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

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.

1 Introduction

Let K be a nonempty subset of an arbitrary Banach space X and $X^{\ast }$ be its dual space. The symbols $D(T)$, $R(T)$ and $F(T)$ 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 $T(x)=x$). We denote by J the normalized duality mapping from E to $2^{E^{\ast }}$ defined by

$$J(x)=\{f^{\ast }\in X^{\ast }:〈x,f^{\ast }〉={\parallel x\parallel }^2={\parallel f^{\ast }\parallel }^2\}.$$

Let $T:D(T)\subseteq X\to X$ be an operator. The following definitions can be found in [1–15] 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 $0<L<1$, T is called contraction.

Definition 2

• (i) T is said to be strongly pseudocontractive if there exists a $t>1$ such that for each $x,y\in D(T)$, there exists $j(x-y)\in J(x-y)$ satisfying

  $$Re〈Tx-Ty,j(x-y)〉\le \frac{1}{t}{\parallel x-y\parallel }^2.$$

• (ii) T is said to be strictly hemicontractive if $F(T)$ is nonempty and if there exists a $t>1$ such that for each $x\in D(T)$ and $q\in F(T)$, there exists $j(x-y)\in J(x-y)$ satisfying

  $$Re〈Tx-q,j(x-q)〉\le \frac{1}{t}{\parallel x-q\parallel }^2.$$

• (iii) T is said to be ϕ-strongly pseudocontractive if there exists a strictly increasing function $\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\varphi (0)=0$ such that for each $x,y\in D(T)$, there exists $j(x-y)\in J(x-y)$ satisfying

  $$Re〈Tx-Ty,j(x-y)〉\le {\parallel x-y\parallel }^2-\varphi (\parallel x-y\parallel )\parallel x-y\parallel .$$

• (iv) T is said to be ϕ-hemicontractive if $F(T)$ is nonempty and if there exists a strictly increasing function $\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\varphi (0)=0$ such that for each $x\in D(T)$ and $q\in F(T)$, there exists $j(x-y)\in J(x-y)$ satisfying

  $$Re〈Tx-q,j(x-q)〉\le {\parallel x-q\parallel }^2-\varphi (\parallel x-q\parallel )\parallel x-q\parallel .$$

Clearly, each strictly hemicontractive operator is ϕ-hemicontractive.

Definition 3

• (i) T is called accretive if the inequality

  $$\parallel x-y\parallel \le \parallel x-y+s(Tx-Ty)\parallel$$

holds for every $x,y\in D(T)$ and for all $s>0$.

• (ii) T is called strongly accretive if, for all $x,y\in D(T)$, there exists a constant $k>0$ and $j(x-y)\in J(x-y)$ such that

  $$〈Tx-Ty,j(x-y)〉\ge k{\parallel x-y\parallel }^2.$$

• (iii) T is called ϕ-strongly accretive if there exists $j(x-y)\in J(x-y)$ and a strictly increasing function $\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\varphi (0)=0$ such that for each $x,y\in X$,

  $$〈Tx-Ty,j(x-y)〉\ge \varphi (\parallel x-y\parallel )\parallel x-y\parallel .$$

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 $(I-T)$ 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 $p\in [2,\mathrm{\infty })$. 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, [7–25]). 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 $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ 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

$$a_{n+1}\le (1+b_n)a_n+c_n,n\ge 1,$$

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 $f\in J(x)$ 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 $T_1,T_2,T_3:X\to X$, $L_{\ast }=(1+L)$ and $R(T_1)$, $R(T_2)$ and $R(T_3)$ 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 $f\in R(T_1)\cap R(T_2)\cap R(T_3)$ and generate $\{x_n\}$ from an arbitrary $x_0\in X$ by

$$\begin{array}{r}{x}_{n+1}=a_n{x}_n+b_n(f+(I-T_1)y_n)+c_n{v}_n,\\ y_n=a_n^{\mathrm{\prime }}{x}_n+b_n^{\mathrm{\prime }}(f+(I-T_2)z_n)+c_n^{\mathrm{\prime }}{u}_n,\\ z_n=a_n^{\mathrm{\prime }\mathrm{\prime }}{x}_n+b_n^{\mathrm{\prime }\mathrm{\prime }}(f+(I-T_3)x_n)+c_n^{\mathrm{\prime }\mathrm{\prime }}{w}_n,n\ge 0,\end{array}$$

(3.1)

where ${\{v_n\}}_{n=0}^{\mathrm{\infty }}$, ${\{u_n\}}_{n=0}^{\mathrm{\infty }}$ and ${\{w_n\}}_{n=0}^{\mathrm{\infty }}$ are bounded sequences in X and $\{a_n\}$, $\{c_n\}$, $\{a_n^{\mathrm{\prime }}\}$, $\{b_n^{\mathrm{\prime }}\}$, $\{c_n^{\mathrm{\prime }}\}$, $\{a_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{b_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{c_n^{\mathrm{\prime }\mathrm{\prime }}\}$ are sequences in $[0,1]$ and $\{b_n\}$ in $(0,1)$ 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 $\{x_n\}$ 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, 8–12, 26, 29], define $S_i:X\to X$ by $S_{i}x=f+(I-T_i)x$; $i=1,2,3$; 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

$$\begin{array}{c}〈(I-S_i)x-(I-S_i)y,j(x-y)〉\hfill \\ \ge {\varphi }_i(\parallel x-y\parallel )\parallel x-y\parallel \hfill \\ \ge \frac{{\varphi }_i(\parallel x-y\parallel )}{(1+{\varphi }_i(\parallel x-y\parallel )+\parallel x-y\parallel )}{\parallel x-y\parallel }^2\hfill \\ ={\theta }_i(x,y){\parallel x-y\parallel }^2,\hfill \end{array}$$

where ${\theta }_i(x,y)=\frac{{\varphi }_i(\parallel x-y\parallel )}{(1+{\varphi }_i(\parallel x-y\parallel )+\parallel x-y\parallel )}\in [0,1)$ for all $x,y\in X$; $i=1,2,3$. Let $x^{\ast }\in {\bigcap }_{i=1}^{3}F(S_i)$ be the fixed point set of $S_i$, and let $\theta (x,y)={inf\hspace{0.17em}min}_i\{{\theta }_i(x,y)\}\in [0,1]$. Thus

$$〈(I-S_i)x-(I-S_i)y,j(x-y)〉\ge \theta (x,y){\parallel x-y\parallel }^2;i=1,2,3.$$

(3.2)

It follows from Lemma 6 and inequality (3.2) that

$$\parallel x-y\parallel \le \parallel x-y+\lambda [(I-S_i)x-\theta (x,y)x-((I-S_i)y-\theta (x,y)y)]\parallel ,$$

(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

$$\begin{array}{r}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{S}_1{y}_n+c_n(v_n-S_1{y}_n),\\ y_n=(1-{\beta }_n)x_n+{\beta }_n{S}_2{z}_n+c_n^{\mathrm{\prime }}(u_n-S_2{z}_n),\\ z_n=(1-{\gamma }_n)x_n+{\gamma }_n{S}_3{x}_n+c_n^{\mathrm{\prime }\mathrm{\prime }}(w_n-S_3{x}_n),n\ge 0.\end{array}$$

(3.4)

We have

$$\begin{array}{rcl}{x}_n& =& (1+{\alpha }_n)x_{n+1}+{\alpha }_n[(I-S_1)x_{n+1}-\theta (x_{n+1},x^{\ast })x_{n+1}]\\ -(1-\theta (x_{n+1},x^{\ast })){\alpha }_n{x}_n+(2-\theta (x_{n+1},x^{\ast })){\alpha }_n^2(x_n-S_1{y}_n)\\ +{\alpha }_n(S_1{x}_{n+1}-S_1{y}_n)-[1+(2-\theta (x_{n+1},x^{\ast })){\alpha }_n]c_n(v_n-S_1{y}_n).\end{array}$$

Furthermore,

$$x^{\ast }=(1+{\alpha }_n)x^{\ast }+{\alpha }_n[(I-S_1)x^{\ast }-\theta (x_{n+1},x^{\ast })x^{\ast }]-(1-\theta (x_{n+1},x^{\ast })){\alpha }_n{x}^{\ast },$$

so that

$$\begin{array}{rcl}{x}_n-x^{\ast }& =& (1+{\alpha }_n)(x_{n+1}-x^{\ast })+{\alpha }_n[(I-S_1)x_{n+1}-\theta (x_{n+1},x^{\ast })x_{n+1}\\ -((I-S_1)x^{\ast }-\theta (x_{n+1},x^{\ast })x^{\ast })]\\ -(1-\theta (x_{n+1},x^{\ast })){\alpha }_n(x_n-x^{\ast })+(2-\theta (x_{n+1},x^{\ast })){\alpha }_n^2(x_n-S_1{y}_n)\\ +{\alpha }_n(S_1{x}_{n+1}-S_1{y}_n)-[1+(2-\theta (x_{n+1},x^{\ast })){\alpha }_n]c_n(v_n-S_1{y}_n).\end{array}$$

Hence,

$$\begin{array}{rcl}\parallel x_n-x^{\ast }\parallel & \ge & (1+{\alpha }_n)\parallel x_{n+1}-x^{\ast }+\frac{{\alpha }_n}{(1+{\alpha }_n)}[(I-S_1)x_{n+1}-\theta (x_{n+1},x^{\ast })x_{n+1}\\ -((I-S_1)x^{\ast }-\theta (x_{n+1},x^{\ast })x^{\ast })]\parallel \\ -(1-\theta (x_{n+1},x^{\ast })){\alpha }_n\parallel x_n-x^{\ast }\parallel -(2-\theta (x_{n+1},x^{\ast })){\alpha }_n^2\parallel x_n-S_1{y}_n\parallel \\ -{\alpha }_n\parallel S_1{x}_{n+1}-S_1{y}_n\parallel -[1+(2-\theta (x_{n+1},x^{\ast })){\alpha }_n]c_n\parallel v_n-S_1{y}_n\parallel \\ \ge & (1+{\alpha }_n)\parallel x_{n+1}-x^{\ast }\parallel -(1-\theta (x_{n+1},x^{\ast })){\alpha }_n\parallel x_n-x^{\ast }\parallel \\ -(2-\theta (x_{n+1},x^{\ast })){\alpha }_n^2\parallel x_n-S_1{y}_n\parallel -{\alpha }_n\parallel S_1{x}_{n+1}-S_1{y}_n\parallel \\ -[1+(2-\theta (x_{n+1},x^{\ast })){\alpha }_n]c_n\parallel v_n-S_1{y}_n\parallel .\end{array}$$

Hence,

$$\begin{array}{rcl}\parallel x_{n+1}-x^{\ast }\parallel & \le & \frac{[1+(1-\theta (x_{n+1},x^{\ast })){\alpha }_n]}{(1+{\alpha }_n)}\parallel x_n-x^{\ast }\parallel +2{\alpha }_n^2\parallel x_n-S_1{y}_n\parallel \\ +{\alpha }_n\parallel S_1{x}_{n+1}-S_1{y}_n\parallel +[1+(2-\theta (x_{n+1},x^{\ast })){\alpha }_n]c_n\parallel v_n-S_1{y}_n\parallel \\ \le & [1+(1-\theta (x_{n+1},x^{\ast })){\alpha }_n][1-{\alpha }_n+{\alpha }_n^2]\parallel x_n-x^{\ast }\parallel \\ +2{\alpha }_n^2\parallel x_n-S_1{y}_n\parallel +{\alpha }_n\parallel S_1{x}_{n+1}-S_1{y}_n\parallel +3c_n\parallel v_n-S_1{y}_n\parallel \\ \le & [1-\theta (x_{n+1},x^{\ast }){\alpha }_n+{\alpha }_n^2]\parallel x_n-x^{\ast }\parallel +2{\alpha }_n^2\parallel x_n-S_1{y}_n\parallel \\ +{\alpha }_n\parallel S_1{x}_{n+1}-S_1{y}_n\parallel +3c_n\parallel v_n-S_1{y}_n\parallel .\end{array}$$

(3.5)

Furthermore, we have the following estimates:

(3.6)

(3.7)

(3.8)

(3.9)

Using (3.4) and (3.6),

$$\begin{array}{rcl}\parallel x_n-y_n\parallel & =& \parallel {\beta }_n(x_n-S_2{z}_n)-c_n^{\mathrm{\prime }}(u_n-S_2{z}_n)\parallel \\ \le & {\beta }_n\parallel x_n-S_2{z}_n\parallel +c_n^{\mathrm{\prime }}\parallel u_n-S_2{z}_n\parallel \\ \le & [[1+L_{\ast }(3L_{\ast }-1)]{\beta }_n+L_{\ast }(3L_{\ast }-1)c_n^{\mathrm{\prime }}]\parallel x_n-x^{\ast }\parallel \\ +L_{\ast }({\beta }_n+c_n^{\mathrm{\prime }})c_n^{\mathrm{\prime }\mathrm{\prime }}\parallel w_n-x^{\ast }\parallel +c_n^{\mathrm{\prime }}\parallel u_n-x^{\ast }\parallel \\ \le & [[1+L_{\ast }(3L_{\ast }-1)]{\beta }_n+L_{\ast }(3L_{\ast }-1)c_n^{\mathrm{\prime }}]\parallel x_n-x^{\ast }\parallel \\ +3L_{\ast }{c}_n^{\mathrm{\prime }\mathrm{\prime }}\parallel w_n-x^{\ast }\parallel +c_n^{\mathrm{\prime }}\parallel u_n-x^{\ast }\parallel .\end{array}$$

(3.10)

Using (3.7),

$$\begin{array}{rcl}\parallel S_1{y}_n-y_n\parallel & \le & \parallel S_1{y}_n-x^{\ast }\parallel +\parallel y_n-x^{\ast }\parallel \\ \le & (1+L_{\ast })\parallel y_n-x^{\ast }\parallel \\ \le & (1+L_{\ast })[3L_{\ast }(3L_{\ast }-1)-1]\parallel x_n-x^{\ast }\parallel \\ +3L_{\ast }(1+L_{\ast })c_n^{\mathrm{\prime }\mathrm{\prime }}\parallel w_n-x^{\ast }\parallel +(1+L_{\ast })c_n^{\mathrm{\prime }}\parallel u_n-x^{\ast }\parallel .\end{array}$$

(3.11)

Again, using (3.7),

$$\begin{array}{rcl}\parallel v_n-S_1{y}_n\parallel & \le & \parallel v_n-x^{\ast }\parallel +L_{\ast }\parallel y_n-x^{\ast }\parallel \\ \le & L_{\ast }[3L_{\ast }(3L_{\ast }-1)-1]\parallel x_n-x^{\ast }\parallel +\parallel v_n-x^{\ast }\parallel \\ +3L_{\ast }^2{c}_n^{\mathrm{\prime }\mathrm{\prime }}\parallel w_n-x^{\ast }\parallel +L_{\ast }{c}_n^{\mathrm{\prime }}\parallel u_n-x^{\ast }\parallel .\end{array}$$

(3.12)

Substituting (3.10)-(3.12) in (3.9), we obtain

$$\begin{array}{rcl}\parallel S_1{x}_{n+1}-S_1{y}_n\parallel & \le & L_{\ast }[1+L_{\ast }(3L_{\ast }-1)]{\beta }_n+L_{\ast }(3L_{\ast }-1)c_n^{\mathrm{\prime }}\\ +[3L_{\ast }(3L_{\ast }-1)-1][(1+L_{\ast }){\alpha }_n+L_{\ast }{c}_n]\parallel x_n-x^{\ast }\parallel \\ +3L_{\ast }[L_{\ast }{c}_n^{\mathrm{\prime }\mathrm{\prime }}+[(1+L_{\ast }){\alpha }_n+L_{\ast }{c}_n]c_n^{\mathrm{\prime }\mathrm{\prime }}]\parallel w_n-x^{\ast }\parallel \\ +L_{\ast }[c_n^{\mathrm{\prime }}+[(1+L_{\ast }){\alpha }_n+L_{\ast }{c}_n]c_n^{\mathrm{\prime }}]\parallel u_n-x^{\ast }\parallel \\ +L_{\ast }{c}_n\parallel v_n-x^{\ast }\parallel .\end{array}$$

(3.13)

Substituting (3.8), (3.12) and (3.13) in (3.5), we obtain

$$\begin{array}{rcl}\parallel x_{n+1}-x^{\ast }\parallel & \le & [1+[3+L_{\ast }(3+L_{\ast })3L_{\ast }(3L_{\ast }-1)-1]]{\alpha }_n^2\\ +L_{\ast }[3L_{\ast }(3L_{\ast }-1)-1]{\alpha }_n{\beta }_n+L_{\ast }^2(3L_{\ast }-1){\alpha }_n{c}_n^{\mathrm{\prime }}\\ +L_{\ast }[3L_{\ast }(3L_{\ast }-1)-1]{\alpha }_n{c}_n+3L_{\ast }[3L_{\ast }(3L_{\ast }-1)-1]c_n\parallel x_n-x^{\ast }\parallel \\ -\theta (x_{n+1},x^{\ast }){\alpha }_n\parallel x_n-x^{\ast }\parallel +[3L_{\ast }(1+3L_{\ast }){\alpha }_n^2{c}_n^{\mathrm{\prime }\mathrm{\prime }}+3L_{\ast }^2{\alpha }_n{c}_n^{\mathrm{\prime }\mathrm{\prime }}+3L_{\ast }^2{\alpha }_n{c}_n{c}_n^{\mathrm{\prime }\mathrm{\prime }}\\ +9L_{\ast }^2{c}_n{c}_n^{\mathrm{\prime }\mathrm{\prime }}]\parallel w_n-x^{\ast }\parallel +[L_{\ast }(3+L_{\ast }){\alpha }_n^2{c}_n^{\mathrm{\prime }}+L_{\ast }{\alpha }_n{c}_n^{\mathrm{\prime }}\\ +L_{\ast }^2{\alpha }_n{c}_n{c}_n^{\mathrm{\prime }}+3L_{\ast }{c}_n{c}_n^{\mathrm{\prime }}]\parallel u_n-x^{\ast }\parallel +(2L_{\ast }+3)c_n\parallel v_n-x^{\ast }\parallel .\end{array}$$

(3.14)

Since $\{v_n\}$, $\{u_n\}$ and $\{w_n\}$ 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

$$\begin{array}{rcl}\parallel x_{n+1}-x^{\ast }\parallel & \le & [1+[3+L_{\ast }(3+L_{\ast })[3L_{\ast }(3L_{\ast }-1)-1]]{\alpha }_n^2\\ +L_{\ast }[3L_{\ast }(3L_{\ast }-1)-1]{\alpha }_n{\beta }_n+L_{\ast }^2(3L_{\ast }-1){\alpha }_n{c}_n^{\mathrm{\prime }}\\ +L_{\ast }[3L_{\ast }(3L_{\ast }-1)-1]{\alpha }_n{c}_n+3L_{\ast }[3L_{\ast }(3L_{\ast }-1)-1]c_n]\parallel x_n-x^{\ast }\parallel \\ -\theta (x_{n+1},x^{\ast }){\alpha }_n\parallel x_n-x^{\ast }\parallel +[3L_{\ast }(1+3L_{\ast }){\alpha }_n^2{c}_n^{\mathrm{\prime }\mathrm{\prime }}+3L_{\ast }^2{\alpha }_n{c}_n^{\mathrm{\prime }\mathrm{\prime }}+3L_{\ast }^2{\alpha }_n{c}_n{c}_n^{\mathrm{\prime }\mathrm{\prime }}\\ +9L_{\ast }^2{c}_n{c}_n^{\mathrm{\prime }\mathrm{\prime }}]M+[L_{\ast }(3+L_{\ast }){\alpha }_n^2{c}_n^{\mathrm{\prime }}+L_{\ast }{\alpha }_n{c}_n^{\mathrm{\prime }}\\ +L_{\ast }^2{\alpha }_n{c}_n{c}_n^{\mathrm{\prime }}+3L_{\ast }{c}_n{c}_n^{\mathrm{\prime }}]M+(2L_{\ast }+3)c_{n}M\\ =& (1+{\delta }_n)\parallel x_n-x^{\ast }\parallel -\theta (x_{n+1},x^{\ast }){\alpha }_n\parallel x_n-x^{\ast }\parallel +{\sigma }_n\\ \le & (1+{\delta }_n)\parallel x_n-x^{\ast }\parallel +{\sigma }_n,\end{array}$$

(3.15)

where

Since $b_n\in (0,1)$, 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

$$\theta (x_{n+1},x^{\ast })\parallel x_n-x^{\ast }\parallel =\frac{\varphi (\parallel x_{n+1}-x^{\ast }\parallel )}{1+\varphi (\parallel x_{n+1}-x^{\ast }\parallel )+\parallel x_{n+1}-x^{\ast }\parallel }\parallel x_n-x^{\ast }\parallel \ge \frac{\varphi (\frac{\delta }{2})\delta }{2(1+\varphi (D)+D)},$$

for all $n\ge N_0$, it follows from (3.15) that

$$\parallel x_{n+1}-x^{\ast }\parallel \le \parallel x_n-x^{\ast }\parallel -\frac{\varphi (\frac{\delta }{2})\delta }{2(1+\varphi (D)+D)}{\alpha }_n+{\lambda }_n\text{for all }n\ge N_0.$$

Hence,

$$\frac{\varphi (\frac{\delta }{2})\delta }{2(1+\varphi (D)+D)}{\alpha }_n\le \parallel x_n-x^{\ast }\parallel -\parallel x_{n+1}-x^{\ast }\parallel +{\lambda }_n\text{for all }n\ge N_0.$$

This implies that

$$\frac{\varphi (\frac{\delta }{2})\delta }{2(1+\varphi (D)+D)}\sum _{j=N_0}^n{\alpha }_j\le \parallel x_{N_0}-x^{\ast }\parallel +\sum _{j=N_0}^n{\lambda }_j.$$

Since $b_n\le {\alpha }_n$,

$$\frac{\varphi (\frac{\delta }{2})\delta }{2(1+\varphi (D)+D)}\sum _{j=N_0}^n{b}_j\le \parallel x_{N_0}-x^{\ast }\parallel +\sum _{j=N_0}^n{\lambda }_j$$

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 ${lim\hspace{0.17em}inf}_{r\to \mathrm{\infty }}\varphi (r)>0$ or $\parallel T_{i}x\parallel \to \mathrm{\infty }$ as $\parallel x\parallel \to \mathrm{\infty }$; $i=1,2,3$. Let $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{a_n^{\mathrm{\prime }}\}$, $\{b_n^{\mathrm{\prime }}\}$, $\{c_n^{\mathrm{\prime }}\}$, $\{a_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{b_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{c_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{w_n\}$, $\{u_n\}$, $\{v_n\}$, $\{y_n\}$ and $\{x_n\}$ be as in Theorem  8. Then, for any given $f\in X$, the sequence $\{x_n\}$ 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 $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{a_n^{\mathrm{\prime }}\}$, $\{b_n^{\mathrm{\prime }}\}$, $\{c_n^{\mathrm{\prime }}\}$, $\{a_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{b_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{c_n^{\mathrm{\prime }\mathrm{\prime }}\}$, $\{w_n\}$, $\{u_n\}$ and $\{v_n\}$ be as in Theorem  8. Let $\{x_n\}$ be the sequence generated iteratively from an arbitrary $x_0\in K$ by

$$\begin{array}{c}{x}_{n+1}=a_n{x}_n+b_n{T}_1{y}_n+c_n{v}_n,\hfill \\ y_n=a_n^{\mathrm{\prime }}{x}_n+b_n^{\mathrm{\prime }}{T}_2{z}_n+c_n^{\mathrm{\prime }}{u}_n,\hfill \\ z_n=a_n^{\mathrm{\prime }\mathrm{\prime }}{x}_n+b_n^{\mathrm{\prime }\mathrm{\prime }}{T}_3{x}_n+c_n^{\mathrm{\prime }\mathrm{\prime }}{w}_n,n\ge 0.\hfill \end{array}$$

Then $\{x_n\}$ 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

$$\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{T}_1{y}_n+c_n(v_n-T_1{y}_n),\hfill \\ y_n=(1-{\beta }_n)x_n+{\beta }_n{T}_2{z}_n+c_n(u_n-T_2{z}_n),\hfill \\ z_n=(1-{\gamma }_n)x_n+{\gamma }_n{T}_3{x}_n+c_n(w_n-T_3{x}_n),n\ge 0.\hfill \end{array}$$

Since each $T_i$; $i=1,2,3$ is a ϕ-strong pseudocontraction, $(I-T_i)$ is ϕ-strongly accretive so that for all $x,y\in X$, there exist $j(x-y)\in J(x-y)$ and a strictly increasing function $\varphi :(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ with $\varphi (0)=0$ such that

$$〈(I-T_i)x-(I-T_i)y,j(x-y)〉\ge \varphi (\parallel x-y\parallel )\parallel x-y\parallel \ge \theta (x,y){\parallel x-y\parallel }^2;i=1,2,3.$$

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

• (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].

• (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 $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be real sequences satisfying the following conditions:

• (i) $0\le {\alpha }_n,{\beta }_n\le 1$, $n\ge 0$,

• (ii) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n={lim}_{n\to \mathrm{\infty }}{\beta }_n=0$,

• (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$,

• (iv) ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n<\mathrm{\infty }$, and

• (v) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n^2<\mathrm{\infty }$.

If we set $a_n^{\mathrm{\prime }}=(1-{\beta }_n)$, $b_n^{\mathrm{\prime }}={\beta }_n$, $c_n^{\mathrm{\prime }}=0$, $a_n=(1-{\alpha }_n)$, $b_n={\alpha }_n$, $c_n=0$, $b_n^{\mathrm{\prime }\mathrm{\prime }}=0=c_n^{\mathrm{\prime }\mathrm{\prime }}$ 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

• (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].

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