresan [23], Rus et al. [24, 25], Berinde [26], Espínola and Petruşel [27], Markin [28], Chifu and Petruşel [29], Olantiwo [30, 31], Şoltuz [32, 33], Şoltuz and Grosan [34], Chugh and Kumar [35] and the references therein.

This paper is organized as follows. In Section 1 we present a brief survey of some known contractive mappings and iterative schemes and collect some preliminaries that will be used in the proofs of our main results. In Section 2 we show that the convergence of a new multi-step iteration, which is a special case of the Jungck multistep-SP iterative process defined in [36], and S-iteration (due to Agarwal et al.) can be used to approximate the fixed points of contractive-like operators. Motivated by the works of Şoltuz [32, 33], Şoltuz and Grosan [34], and Chugh and Kumar [35], we prove two data dependence results for the new multi-step iteration and S-iteration schemes by employing contractive-like operators.

As a background of our exposition, we now mention some contractive mappings and iteration schemes.

In [37] Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition. For a mapping $T:E\to E$, there exist real numbers a, b, c satisfying $0<a<1$, $0<b,c<1/2$ such that, for each pair $x,y\in X$, at least one of the following is true:

$$\{\begin{array}{c}({\text{z}}_1)\parallel Tx-Ty\parallel \le a\parallel x-y\parallel ,\hfill \\ ({\text{z}}_2)\parallel Tx-Ty\parallel \le b(\parallel x-Tx\parallel +\parallel y-Ty\parallel ),\hfill \\ ({\text{z}}_3)\parallel Tx-Ty\parallel \le c(\parallel x-Ty\parallel +\parallel y-Tx\parallel ).\hfill \end{array}$$

(1.1)

A mapping T satisfying the contractive conditions (z₁), (z₂) and (z₃) in (1.1) is called a Zamfirescu operator. An operator satisfying condition (z₂) is called a Kannan operator, while the mapping satisfying condition (z₃) is called a Chatterjea operator. As shown in [13], the contractive condition (1.1) leads to

$$\{\begin{array}{c}({\text{b}}_1)\parallel Tx-Ty\parallel \le \delta \parallel x-y\parallel +2\delta \parallel x-Tx\parallel \text{ if one uses }({\text{z}}_2),\text{ and}\hfill \\ ({\text{b}}_2)\parallel Tx-Ty\parallel \le \delta \parallel x-y\parallel +2\delta \parallel x-Ty\parallel \text{ if one uses }({\text{z}}_3),\hfill \end{array}$$

(1.2)

for all $x,y\in E$, where $\delta :=max\{a,\frac{b}{1-b},\frac{c}{1-c}\}$, $\delta \in [0,1)$, and it was shown that this class of operators is wider than the class of Zamfirescu operators. Any mapping satisfying condition (b₁) or (b₂) is called a quasi-contractive operator.

Extending the above definition, Osilike and Udomene [20] considered operators T for which there exist real numbers $L\ge 0$ and $\delta \in [0,1)$ such that for all $x,y\in E$,

$$\parallel Tx-Ty\parallel \le \delta \parallel x-y\parallel +L\parallel x-Tx\parallel .$$

(1.3)

Imoru and Olantiwo [38] gave a more general definition: An operator T is called a contractive-like operator if there exists a constant $\delta \in [0,1)$ and a strictly increasing and continuous function $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$, with $\phi (0)=0$, such that for each $x,y\in E$,

$$\parallel Tx-Ty\parallel \le \delta \parallel x-y\parallel +\phi (\parallel x-Tx\parallel ).$$

(1.4)

A map satisfying (1.4) need not have a fixed point, even if E is complete. For example, let $E=[0,\mathrm{\infty })$ and define T by

$$Tx=\{\begin{array}{cc}1.0,\hfill & 0\le x\le 0.8,\hfill \\ 0.6,\hfill & 0.8<x.\hfill \end{array}$$

WLOG, assume that $x<y$. Then, for $0\le x<y\le 0.8$ or $0.8<x<y$, $\parallel Tx-Ty\parallel =0$, and (1.4) is automatically satisfied.

If $0\le x\le 0.8<y$, then $\parallel Tx-Ty\parallel =0.4$.

Define φ by $\phi (t)=Lt$ for any $L\ge 2$. Then φ is increasing, continuous, and $\phi (0)=0$. Also, $\parallel x-Tx\parallel =1-x$ so that $\phi (\parallel x-Tx\parallel )=L(1-x)\ge 0.2L\ge 0.4$.

Therefore

$$0.4=\parallel Tx-Ty\parallel \le L\parallel x-Tx\parallel \le \delta \parallel x-y\parallel +L\parallel x-Tx\parallel$$

for any $0\le \delta <1$, and (1.4) is satisfied for $0\le x\le 0.8<y$. But T has no fixed point.

However, using (1.4) it is obvious that if T has a fixed point, then it is unique.

From now on, we demand that ℕ denotes the set of all nonnegative integers. Let X be a Banach space, let $E\subset X$ be a nonempty closed, convex subset of X, and let T be a self-map on E. Define $F_T:=\{p\in X:p=Tp\}$ to be the set of fixed points of T. Let ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\beta }_n\}}_{n=0}^{\mathrm{\infty }}$, ${\{{\gamma }_n\}}_{n=0}^{\mathrm{\infty }}$ and ${\{{\beta }_n^i\}}_{n=0}^{\mathrm{\infty }}$, $i=\overline{1,k-2}$, $k\ge 2$ be real sequences in $[0,1)$ satisfying certain conditions.

In [5] Rhoades and Şoltuz introduced a multi-step iterative procedure given by

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ y_n^{k-1}=(1-{\beta }_n^{k-1})x_n+{\beta }_n^{k-1}T{x}_n,k\ge 2,\hfill \\ y_n^i=(1-{\beta }_n^i)x_n+{\beta }_n^{i}T{y}_n^{i+1},i=\overline{1,k-2},\hfill \\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{y}_n^1,n\in \mathbb{N}.\hfill \end{array}$$

(1.5)

The sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ defined by

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ x_{n+1}=(1-{\alpha }_n)T{x}_n+{\alpha }_{n}T{y}_n,\hfill \\ y_n=(1-{\beta }_n)x_n+{\beta }_{n}T{x}_n,n\in \mathbb{N},\hfill \end{array}$$

(1.6)

is known as the S-iteration process (see [12, 17, 39]).

Thianwan [6] defined a two-step iteration ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ by

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ x_{n+1}=(1-{\alpha }_n)y_n+{\alpha }_{n}T{y}_n,\hfill \\ y_n=(1-{\beta }_n)x_n+{\beta }_{n}T{x}_n,n\in \mathbb{N}.\hfill \end{array}$$

(1.7)

Recently Phuengrattana and Suantai [7] introduced an SP iteration method defined by

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ x_{n+1}=(1-{\alpha }_n)y_n+{\alpha }_{n}T{y}_n,\hfill \\ y_n=(1-{\beta }_n)z_n+{\beta }_{n}T{z}_n,\hfill \\ z_n=(1-{\gamma }_n)x_n+{\gamma }_{n}T{x}_n,n\in \mathbb{N}.\hfill \end{array}$$

(1.8)

We shall employ the following iterative process. For an arbitrary fixed order $k\ge 2$,

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ x_{n+1}=(1-{\alpha }_n)y_n^1+{\alpha }_{n}T{y}_n^1,\hfill \\ y_n^1=(1-{\beta }_n^1)y_n^2+{\beta }_n^{1}T{y}_n^2,\hfill \\ y_n^2=(1-{\beta }_n^2)y_n^3+{\beta }_n^{2}T{y}_n^3,\hfill \\ \cdots ,\hfill \\ y_n^{k-2}=(1-{\beta }_n^{k-2})y_n^{k-1}+{\beta }_n^{k-2}T{y}_n^{k-1},\hfill \\ y_n^{k-1}=(1-{\beta }_n^{k-1})x_n+{\beta }_n^{k-1}T{x}_n,n\in \mathbb{N},\hfill \end{array}$$

(1.9)

or, in short,

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ x_{n+1}=(1-{\alpha }_n)y_n^1+{\alpha }_{n}T{y}_n^1,\hfill \\ y_n^i=(1-{\beta }_n^i)y_n^{i+1}+{\beta }_n^{i}T{y}_n^{i+1},i=\overline{1,k-2},\hfill \\ y_n^{k-1}=(1-{\beta }_n^{k-1})x_n+{\beta }_n^{k-1}T{x}_n,n\in \mathbb{N},\hfill \end{array}$$

(1.10)

where

$${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}\subset [0,1),\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },$$

(1.11)

and

$${\left\{{\beta }_n^i\right\}}_{n=0}^{\mathrm{\infty }}\subset [0,1),i=\overline{1,k-1}.$$

(1.12)

Remark 1 If each ${\gamma }_n=0$, then SP iteration (1.8) reduces to two-step iteration (1.7). By taking $k=3$ and $k=2$ in (1.10), we obtain iterations (1.8) and (1.7), respectively.

We shall need the following definition and lemma in the sequel.

Definition 1 [40]

Let $T,\tilde{T}:X\to X$ be two operators. We say that $\tilde{T}$ is an approximate operator for T if, for some $\epsilon >0$, we have

$$\parallel Tx-\tilde{T}x\parallel \le \epsilon$$

for all $x\in X$.

Lemma 1 [34]

Let ${\{a_n\}}_{n=0}^{\mathrm{\infty }}$ be a nonnegative sequence for which one assumes that there exists an $n_0\in \mathbb{N}$ such that for all $n\ge n_0$,

$$a_{n+1}\le (1-{\mu }_n)a_n+{\mu }_n{\eta }_n$$

is satisfied, where ${\mu }_n\in (0,1)$ for all $n\in \mathbb{N}$, ${\sum }_{n=0}^{\mathrm{\infty }}{\mu }_n=\mathrm{\infty }$ and ${\eta }_n\ge 0$, $\mathrm{\forall }n\in \mathbb{N}$. Then the following holds:

$$0\le \underset{n\to \mathrm{\infty }}{lim}sup{a}_n\le \underset{n\to \mathrm{\infty }}{lim}sup{\eta }_n.$$

2 Main results

For simplicity we use the following notation throughout this section.

For any iterative process, ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ and ${\{u_n\}}_{n=0}^{\mathrm{\infty }}$ denote iterative sequences associated to T and $\tilde{T}$, respectively.

Theorem 1 Let $T:E\to E$ be a map satisfying (1.4) with $F_T\ne \mathrm{\varnothing }$, and let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be a sequence defined by (1.10), then the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ converges to the unique fixed point of T.

Proof The proof can be easily obtained by using the argument in the proof of ([36], Theorem 3.1). □

This result allows us to give the next theorem.

Theorem 2 Let $T:E\to E$ be a map satisfying (1.4) with $F_T\ne \mathrm{\varnothing }$, and let $\tilde{T}$ be an approximate operator of T as in Definition  1. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$, ${\{u_n\}}_{n=0}^{\mathrm{\infty }}$ be two iterative sequences defined by (1.10) with real sequences ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }},{\{{\beta }_n^i\}}_{n=0}^{\mathrm{\infty }}\subset [0,1)$ satisfying (i) $0\le {\beta }_n^i<{\alpha }_n\le 1$, $i=\overline{1,k-1}$, (ii) $\sum {\alpha }_n=\mathrm{\infty }$. If $p=Tp$ and $q=\tilde{T}q$, then we have

$$\parallel p-q\parallel \le \frac{k\epsilon }{1-\delta }.$$

Proof For given $x_0\in E$ and $u_0\in E$, we consider the following multi-step iteration for T and $\tilde{T}$:

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ x_{n+1}=(1-{\alpha }_n)y_n^1+{\alpha }_{n}T{y}_n^1,\hfill \\ y_n^i=(1-{\beta }_n^i)y_n^{i+1}+{\beta }_n^{i}T{y}_n^{i+1},i=\overline{1,k-2},\hfill \\ y_n^{k-1}=(1-{\beta }_n^{k-1})x_n+{\beta }_n^{k-1}T{x}_n,k\ge 2,n\in \mathbb{N},\hfill \end{array}$$

(2.1)

and

$$\{\begin{array}{c}{u}_0\in E,\hfill \\ u_{n+1}=(1-{\alpha }_n)v_n^1+{\alpha }_n\tilde{T}{v}_n^1,\hfill \\ v_n^i=(1-{\beta }_n^i)v_n^{i+1}+{\beta }_n^i\tilde{T}{v}_n^{i+1},i=\overline{1,k-2},\hfill \\ v_n^{k-1}=(1-{\beta }_n^{k-1})u_n+{\beta }_n^{k-1}\tilde{T}{u}_n,k\ge 2,n\in \mathbb{N}.\hfill \end{array}$$

(2.2)

Thus, from (1.4), (2.1) and (2.2), we have the following inequalities.

(2.3)

(2.4)

(2.5)

Combining (2.3), (2.4) and (2.5), we obtain

$$\begin{array}{rcl}\parallel x_{n+1}-u_{n+1}\parallel & \le & [1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )][1-{\beta }_n^2(1-\delta )]\parallel y_n^3-v_n^3\parallel \\ +[1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )]{\beta }_n^2\phi \left(\parallel y_n^3-T{y}_n^3\parallel \right)\\ +[1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )]{\beta }_n^2\epsilon \\ +[1-{\alpha }_n(1-\delta )]{\beta }_n^1\phi \left(\parallel y_n^2-T{y}_n^2\parallel \right)\\ +[1-{\alpha }_n(1-\delta )]{\beta }_n^1\epsilon +{\alpha }_n\phi \left(\parallel y_n^1-T{y}_n^1\parallel \right)+{\alpha }_n\epsilon .\end{array}$$

(2.6)

Thus, by induction, we get

$$\begin{array}{rcl}\parallel x_{n+1}-u_{n+1}\parallel & \le & [1-{\alpha }_n(1-\delta )]\\ [1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-2}(1-\delta )]\parallel y_n^{k-1}-v_n^{k-1}\parallel \\ +[1-{\alpha }_n(1-\delta )]\\ [1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-3}(1-\delta )]{\beta }_n^{k-2}\phi \left(\parallel y_n^{k-1}-T{y}_n^{k-1}\parallel \right)\\ +\cdots +[1-{\alpha }_n(1-\delta )]{\beta }_n^1\phi \left(\parallel y_n^2-T{y}_n^2\parallel \right)+{\alpha }_n\phi \left(\parallel y_n^1-T{y}_n^1\parallel \right)\\ +[1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-3}(1-\delta )]{\beta }_n^{k-2}\epsilon \\ +\cdots +[1-{\alpha }_n(1-\delta )]{\beta }_n^1\epsilon +{\alpha }_n\epsilon .\end{array}$$

(2.7)

Again, using (1.4), (2.1) and (2.2), we get

$$\begin{array}{rcl}\parallel y_n^{k-1}-v_n^{k-1}\parallel & =& \parallel (1-{\beta }_n^{k-1})(x_n-u_n)+{\beta }_n^{k-1}(T{x}_n-\tilde{T}{u}_n)\parallel \\ \le & (1-{\beta }_n^{k-1})\parallel x_n-u_n\parallel +{\beta }_n^{k-1}\parallel T{x}_n-\tilde{T}{u}_n\parallel \\ \le & (1-{\beta }_n^{k-1})\parallel x_n-u_n\parallel +{\beta }_n^{k-1}\parallel T{x}_n-T{u}_n\parallel +{\beta }_n^{k-1}\parallel T{u}_n-\tilde{T}{u}_n\parallel \\ \le & [1-{\beta }_n^{k-1}(1-\delta )]\parallel x_n-u_n\parallel +{\beta }_n^{k-1}\phi (\parallel x_n-T{x}_n\parallel )+{\beta }_n^{k-1}\epsilon .\end{array}$$

(2.8)

Substituting (2.8) in (2.7), we have

$$\begin{array}{rcl}\parallel x_{n+1}-u_{n+1}\parallel & \le & [1-{\alpha }_n(1-\delta )]\\ [1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-1}(1-\delta )]\parallel x_n-u_n\parallel \\ +[1-{\alpha }_n(1-\delta )]\\ [1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-2}(1-\delta )]{\beta }_n^{k-1}\phi (\parallel x_n-T{x}_n\parallel )\\ +[1-{\alpha }_n(1-\delta )]\\ [1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-3}(1-\delta )]{\beta }_n^{k-2}\phi \left(\parallel y_n^{k-1}-T{y}_n^{k-1}\parallel \right)\\ +\cdots +[1-{\alpha }_n(1-\delta )]{\beta }_n^1\phi \left(\parallel y_n^2-T{y}_n^2\parallel \right)+{\alpha }_n\phi \left(\parallel y_n^1-T{y}_n^1\parallel \right)\\ +[1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-2}(1-\delta )]{\beta }_n^{k-1}\epsilon \\ +\cdots +[1-{\alpha }_n(1-\delta )]{\beta }_n^1\epsilon +{\alpha }_n\epsilon .\end{array}$$

(2.9)

Since $\delta \in [0,1)$ and ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }},{\{{\beta }_n^i\}}_{n=0}^{\mathrm{\infty }}\subset [0,1)$, for $i=\overline{1,k-1}$, we have

$$[1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^i(1-\delta )]\le [1-{\alpha }_n(1-\delta )].$$

(2.10)

From inequality (2.10) and assumption (i) in (2.9), it follows

$$\begin{array}{rcl}\parallel x_{n+1}-u_{n+1}\parallel & \le & [1-{\alpha }_n(1-\delta )]\parallel x_n-u_n\parallel \\ +{\alpha }_n\phi (\parallel x_n-T{x}_n\parallel )+{\alpha }_n\phi \left(\parallel y_n^{k-1}-T{y}_n^{k-1}\parallel \right)\\ +\cdots +{\alpha }_n\phi \left(\parallel y_n^2-T{y}_n^2\parallel \right)+{\alpha }_n\phi \left(\parallel y_n^1-T{y}_n^1\parallel \right)\\ +{\alpha }_n\epsilon +{\alpha }_n\epsilon +\cdots +{\alpha }_n\epsilon +{\alpha }_n\epsilon \\ =& [1-{\alpha }_n(1-\delta )]\parallel x_n-u_n\parallel \\ +{\alpha }_n(1-\delta )\{\frac{\phi (\parallel x_n-T{x}_n\parallel )+\phi (\parallel y_n^{k-1}-T{y}_n^{k-1}\parallel )}{1-\delta }\\ +\cdots +\frac{\phi (\parallel y_n^1-T{y}_n^1\parallel )+k\epsilon }{1-\delta }\}.\end{array}$$

(2.11)

Define

From Theorem 1 it follows that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel =0$. Since T satisfies condition (1.4) and $Tp=p\in F_T$,

$$\begin{array}{rcl}0& \le & \parallel x_n-T{x}_n\parallel \\ \le & \parallel x_n-p\parallel +\parallel Tp-T{x}_n\parallel \\ \le & \parallel x_n-p\parallel +\delta \parallel p-x_n\parallel +\phi (\parallel p-Tp\parallel )\\ =& (1+\delta )\parallel x_n-p\parallel \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(2.12)

Since ${\beta }_n^i\in [0,1)$, $\mathrm{\forall }n\in \mathbb{N}$, $i=\overline{1,k-1}$, using (1.4) and (1.10), we have

$$\begin{array}{rcl}0& \le & \parallel y_n^1-T{y}_n^1\parallel =\parallel y_n^1-p+p-T{y}_n^1\parallel \\ \le & \parallel y_n^1-p\parallel +\parallel Tp-T{y}_n^1\parallel \\ \le & \parallel y_n^1-p\parallel +\delta \parallel p-y_n^1\parallel +\phi (\parallel p-Tp\parallel )\\ =& (1+\delta )\parallel y_n^1-p\parallel \\ =& (1+\delta )\parallel (1-{\beta }_n^1)y_n^2+{\beta }_n^{1}T{y}_n^2-p(1-{\beta }_n^1+{\beta }_n^1)\parallel \\ \le & (1+\delta )\{(1-{\beta }_n^1)\parallel y_n^2-p\parallel +{\beta }_n^1\parallel T{y}_n^2-Tp\parallel \}\\ \le & (1+\delta )\{(1-{\beta }_n^1)\parallel y_n^2-p\parallel +{\beta }_n^1\delta \parallel y_n^2-p\parallel \}\\ =& (1+\delta )[1-{\beta }_n^1(1-\delta )]\parallel y_n^2-p\parallel \\ \le & \cdots \\ \le & (1+\delta )[1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-2}(1-\delta )]\parallel y_n^{k-1}-p\parallel \\ \le & (1+\delta )[1-{\beta }_n^1(1-\delta )]\cdots [1-{\beta }_n^{k-1}(1-\delta )]\parallel x_n-p\parallel \\ \le & (1+\delta )\parallel x_n-p\parallel \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

(2.13)

It is easy to see from (2.13) that this result is also valid for $\parallel T{y}_n^2-y_n^2\parallel ,\dots ,\parallel T{y}_n^{k-1}-y_n^{k-1}\parallel$.

Since φ is continuous, we have

$$\underset{n\to \mathrm{\infty }}{lim}\phi (\parallel x_n-T{x}_n\parallel )=\underset{n\to \mathrm{\infty }}{lim}\phi \left(\parallel y_n^1-T{y}_n^1\parallel \right)=\cdots =\underset{n\to \mathrm{\infty }}{lim}\phi \left(\parallel y_n^{k-1}-T{y}_n^{k-1}\parallel \right)=0.$$

(2.14)

Hence an application of Lemma 1 to (2.11) leads to

$$\parallel p-q\parallel \le \frac{k\epsilon }{1-\delta }.$$

(2.15)

 □

As shown by Hussain et al. ([22], Theorem 8), in an arbitrary Banach space X, the S-iteration ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ given by (1.6) converges to the fixed point of T, where $T:E\to E$ is a mapping satisfying condition (1.3).

Theorem 3 Let $T:E\to E$ be a map satisfying (1.4) with $F_T\ne \mathrm{\varnothing }$, and let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be defined by (1.6) with real sequences ${\{{\beta }_n\}}_{n=0}^{\mathrm{\infty }},{\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}\subset [0,1)$ satisfying ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$. Then the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ converges to the unique fixed point of T.

Proof The argument is similar to the proof of Theorem 8 of [22], and is thus omitted. □

We now prove the result on data dependence for the S-iterative procedure by utilizing Theorem 3.

Theorem 4 Let T, $\tilde{T}$ be two operators as in Theorem  2. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$, ${\{u_n\}}_{n=0}^{\mathrm{\infty }}$ be S-iterations defined by (1.6) with real sequences ${\{{\beta }_n\}}_{n=0}^{\mathrm{\infty }},{\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}\subset [0,1)$ satisfying (i) $\frac{1}{2}\le {\alpha }_n$, $\mathrm{\forall }n\in \mathbb{N}$, and (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$. If $p=Tp$ and $q=\tilde{T}q$, then we have

$$\parallel p-q\parallel \le \frac{3\epsilon }{1-\delta }.$$

Proof For a given $x_0\in E$ and $u_0\in E$, we consider the following iteration for T and $\tilde{T}$:

$$\{\begin{array}{c}{x}_0\in E,\hfill \\ x_{n+1}=(1-{\alpha }_n)T{x}_n+{\alpha }_{n}T{y}_n,\hfill \\ y_n=(1-{\beta }_n)x_n+{\beta }_{n}T{x}_n,n\in \mathbb{N}\hfill \end{array}$$

(2.16)

and

$$\{\begin{array}{c}{u}_0\in E,\hfill \\ u_{n+1}=(1-{\alpha }_n)\tilde{T}{u}_n+{\alpha }_n\tilde{T}{v}_n,\hfill \\ v_n=(1-{\beta }_n)u_n+{\beta }_n\tilde{T}{u}_n,n\in \mathbb{N}.\hfill \end{array}$$

(2.17)

Using (1.4), (2.16) and (2.17), we obtain the following estimates:

(2.18)

(2.19)

Combining (2.18) and (2.19), we get

$$\begin{array}{rcl}\parallel x_{n+1}-u_{n+1}\parallel & \le & \{(1-{\alpha }_n)\delta +{\alpha }_n\delta [1-{\beta }_n(1-\delta )]\}\parallel x_n-u_n\parallel \\ +\{1-{\alpha }_n+{\alpha }_n\delta {\beta }_n\}\phi (\parallel x_n-T{x}_n\parallel )+{\alpha }_n\phi (\parallel y_n-T{y}_n\parallel )\\ +{\alpha }_n\delta {\beta }_n\epsilon +(1-{\alpha }_n)\epsilon +{\alpha }_n\epsilon .\end{array}$$

(2.20)

For ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }},{\{{\beta }_n\}}_{n=0}^{\mathrm{\infty }}\subset [0,1)$ and $\delta \in [0,1)$,

$$(1-{\alpha }_n)\delta <1-{\alpha }_n,1-{\beta }_n(1-\delta )<1,{\alpha }_n\delta {\beta }_n<{\alpha }_n.$$

(2.21)

It follows from assumption (i) that

$$1-{\alpha }_n<{\alpha }_n,\mathrm{\forall }n\in \mathbb{N}.$$

(2.22)

Therefore, combining (2.22) and (2.21) to (2.20) gives

$$\begin{array}{rcl}\parallel x_{n+1}-u_{n+1}\parallel & \le & [1-{\alpha }_n(1-\delta )]\parallel x_n-u_n\parallel \\ +2{\alpha }_n\phi (\parallel x_n-T{x}_n\parallel )+{\alpha }_n\phi (\parallel y_n-T{y}_n\parallel )\\ +{\alpha }_n\epsilon +{\alpha }_n\epsilon +{\alpha }_n\epsilon ,\end{array}$$

(2.23)

or, equivalently,

$$\begin{array}{rcl}\parallel x_{n+1}-u_{n+1}\parallel & \le & [1-{\alpha }_n(1-\delta )]\parallel x_n-u_n\parallel \\ +{\alpha }_n(1-\delta )\frac{\{2\phi (\parallel x_n-T{x}_n\parallel )+\phi (\parallel y_n-T{y}_n\parallel )+3\epsilon \}}{1-\delta }.\end{array}$$

(2.24)

Now define

From Theorem 3, we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel =0$. Since T satisfies condition (1.4), and $Tp=p\in F_T$, using an argument similar to that in the proof of Theorem 2,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T{x}_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-T{y}_n\parallel =0.$$

(2.25)

Using the fact that φ is continuous, we have

$$\underset{n\to \mathrm{\infty }}{lim}\phi (\parallel x_n-T{x}_n\parallel )=\underset{n\to \mathrm{\infty }}{lim}\phi (\parallel y_n-T{y}_n\parallel )=0.$$

(2.26)

An application of Lemma 1 to (2.24) leads to

$$\parallel p-q\parallel \le \frac{3\epsilon }{1-\delta }.$$

(2.27)

 □

3 Conclusion

Since the iterative schemes (1.7) and (1.8) are special cases of the iterative process (1.10), Theorem 1 generalizes Theorem 2.1 of [19] and Theorem 2.1 of [18]. By taking $k=3$ and $k=2$ in Theorem 2, data dependence results for the iterative schemes (1.8) and (1.7) can be easily obtained. For $k=3$, Theorem 2 reduces to Theorem 3.2 of [35]. Since condition (1.4) is more general than condition (1.3), Theorem 3 generalizes Theorem 8 of [22].