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# A three-step iterative scheme for solving nonlinear ϕ-strongly accretive operator equations in Banach spaces

Fixed Point Theory and Applications20122012:149

https://doi.org/10.1186/1687-1812-2012-149

• Received: 30 June 2012
• Accepted: 29 August 2012
• Published:

## Abstract

In this paper, we study a three-step iterative scheme with error terms for solving nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.

## Keywords

• three-step iterative scheme
• ϕ-strongly accretive operator
• ϕ-hemicontractive operator

## 1 Introduction

Let K be a nonempty subset of an arbitrary Banach space X and ${X}^{\ast }$ be its dual space. The symbols $D\left(T\right)$, $R\left(T\right)$ and $F\left(T\right)$ stand for the domain, the range and the set of fixed points of T respectively (for a single-valued map $T:X\to X$, $x\in X$ is called a fixed point of T iff $T\left(x\right)=x$). We denote by J the normalized duality mapping from E to ${2}^{{E}^{\ast }}$ defined by
$J\left(x\right)=\left\{{f}^{\ast }\in {X}^{\ast }:〈x,{f}^{\ast }〉={\parallel x\parallel }^{2}={\parallel {f}^{\ast }\parallel }^{2}\right\}.$

Let $T:D\left(T\right)\subseteq X\to X$ be an operator. The following definitions can be found in [115] for example.

Definition 1 T is called Lipshitzian if there exists $L>0$ such that
$\parallel Tx-Ty\parallel ⩽L\parallel x-y\parallel ,$

for all $x,y\in K$. If $L=1$, then T is called nonexpansive, and if $0, T is called contraction.

Definition 2
1. (i)
T is said to be strongly pseudocontractive if there exists a $t>1$ such that for each $x,y\in D\left(T\right)$, there exists $j\left(x-y\right)\in J\left(x-y\right)$ satisfying
$Re〈Tx-Ty,j\left(x-y\right)〉\le \frac{1}{t}{\parallel x-y\parallel }^{2}.$

2. (ii)
T is said to be strictly hemicontractive if $F\left(T\right)$ is nonempty and if there exists a $t>1$ such that for each $x\in D\left(T\right)$ and $q\in F\left(T\right)$, there exists $j\left(x-y\right)\in J\left(x-y\right)$ satisfying
$Re〈Tx-q,j\left(x-q\right)〉\le \frac{1}{t}{\parallel x-q\parallel }^{2}.$

3. (iii)
T is said to be ϕ-strongly pseudocontractive if there exists a strictly increasing function $\varphi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ with $\varphi \left(0\right)=0$ such that for each $x,y\in D\left(T\right)$, there exists $j\left(x-y\right)\in J\left(x-y\right)$ satisfying
$Re〈Tx-Ty,j\left(x-y\right)〉\le {\parallel x-y\parallel }^{2}-\varphi \left(\parallel x-y\parallel \right)\parallel x-y\parallel .$

4. (iv)
T is said to be ϕ-hemicontractive if $F\left(T\right)$ is nonempty and if there exists a strictly increasing function $\varphi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ with $\varphi \left(0\right)=0$ such that for each $x\in D\left(T\right)$ and $q\in F\left(T\right)$, there exists $j\left(x-y\right)\in J\left(x-y\right)$ satisfying
$Re〈Tx-q,j\left(x-q\right)〉\le {\parallel x-q\parallel }^{2}-\varphi \left(\parallel x-q\parallel \right)\parallel x-q\parallel .$

Clearly, each strictly hemicontractive operator is ϕ-hemicontractive.

Definition 3
1. (i)
T is called accretive if the inequality
$\parallel x-y\parallel \le \parallel x-y+s\left(Tx-Ty\right)\parallel$

holds for every $x,y\in D\left(T\right)$ and for all $s>0$.
1. (ii)
T is called strongly accretive if, for all $x,y\in D\left(T\right)$, there exists a constant $k>0$ and $j\left(x-y\right)\in J\left(x-y\right)$ such that
$〈Tx-Ty,j\left(x-y\right)〉\ge k{\parallel x-y\parallel }^{2}.$

2. (iii)
T is called ϕ-strongly accretive if there exists $j\left(x-y\right)\in J\left(x-y\right)$ and a strictly increasing function $\varphi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ with $\varphi \left(0\right)=0$ such that for each $x,y\in X$,
$〈Tx-Ty,j\left(x-y\right)〉\ge \varphi \left(\parallel x-y\parallel \right)\parallel x-y\parallel .$

Remark 4 It has been shown in [11, 12] that the class of strongly accretive operators is a proper subclass of the class of ϕ-strongly accretive operators. If I denotes the identity operator, then T is called strongly pseudocontractive (respectively, ϕ-strongly pseudocontractive) if and only if $\left(I-T\right)$ is strongly accretive (respectively, ϕ-strongly accretive).

Chidume [1] showed that the Mann iterative method can be used to approximate fixed points of Lipschitz strongly pseudocontractive operators in ${L}_{p}$ (or ${l}_{p}$) spaces for $p\in \left[2,\mathrm{\infty }\right)$. Chidume and Osilike [4] proved that each strongly pseudocontractive operator with a fixed point is strictly hemicontractive, but the converse does not hold in general. They also proved that the class of strongly pseudocontractive operators is a proper subclass of the class of ϕ-strongly pseudocontractive operators and pointed out that the class of ϕ-strongly pseudocontractive operators with a fixed point is a proper subclass of the class of ϕ-hemicontractive operators. These classes of nonlinear operators have been studied by various researchers (see, for example, [725]). Liu et al. [26] proved that, under certain conditions, a three-step iteration scheme with error terms converges strongly to the unique fixed point of ϕ-hemicontractive mappings.

In this paper, we study a three-step iterative scheme with error terms for nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.

## 2 Preliminaries

We need the following results.

Lemma 5 [27]

Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$ and $\left\{{c}_{n}\right\}$ be three sequences of nonnegative real numbers with ${\sum }_{n=1}^{\mathrm{\infty }}{b}_{n}<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{c}_{n}<\mathrm{\infty }$. If
${a}_{n+1}\le \left(1+{b}_{n}\right){a}_{n}+{c}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,$

then the limit ${lim}_{n\to \mathrm{\infty }}{a}_{n}$ exists.

Lemma 6 [28]

Let $x,y\in X$. Then $\parallel x\parallel \le \parallel x+ry\parallel$ for every $r>0$ if and only if there is $f\in J\left(x\right)$ such that $Re〈y,f〉\ge 0$.

Lemma 7 [9]

Suppose that X is an arbitrary Banach space and $A:E\to E$ is a continuous ϕ-strongly accretive operator. Then the equation $Ax=f$ has a unique solution for any $f\in E$.

## 3 Strong convergence of a three-step iterative scheme to a solution of the system of nonlinear operator equations

For the rest of this section, L denotes the Lipschitz constant of ${T}_{1},{T}_{2},{T}_{3}:X\to X$, ${L}_{\ast }=\left(1+L\right)$ and $R\left({T}_{1}\right)$, $R\left({T}_{2}\right)$ and $R\left({T}_{3}\right)$ denote the ranges of ${T}_{1}$, ${T}_{2}$ and ${T}_{3}$ respectively. We now prove our main results.

Theorem 8 Let X be an arbitrary real Banach space and ${T}_{1},{T}_{2},{T}_{3}:X\to X$ Lipschitz ϕ-strongly accretive operators. Let $f\in R\left({T}_{1}\right)\cap R\left({T}_{2}\right)\cap R\left({T}_{3}\right)$ and generate $\left\{{x}_{n}\right\}$ from an arbitrary ${x}_{0}\in X$ by
$\begin{array}{r}{x}_{n+1}={a}_{n}{x}_{n}+{b}_{n}\left(f+\left(I-{T}_{1}\right){y}_{n}\right)+{c}_{n}{v}_{n},\\ {y}_{n}={a}_{n}^{\mathrm{\prime }}{x}_{n}+{b}_{n}^{\mathrm{\prime }}\left(f+\left(I-{T}_{2}\right){z}_{n}\right)+{c}_{n}^{\mathrm{\prime }}{u}_{n},\\ {z}_{n}={a}_{n}^{\mathrm{\prime }\mathrm{\prime }}{x}_{n}+{b}_{n}^{\mathrm{\prime }\mathrm{\prime }}\left(f+\left(I-{T}_{3}\right){x}_{n}\right)+{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}{w}_{n},\phantom{\rule{1em}{0ex}}n\ge 0,\end{array}$
(3.1)

where ${\left\{{v}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{u}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ and ${\left\{{w}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ are bounded sequences in X and $\left\{{a}_{n}\right\}$, $\left\{{c}_{n}\right\}$, $\left\{{a}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{b}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{c}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{a}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{b}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$ are sequences in $\left[0,1\right]$ and $\left\{{b}_{n}\right\}$ in $\left(0,1\right)$ satisfying the following conditions: (i) ${a}_{n}+{b}_{n}+{c}_{n}=1={a}_{n}^{\mathrm{\prime }}+{b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}={a}_{n}^{\mathrm{\prime }\mathrm{\prime }}+{b}_{n}^{\mathrm{\prime }\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}$, (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{b}_{n}=\mathrm{\infty }$, (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{b}_{n}^{2}<\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}{b}_{n}^{\mathrm{\prime }}<\mathrm{\infty }$, (iv) ${\sum }_{n=0}^{\mathrm{\infty }}{c}_{n}<\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}{c}_{n}^{\mathrm{\prime }}<\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}<\mathrm{\infty }$. Then the sequence $\left\{{x}_{n}\right\}$ converges strongly to the solution of the system ${T}_{i}x=f$; $i=1,2,3$.

Proof By Lemma 7, the system ${T}_{i}x=f$; $i=1,2,3$ has the unique solution ${x}^{\ast }\in X$. Following the techniques of [5, 812, 26, 29], define ${S}_{i}:X\to X$ by ${S}_{i}x=f+\left(I-{T}_{i}\right)x$; $i=1,2,3$; then each ${S}_{i}$ is demicontinuous and ${x}^{\ast }$ is the unique fixed point of ${S}_{i}$; $i=1,2,3$, and for all $x,y\in X$, we have
$\begin{array}{c}〈\left(I-{S}_{i}\right)x-\left(I-{S}_{i}\right)y,j\left(x-y\right)〉\hfill \\ \phantom{\rule{1em}{0ex}}\ge {\varphi }_{i}\left(\parallel x-y\parallel \right)\parallel x-y\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\ge \frac{{\varphi }_{i}\left(\parallel x-y\parallel \right)}{\left(1+{\varphi }_{i}\left(\parallel x-y\parallel \right)+\parallel x-y\parallel \right)}{\parallel x-y\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\theta }_{i}\left(x,y\right){\parallel x-y\parallel }^{2},\hfill \end{array}$
where ${\theta }_{i}\left(x,y\right)=\frac{{\varphi }_{i}\left(\parallel x-y\parallel \right)}{\left(1+{\varphi }_{i}\left(\parallel x-y\parallel \right)+\parallel x-y\parallel \right)}\in \left[0,1\right)$ for all $x,y\in X$; $i=1,2,3$. Let ${x}^{\ast }\in {\bigcap }_{i=1}^{3}F\left({S}_{i}\right)$ be the fixed point set of ${S}_{i}$, and let $\theta \left(x,y\right)={inf min}_{i}\left\{{\theta }_{i}\left(x,y\right)\right\}\in \left[0,1\right]$. Thus
$〈\left(I-{S}_{i}\right)x-\left(I-{S}_{i}\right)y,j\left(x-y\right)〉\ge \theta \left(x,y\right){\parallel x-y\parallel }^{2};\phantom{\rule{1em}{0ex}}i=1,2,3.$
(3.2)
It follows from Lemma 6 and inequality (3.2) that
$\parallel x-y\parallel \le \parallel x-y+\lambda \left[\left(I-{S}_{i}\right)x-\theta \left(x,y\right)x-\left(\left(I-{S}_{i}\right)y-\theta \left(x,y\right)y\right)\right]\parallel ,$
(3.3)

for all $x,y\in X$ and for all $\lambda >0$; $i=1,2,3$.

Set ${\alpha }_{n}={b}_{n}+{c}_{n}$, ${\beta }_{n}={b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}$ and ${\gamma }_{n}={b}_{n}^{\mathrm{\prime }\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}$, then (3.1) becomes
$\begin{array}{r}{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}{S}_{1}{y}_{n}+{c}_{n}\left({v}_{n}-{S}_{1}{y}_{n}\right),\\ {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}{S}_{2}{z}_{n}+{c}_{n}^{\mathrm{\prime }}\left({u}_{n}-{S}_{2}{z}_{n}\right),\\ {z}_{n}=\left(1-{\gamma }_{n}\right){x}_{n}+{\gamma }_{n}{S}_{3}{x}_{n}+{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\left({w}_{n}-{S}_{3}{x}_{n}\right),\phantom{\rule{1em}{0ex}}n\ge 0.\end{array}$
(3.4)
We have
$\begin{array}{rcl}{x}_{n}& =& \left(1+{\alpha }_{n}\right){x}_{n+1}+{\alpha }_{n}\left[\left(I-{S}_{1}\right){x}_{n+1}-\theta \left({x}_{n+1},{x}^{\ast }\right){x}_{n+1}\right]\\ -\left(1-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}{x}_{n}+\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}^{2}\left({x}_{n}-{S}_{1}{y}_{n}\right)\\ +{\alpha }_{n}\left({S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\right)-\left[1+\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\right]{c}_{n}\left({v}_{n}-{S}_{1}{y}_{n}\right).\end{array}$
Furthermore,
${x}^{\ast }=\left(1+{\alpha }_{n}\right){x}^{\ast }+{\alpha }_{n}\left[\left(I-{S}_{1}\right){x}^{\ast }-\theta \left({x}_{n+1},{x}^{\ast }\right){x}^{\ast }\right]-\left(1-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}{x}^{\ast },$
so that
$\begin{array}{rcl}{x}_{n}-{x}^{\ast }& =& \left(1+{\alpha }_{n}\right)\left({x}_{n+1}-{x}^{\ast }\right)+{\alpha }_{n}\left[\left(I-{S}_{1}\right){x}_{n+1}-\theta \left({x}_{n+1},{x}^{\ast }\right){x}_{n+1}\\ -\left(\left(I-{S}_{1}\right){x}^{\ast }-\theta \left({x}_{n+1},{x}^{\ast }\right){x}^{\ast }\right)\right]\\ -\left(1-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\left({x}_{n}-{x}^{\ast }\right)+\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}^{2}\left({x}_{n}-{S}_{1}{y}_{n}\right)\\ +{\alpha }_{n}\left({S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\right)-\left[1+\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\right]{c}_{n}\left({v}_{n}-{S}_{1}{y}_{n}\right).\end{array}$
Hence,
$\begin{array}{rcl}\parallel {x}_{n}-{x}^{\ast }\parallel & \ge & \left(1+{\alpha }_{n}\right)\parallel {x}_{n+1}-{x}^{\ast }+\frac{{\alpha }_{n}}{\left(1+{\alpha }_{n}\right)}\left[\left(I-{S}_{1}\right){x}_{n+1}-\theta \left({x}_{n+1},{x}^{\ast }\right){x}_{n+1}\\ -\left(\left(I-{S}_{1}\right){x}^{\ast }-\theta \left({x}_{n+1},{x}^{\ast }\right){x}^{\ast }\right)\right]\parallel \\ -\left(1-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel -\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}^{2}\parallel {x}_{n}-{S}_{1}{y}_{n}\parallel \\ -{\alpha }_{n}\parallel {S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\parallel -\left[1+\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\right]{c}_{n}\parallel {v}_{n}-{S}_{1}{y}_{n}\parallel \\ \ge & \left(1+{\alpha }_{n}\right)\parallel {x}_{n+1}-{x}^{\ast }\parallel -\left(1-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel \\ -\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}^{2}\parallel {x}_{n}-{S}_{1}{y}_{n}\parallel -{\alpha }_{n}\parallel {S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\parallel \\ -\left[1+\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\right]{c}_{n}\parallel {v}_{n}-{S}_{1}{y}_{n}\parallel .\end{array}$
Hence,
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}^{\ast }\parallel & \le & \frac{\left[1+\left(1-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\right]}{\left(1+{\alpha }_{n}\right)}\parallel {x}_{n}-{x}^{\ast }\parallel +2{\alpha }_{n}^{2}\parallel {x}_{n}-{S}_{1}{y}_{n}\parallel \\ +{\alpha }_{n}\parallel {S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\parallel +\left[1+\left(2-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\right]{c}_{n}\parallel {v}_{n}-{S}_{1}{y}_{n}\parallel \\ \le & \left[1+\left(1-\theta \left({x}_{n+1},{x}^{\ast }\right)\right){\alpha }_{n}\right]\left[1-{\alpha }_{n}+{\alpha }_{n}^{2}\right]\parallel {x}_{n}-{x}^{\ast }\parallel \\ +2{\alpha }_{n}^{2}\parallel {x}_{n}-{S}_{1}{y}_{n}\parallel +{\alpha }_{n}\parallel {S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\parallel +3{c}_{n}\parallel {v}_{n}-{S}_{1}{y}_{n}\parallel \\ \le & \left[1-\theta \left({x}_{n+1},{x}^{\ast }\right){\alpha }_{n}+{\alpha }_{n}^{2}\right]\parallel {x}_{n}-{x}^{\ast }\parallel +2{\alpha }_{n}^{2}\parallel {x}_{n}-{S}_{1}{y}_{n}\parallel \\ +{\alpha }_{n}\parallel {S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\parallel +3{c}_{n}\parallel {v}_{n}-{S}_{1}{y}_{n}\parallel .\end{array}$
(3.5)
Furthermore, we have the following estimates:
(3.6)
(3.7)
(3.8)
(3.9)
Using (3.4) and (3.6),
$\begin{array}{rcl}\parallel {x}_{n}-{y}_{n}\parallel & =& \parallel {\beta }_{n}\left({x}_{n}-{S}_{2}{z}_{n}\right)-{c}_{n}^{\mathrm{\prime }}\left({u}_{n}-{S}_{2}{z}_{n}\right)\parallel \\ \le & {\beta }_{n}\parallel {x}_{n}-{S}_{2}{z}_{n}\parallel +{c}_{n}^{\mathrm{\prime }}\parallel {u}_{n}-{S}_{2}{z}_{n}\parallel \\ \le & \left[\left[1+{L}_{\ast }\left(3{L}_{\ast }-1\right)\right]{\beta }_{n}+{L}_{\ast }\left(3{L}_{\ast }-1\right){c}_{n}^{\mathrm{\prime }}\right]\parallel {x}_{n}-{x}^{\ast }\parallel \\ +{L}_{\ast }\left({\beta }_{n}+{c}_{n}^{\mathrm{\prime }}\right){c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\parallel {w}_{n}-{x}^{\ast }\parallel +{c}_{n}^{\mathrm{\prime }}\parallel {u}_{n}-{x}^{\ast }\parallel \\ \le & \left[\left[1+{L}_{\ast }\left(3{L}_{\ast }-1\right)\right]{\beta }_{n}+{L}_{\ast }\left(3{L}_{\ast }-1\right){c}_{n}^{\mathrm{\prime }}\right]\parallel {x}_{n}-{x}^{\ast }\parallel \\ +3{L}_{\ast }{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\parallel {w}_{n}-{x}^{\ast }\parallel +{c}_{n}^{\mathrm{\prime }}\parallel {u}_{n}-{x}^{\ast }\parallel .\end{array}$
(3.10)
Using (3.7),
$\begin{array}{rcl}\parallel {S}_{1}{y}_{n}-{y}_{n}\parallel & \le & \parallel {S}_{1}{y}_{n}-{x}^{\ast }\parallel +\parallel {y}_{n}-{x}^{\ast }\parallel \\ \le & \left(1+{L}_{\ast }\right)\parallel {y}_{n}-{x}^{\ast }\parallel \\ \le & \left(1+{L}_{\ast }\right)\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]\parallel {x}_{n}-{x}^{\ast }\parallel \\ +3{L}_{\ast }\left(1+{L}_{\ast }\right){c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\parallel {w}_{n}-{x}^{\ast }\parallel +\left(1+{L}_{\ast }\right){c}_{n}^{\mathrm{\prime }}\parallel {u}_{n}-{x}^{\ast }\parallel .\end{array}$
(3.11)
Again, using (3.7),
$\begin{array}{rcl}\parallel {v}_{n}-{S}_{1}{y}_{n}\parallel & \le & \parallel {v}_{n}-{x}^{\ast }\parallel +{L}_{\ast }\parallel {y}_{n}-{x}^{\ast }\parallel \\ \le & {L}_{\ast }\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]\parallel {x}_{n}-{x}^{\ast }\parallel +\parallel {v}_{n}-{x}^{\ast }\parallel \\ +3{L}_{\ast }^{2}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\parallel {w}_{n}-{x}^{\ast }\parallel +{L}_{\ast }{c}_{n}^{\mathrm{\prime }}\parallel {u}_{n}-{x}^{\ast }\parallel .\end{array}$
(3.12)
Substituting (3.10)-(3.12) in (3.9), we obtain
$\begin{array}{rcl}\parallel {S}_{1}{x}_{n+1}-{S}_{1}{y}_{n}\parallel & \le & {L}_{\ast }\left[1+{L}_{\ast }\left(3{L}_{\ast }-1\right)\right]{\beta }_{n}+{L}_{\ast }\left(3{L}_{\ast }-1\right){c}_{n}^{\mathrm{\prime }}\\ +\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]\left[\left(1+{L}_{\ast }\right){\alpha }_{n}+{L}_{\ast }{c}_{n}\right]\parallel {x}_{n}-{x}^{\ast }\parallel \\ +3{L}_{\ast }\left[{L}_{\ast }{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}+\left[\left(1+{L}_{\ast }\right){\alpha }_{n}+{L}_{\ast }{c}_{n}\right]{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right]\parallel {w}_{n}-{x}^{\ast }\parallel \\ +{L}_{\ast }\left[{c}_{n}^{\mathrm{\prime }}+\left[\left(1+{L}_{\ast }\right){\alpha }_{n}+{L}_{\ast }{c}_{n}\right]{c}_{n}^{\mathrm{\prime }}\right]\parallel {u}_{n}-{x}^{\ast }\parallel \\ +{L}_{\ast }{c}_{n}\parallel {v}_{n}-{x}^{\ast }\parallel .\end{array}$
(3.13)
Substituting (3.8), (3.12) and (3.13) in (3.5), we obtain
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}^{\ast }\parallel & \le & \left[1+\left[3+{L}_{\ast }\left(3+{L}_{\ast }\right)3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]\right]{\alpha }_{n}^{2}\\ +{L}_{\ast }\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]{\alpha }_{n}{\beta }_{n}+{L}_{\ast }^{2}\left(3{L}_{\ast }-1\right){\alpha }_{n}{c}_{n}^{\mathrm{\prime }}\\ +{L}_{\ast }\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]{\alpha }_{n}{c}_{n}+3{L}_{\ast }\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]{c}_{n}\parallel {x}_{n}-{x}^{\ast }\parallel \\ -\theta \left({x}_{n+1},{x}^{\ast }\right){\alpha }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +\left[3{L}_{\ast }\left(1+3{L}_{\ast }\right){\alpha }_{n}^{2}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}+3{L}_{\ast }^{2}{\alpha }_{n}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}+3{L}_{\ast }^{2}{\alpha }_{n}{c}_{n}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\\ +9{L}_{\ast }^{2}{c}_{n}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right]\parallel {w}_{n}-{x}^{\ast }\parallel +\left[{L}_{\ast }\left(3+{L}_{\ast }\right){\alpha }_{n}^{2}{c}_{n}^{\mathrm{\prime }}+{L}_{\ast }{\alpha }_{n}{c}_{n}^{\mathrm{\prime }}\\ +{L}_{\ast }^{2}{\alpha }_{n}{c}_{n}{c}_{n}^{\mathrm{\prime }}+3{L}_{\ast }{c}_{n}{c}_{n}^{\mathrm{\prime }}\right]\parallel {u}_{n}-{x}^{\ast }\parallel +\left(2{L}_{\ast }+3\right){c}_{n}\parallel {v}_{n}-{x}^{\ast }\parallel .\end{array}$
(3.14)
Since $\left\{{v}_{n}\right\}$, $\left\{{u}_{n}\right\}$ and $\left\{{w}_{n}\right\}$ are bounded, we set
$M=\underset{n\ge 0}{sup}\parallel {v}_{n}-{x}^{\ast }\parallel +\underset{n\ge 0}{sup}\parallel {u}_{n}-{x}^{\ast }\parallel +\underset{n\ge 0}{sup}\parallel {w}_{n}-{x}^{\ast }\parallel <\mathrm{\infty }.$
Then it follows from (3.14) that
$\begin{array}{rcl}\parallel {x}_{n+1}-{x}^{\ast }\parallel & \le & \left[1+\left[3+{L}_{\ast }\left(3+{L}_{\ast }\right)\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]\right]{\alpha }_{n}^{2}\\ +{L}_{\ast }\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]{\alpha }_{n}{\beta }_{n}+{L}_{\ast }^{2}\left(3{L}_{\ast }-1\right){\alpha }_{n}{c}_{n}^{\mathrm{\prime }}\\ +{L}_{\ast }\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]{\alpha }_{n}{c}_{n}+3{L}_{\ast }\left[3{L}_{\ast }\left(3{L}_{\ast }-1\right)-1\right]{c}_{n}\right]\parallel {x}_{n}-{x}^{\ast }\parallel \\ -\theta \left({x}_{n+1},{x}^{\ast }\right){\alpha }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +\left[3{L}_{\ast }\left(1+3{L}_{\ast }\right){\alpha }_{n}^{2}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}+3{L}_{\ast }^{2}{\alpha }_{n}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}+3{L}_{\ast }^{2}{\alpha }_{n}{c}_{n}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\\ +9{L}_{\ast }^{2}{c}_{n}{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right]M+\left[{L}_{\ast }\left(3+{L}_{\ast }\right){\alpha }_{n}^{2}{c}_{n}^{\mathrm{\prime }}+{L}_{\ast }{\alpha }_{n}{c}_{n}^{\mathrm{\prime }}\\ +{L}_{\ast }^{2}{\alpha }_{n}{c}_{n}{c}_{n}^{\mathrm{\prime }}+3{L}_{\ast }{c}_{n}{c}_{n}^{\mathrm{\prime }}\right]M+\left(2{L}_{\ast }+3\right){c}_{n}M\\ =& \left(1+{\delta }_{n}\right)\parallel {x}_{n}-{x}^{\ast }\parallel -\theta \left({x}_{n+1},{x}^{\ast }\right){\alpha }_{n}\parallel {x}_{n}-{x}^{\ast }\parallel +{\sigma }_{n}\\ \le & \left(1+{\delta }_{n}\right)\parallel {x}_{n}-{x}^{\ast }\parallel +{\sigma }_{n},\end{array}$
(3.15)
where
Since ${b}_{n}\in \left(0,1\right)$, the conditions (iii) and (iv) imply that ${\sum }_{n=0}^{\mathrm{\infty }}{\delta }_{n}<\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\sigma }_{n}<\mathrm{\infty }$. It then follows from Lemma 5 that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{x}^{\ast }\parallel$ exists. Let ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{x}^{\ast }\parallel =\delta \ge 0$. We now prove that $\delta =0$. Assume that $\delta >0$. Then there exists a positive integer ${N}_{0}$ such that $\parallel {x}_{n}-{x}^{\ast }\parallel \ge \frac{\delta }{2}$ for all $n\ge {N}_{0}$. Since
$\theta \left({x}_{n+1},{x}^{\ast }\right)\parallel {x}_{n}-{x}^{\ast }\parallel =\frac{\varphi \left(\parallel {x}_{n+1}-{x}^{\ast }\parallel \right)}{1+\varphi \left(\parallel {x}_{n+1}-{x}^{\ast }\parallel \right)+\parallel {x}_{n+1}-{x}^{\ast }\parallel }\parallel {x}_{n}-{x}^{\ast }\parallel \ge \frac{\varphi \left(\frac{\delta }{2}\right)\delta }{2\left(1+\varphi \left(D\right)+D\right)},$
for all $n\ge {N}_{0}$, it follows from (3.15) that
Hence,
This implies that
$\frac{\varphi \left(\frac{\delta }{2}\right)\delta }{2\left(1+\varphi \left(D\right)+D\right)}\sum _{j={N}_{0}}^{n}{\alpha }_{j}\le \parallel {x}_{{N}_{0}}-{x}^{\ast }\parallel +\sum _{j={N}_{0}}^{n}{\lambda }_{j}.$
Since ${b}_{n}\le {\alpha }_{n}$,
$\frac{\varphi \left(\frac{\delta }{2}\right)\delta }{2\left(1+\varphi \left(D\right)+D\right)}\sum _{j={N}_{0}}^{n}{b}_{j}\le \parallel {x}_{{N}_{0}}-{x}^{\ast }\parallel +\sum _{j={N}_{0}}^{n}{\lambda }_{j}$

yields ${\sum }_{n=0}^{\mathrm{\infty }}{b}_{n}<\mathrm{\infty }$, contradicting the fact that ${\sum }_{n=0}^{\mathrm{\infty }}{b}_{n}=\mathrm{\infty }$. Hence, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-{x}^{\ast }\parallel =0$. □

Corollary 9 Let X be an arbitrary real Banach space and ${T}_{1},{T}_{2},{T}_{3}:X\to X$ be three Lipschitz ϕ-strongly accretive operators, where ϕ is in addition continuous. Suppose ${lim inf}_{r\to \mathrm{\infty }}\varphi \left(r\right)>0$ or $\parallel {T}_{i}x\parallel \to \mathrm{\infty }$ as $\parallel x\parallel \to \mathrm{\infty }$; $i=1,2,3$. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, $\left\{{a}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{b}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{c}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{a}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{b}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{w}_{n}\right\}$, $\left\{{u}_{n}\right\}$, $\left\{{v}_{n}\right\}$, $\left\{{y}_{n}\right\}$ and $\left\{{x}_{n}\right\}$ be as in Theorem  8. Then, for any given $f\in X$, the sequence $\left\{{x}_{n}\right\}$ converges strongly to the solution of the system ${T}_{i}x=f$; $i=1,2,3$.

Proof The existence of a unique solution to the system ${T}_{i}x=f$; $i=1,2,3$ follows from [9] and the result follows from Theorem 8. □

Theorem 10 Let X be a real Banach space and K be a nonempty closed convex subset of X. Let ${T}_{1},{T}_{2},{T}_{3}:K\to K$ be three Lipschitz ϕ-strong pseudocontractions with a nonempty fixed point set. Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, $\left\{{a}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{b}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{c}_{n}^{\mathrm{\prime }}\right\}$, $\left\{{a}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{b}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}\right\}$, $\left\{{w}_{n}\right\}$, $\left\{{u}_{n}\right\}$ and $\left\{{v}_{n}\right\}$ be as in Theorem  8. Let $\left\{{x}_{n}\right\}$ be the sequence generated iteratively from an arbitrary ${x}_{0}\in K$ by
$\begin{array}{c}{x}_{n+1}={a}_{n}{x}_{n}+{b}_{n}{T}_{1}{y}_{n}+{c}_{n}{v}_{n},\hfill \\ {y}_{n}={a}_{n}^{\mathrm{\prime }}{x}_{n}+{b}_{n}^{\mathrm{\prime }}{T}_{2}{z}_{n}+{c}_{n}^{\mathrm{\prime }}{u}_{n},\hfill \\ {z}_{n}={a}_{n}^{\mathrm{\prime }\mathrm{\prime }}{x}_{n}+{b}_{n}^{\mathrm{\prime }\mathrm{\prime }}{T}_{3}{x}_{n}+{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}{w}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.\hfill \end{array}$

Then $\left\{{x}_{n}\right\}$ converges strongly to the common fixed point of ${T}_{1}$, ${T}_{2}$, ${T}_{3}$.

Proof As in the proof of Theorem 8, set ${\alpha }_{n}={b}_{n}+{c}_{n}$, ${\beta }_{n}={b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}$, ${\gamma }_{n}={b}_{n}^{\mathrm{\prime }\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }\mathrm{\prime }}$ to obtain
$\begin{array}{c}{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}{T}_{1}{y}_{n}+{c}_{n}\left({v}_{n}-{T}_{1}{y}_{n}\right),\hfill \\ {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}{T}_{2}{z}_{n}+{c}_{n}\left({u}_{n}-{T}_{2}{z}_{n}\right),\hfill \\ {z}_{n}=\left(1-{\gamma }_{n}\right){x}_{n}+{\gamma }_{n}{T}_{3}{x}_{n}+{c}_{n}\left({w}_{n}-{T}_{3}{x}_{n}\right),\phantom{\rule{1em}{0ex}}n\ge 0.\hfill \end{array}$
Since each ${T}_{i}$; $i=1,2,3$ is a ϕ-strong pseudocontraction, $\left(I-{T}_{i}\right)$ is ϕ-strongly accretive so that for all $x,y\in X$, there exist $j\left(x-y\right)\in J\left(x-y\right)$ and a strictly increasing function $\varphi :\left(0,\mathrm{\infty }\right)\to \left(0,\mathrm{\infty }\right)$ with $\varphi \left(0\right)=0$ such that
$〈\left(I-{T}_{i}\right)x-\left(I-{T}_{i}\right)y,j\left(x-y\right)〉\ge \varphi \left(\parallel x-y\parallel \right)\parallel x-y\parallel \ge \theta \left(x,y\right){\parallel x-y\parallel }^{2};\phantom{\rule{1em}{0ex}}i=1,2,3.$

The rest of the argument now follows as in the proof of Theorem 8. □

Remark 11 The example in [4] shows that the class of ϕ-strongly pseudocontractive operators with nonempty fixed point sets is a proper subclass of the class of ϕ-hemicontractive operators. It is easy to see that Theorem 8 easily extends to the class of ϕ-hemicontractive operators.

Remark 12
1. (i)

If we set ${b}_{n}^{\mathrm{\prime }\mathrm{\prime }}=0={c}_{n}^{\mathrm{\prime }\mathrm{\prime }}$ for all $n\ge 0$ in our results, we obtain the corresponding results for the Ishikawa iteration scheme with error terms in the sense of Xu [15].

2. (ii)

If we set ${b}_{n}^{\mathrm{\prime }\mathrm{\prime }}=0={c}_{n}^{\mathrm{\prime }\mathrm{\prime }}={b}_{n}^{\mathrm{\prime }}=0={c}_{n}^{\mathrm{\prime }}$ for all $n\ge 0$ in our results, we obtain the corresponding results for the Mann iteration scheme with error terms in the sense of Xu [15].

Remark 13 Let $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ be real sequences satisfying the following conditions:
1. (i)

$0\le {\alpha }_{n},{\beta }_{n}\le 1$, $n\ge 0$,

2. (ii)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}={lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0$,

3. (iii)

${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$,

4. (iv)

${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_{n}<\mathrm{\infty }$, and

5. (v)

${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}^{2}<\mathrm{\infty }$.

If we set ${a}_{n}^{\mathrm{\prime }}=\left(1-{\beta }_{n}\right)$, ${b}_{n}^{\mathrm{\prime }}={\beta }_{n}$, ${c}_{n}^{\mathrm{\prime }}=0$, ${a}_{n}=\left(1-{\alpha }_{n}\right)$, ${b}_{n}={\alpha }_{n}$, ${c}_{n}=0$, ${b}_{n}^{\mathrm{\prime }\mathrm{\prime }}=0={c}_{n}^{\mathrm{\prime }\mathrm{\prime }}$ for all $n\ge 0$ in Theorems 8 and 10 respectively, we obtain the corresponding convergence theorems for the original Ishikawa [18] and Mann [30] iteration schemes.

Remark 14
1. (i)

Gurudwan and Sharma [29] studied a strong convergence of multi-step iterative scheme to a common solution for a finite family of ϕ-strongly accretive operator equations in a reflexive Banach space with weakly continuous duality mapping. Some remarks on their work can be seen in [31].

2. (ii)

All the above results can be extended to a finite family of ϕ-strongly accretive operators.

## Declarations

### Acknowledgements

The last author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

## Authors’ Affiliations

(1)
Department of Mathematics, Statistics and Physics, Qatar University, Doha, 2713, Qatar
(2)
Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan
(3)
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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## Copyright

© Khan et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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