# A further remark to paper ‘Convergence theorems for the common solution for a finite family of *ϕ*-strongly accretive operator equations’

- Zhiqun Xue
^{1}Email author and - Haiyun Zhou
^{2}

**2013**:35

https://doi.org/10.1186/1029-242X-2013-35

© Xue and Zhou; licensee Springer 2013

**Received: **2 September 2012

**Accepted: **6 January 2013

**Published: **30 January 2013

## Abstract

In this note, we point out several gaps in Gurudwan and Sharma (Appl. Math. Comput. 217(15):6748-6754, 2011) and Yang (Appl. Math. Comput. 218(21):10367-10369, 2012) and give the main results under weaker conditions.

**MSC:**47H10, 47H09, 46B20.

## Keywords

## 1 Introduction

for approximation of a common solution of a finite family of uniformly continuous Φ-strongly accretive operator equations. Their results are as follows.

**Theorem GS** [[1], Theorem 3.1]

*Let*

*E*

*be an arbitrary real Banach space and let*${\{{A}_{i}\}}_{i=1}^{N}:E\to E$

*be uniformly continuous*

*ϕ*-

*strongly accretive operators and each range of either*${A}_{i}$

*or*$(I-{A}_{i})$

*be bounded*.

*Let*,

*for*$i=1,\dots ,N$, ${\{{u}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$

*be sequences in*

*E*

*and*${\{{a}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$, ${\{{b}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$, ${\{{c}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$

*be real sequences in*$[0,1]$

*satisfying*

- (i)
${a}_{n}^{i}+{b}_{n}^{i}+{c}_{n}^{i}=1$,

- (ii)
${\sum}_{n=0}^{\mathrm{\infty}}{b}_{n}^{N}=\mathrm{\infty}$,

- (iii)
${\sum}_{n=0}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}$,

- (iv)
${lim}_{n\to \mathrm{\infty}}{b}_{n}^{i}={lim}_{n\to \mathrm{\infty}}{c}_{n}^{i}={lim}_{n\to \mathrm{\infty}}\frac{{c}_{n}^{i}}{{b}_{n}^{i}+{c}_{n}^{i}}=0$, $\mathrm{\forall}i=1,\dots ,N$, $n\ge 1$.

*For any given* $f\in E$, *define* ${\{{S}_{i}\}}_{i=1}^{N}:E\to E$ *by* ${S}_{i}x=x-{A}_{i}x+f$, $\mathrm{\forall}i=1,\dots ,N$, $\mathrm{\forall}x\in E$. *Then the multi*-*step iterative sequence with errors* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *defined by the above converges strongly to the unique solution of the operator equations* ${\{{A}_{i}x\}}_{i=1}^{N}=f$.

On the basis of the above result, Yang [2] proved the following convergence theorem.

**Thoerem Yang** [[2], Theorem 2]

*Let*

*E*

*be an arbitrary real Banach space and let*${\{{A}_{i}\}}_{i=1}^{N}:E\to E$

*be uniformly continuous*

*ϕ*-

*strongly accretive operators and each range of either*${A}_{i}$

*or*$(I-{A}_{i})$

*is bounded*.

*Let for*$i=1,\dots ,N$, ${\{{u}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$

*be bounded sequences in*

*E*

*and*${\{{a}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$, ${\{{b}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$, ${\{{c}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$

*be real sequences in*$[0,1]$

*satisfying*

- (i)
${a}_{n}^{i}+{b}_{n}^{i}+{c}_{n}^{i}=1$,

- (ii)
${\sum}_{n=0}^{\mathrm{\infty}}{b}_{n}^{N}=\mathrm{\infty}$,

- (iii)
${lim}_{n\to \mathrm{\infty}}{b}_{n}^{i}={lim}_{n\to \mathrm{\infty}}{c}_{n}^{i}={lim}_{n\to \mathrm{\infty}}\frac{{c}_{n}^{i}}{{b}_{n}^{i}+{c}_{n}^{i}}=0$, $\mathrm{\forall}i=1,\dots ,N$, $n\ge 1$.

*For any given* $f\in E$, *define* ${\{{S}_{i}\}}_{i=1}^{N}:E\to E$ *by* ${S}_{i}x=(I-{A}_{i})x+f$, $\mathrm{\forall}i=1,\dots ,N$, $\mathrm{\forall}x\in E$. *Then the multi*-*step iterative sequence with errors* ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *defined by the above converges strongly to the unique solution of the operator equations* ${\{{A}_{i}x\}}_{i=1}^{N}=f$.

However, after careful reading of their works, we discovered that there exist some problems in references [1] and [2] as follows.

**Problem 1** In the proof course of Theorem 3.1 of Gurudwan and Sharma [1], which happens in line 11 of page 6751. Here, it is defective that they obtained $\parallel x-y\parallel \le {\varphi}_{i}^{-1}(\parallel {A}_{i}x-{A}_{i}y\parallel )$, that is, $\u3008{A}_{i}x-{A}_{i}y,j(x-y)\u3009\ge \varphi (\parallel x-y\parallel )\parallel x-y\parallel \Rightarrow \varphi (\parallel x-y\parallel )\le \parallel {A}_{i}x-{A}_{i}y\parallel $, but we cannot deduce $\parallel x-y\parallel \le {\varphi}_{i}^{-1}(\parallel {A}_{i}x-{A}_{i}y\parallel )$. The reason is that it is possible $\parallel {A}_{i}x-{A}_{i}y\parallel $ does not belong to $R(\varphi )$ (range of *ϕ*). A counterexample is as follows. Let us define $\varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ by $\varphi (\alpha )=\frac{{2}^{\alpha}-1}{{2}^{\alpha}+1}$; then it can be easily seen that *ϕ* is increasing with $\varphi (0)=0$, but ${lim}_{\alpha \to +\mathrm{\infty}}\varphi (\alpha )=1$ and ${\varphi}^{-1}(2)$ makes no sense (see [3]).

**Problem 2**In the paper of Yang [2], he referred to the mistakes of ‘$\parallel {x}_{{n}_{m}+j}^{i}-q\parallel <\u03f5$ for $j\ge 1$ to deduce $\parallel {x}_{n}-q\parallel \to 0$ ($n\to \mathrm{\infty}$)’ in [1] and cited an example,

*i.e.*,

Now, we want to clarify the fact. Let $\{{\gamma}_{n}\}$ be a real sequence, $\{{\gamma}_{{n}_{m}}\}$ be some infinite subsequence of $\{{\gamma}_{n}\}$ and $\{{n}_{m}\}$ be neither odd nor even sequence, then the conclusions are as follows:

(C-i) ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0\iff \mathrm{\forall}\u03f5>0$, ∃ nonnegative integer ${n}_{0}$ such that $|{\gamma}_{{n}_{m}+j}|<\u03f5$ for ${n}_{m}\ge {n}_{0}$, $j\ge 1$.

(C-ii) ${lim}_{n\to \mathrm{\infty}}{\gamma}_{n}=0\Rightarrow {lim}_{m\to \mathrm{\infty}}{\gamma}_{{n}_{m}}=0$ and ${lim}_{m\to \mathrm{\infty}}{\gamma}_{{n}_{m}+j}=0$ for $\mathrm{\forall}j\ge 1$.

Indeed, the above example (∗∗) does not satisfy the conclusion (C-i), it just illustrates the result (C-ii). Therefore, the note given by Yang [2] confused the conclusions (C-i) and (C-ii).

The aim of this paper is to generalize the results of papers [1] and [2]. For this, we need the following knowledge.

## 2 Preliminary

*E*be a real Banach space and ${E}^{\ast}$ be its dual space. The normalized duality mapping $J:E\to {2}^{{E}^{\ast}}$ is defined by

where $\u3008\cdot ,\cdot \u3009$ denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by *j*.

*T*is called

*ϕ*-strongly accretive if for any $x,y\in E$, there exist $j(x-y)\in J(x-y)$ and a strictly increasing continuous function $\varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ with $\varphi (0)=0$ such that

It is obvious that a strongly accretive operator must be the *ϕ*-strongly accretive in the special case in which $\varphi (t)=kt$, but the converse is not true in general. That is, the class of strongly accretive operators is a proper subclass of the class of *ϕ*-strongly accretive operators.

In order to obtain the main conclusion of this paper, we need the following lemmas.

**Lemma 2.1** [1]

*Suppose that* *E* *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* $f\in E$.

**Lemma 2.2** [4]

*Let*

*E*

*be a real Banach space and let*$J:E\to {2}^{{E}^{\ast}}$

*be a normalized duality mapping*.

*Then*

*for all* $x,y\in E$ *and* $j(x+y)\in J(x+y)$.

**Lemma 2.3** [5]

*Let*${\{{\delta}_{n}\}}_{n=0}^{\mathrm{\infty}}$, ${\{{\lambda}_{n}\}}_{n=0}^{\mathrm{\infty}}$

*and*${\{{\gamma}_{n}\}}_{n=0}^{\mathrm{\infty}}$

*be three nonnegative real sequences and*$\varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$

*be a strictly increasing and continuous function with*$\varphi (0)=0$

*satisfying the following inequality*:

*where* ${\lambda}_{n}\in [0,1]$ *with* ${\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty}$, ${\gamma}_{n}=o({\lambda}_{n})$. *Then* ${\delta}_{n}\to 0$ *as* $n\to \mathrm{\infty}$.

## 3 Main results

**Theorem 3.1**

*Let*

*E*

*be an arbitrary real Banach space and*${\{{A}_{i}\}}_{i=1}^{N}:E\to E$

*be*

*N*

*uniformly continuous*

*ϕ*-

*strongly accretive operators*.

*For*$i=1,2,\dots ,N$,

*let*${\{{u}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$

*be bounded sequences in*

*E*

*and*${\{{a}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$, ${\{{b}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$, ${\{{c}_{n}^{i}\}}_{n=1}^{\mathrm{\infty}}$

*be real sequences in*$[0,1]$

*satisfying*

- (i)
${a}_{n}^{i}+{b}_{n}^{i}+{c}_{n}^{i}=1$, $i=1,2,\dots ,N$;

- (ii)
${\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}^{N}=+\mathrm{\infty}$;

- (iii)
${lim}_{n\to \mathrm{\infty}}{b}_{n}^{i}={lim}_{n\to \mathrm{\infty}}{c}_{n}^{i}=0$, $i=1,2,\dots ,N$;

- (iv)
${c}_{n}^{N}=o({b}_{n}^{N})$.

*For any given*$f\in E$,

*define*${\{{S}_{i}\}}_{i=1}^{N}:E\to E$

*with*${\bigcap}_{i=1}^{N}F({S}_{i})\ne \mathrm{\varnothing}$

*by*${S}_{i}x=x-{A}_{i}x+f$, $\mathrm{\forall}i=1,2,\dots ,N$, $\mathrm{\forall}x\in E$,

*where*$F({S}_{i})=\{x\in E:{S}_{i}x=x\}$.

*Then*,

*for some*${x}_{0}\in E$,

*the multi*-

*step iterative sequence with errors*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*defined by*

*converges strongly to the unique solution of the operator equations* ${\{{A}_{i}x\}}_{i=1}^{N}=f$.

*Proof*Since ${\{{A}_{i}\}}_{i=1}^{N}:E\to E$ is

*ϕ*-strongly accretive operator, we obtain that each equation ${A}_{i}x=f$ has the unique solution by Lemma 2.1, denote ${q}_{i}$,

*i.e.*, ${q}_{i}$ is the unique fixed point of ${S}_{i}$ by ${S}_{i}x=x-{A}_{i}x+f$. Since ${\bigcap}_{i=1}^{N}F({S}_{i})\ne \mathrm{\varnothing}$, then ${\bigcap}_{i=1}^{N}F({S}_{i})$ is a single set, let

*q*. Meanwhile, there exists a strictly increasing continuous function $\varphi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ with $\varphi (0)=0$ such that

$R(\mathrm{\Phi})$ is the range of Φ. Indeed, if $\mathrm{\Phi}(r)\to +\mathrm{\infty}$ as $r\to +\mathrm{\infty}$, then ${r}_{0}\in R(\mathrm{\Phi})$; if $sup\{\mathrm{\Phi}(r):r\in [0,+\mathrm{\infty})\}={r}_{1}<+\mathrm{\infty}$ with ${r}_{1}<{r}_{0}$, then for $q\in E$, there exists a sequence $\{{w}_{n}\}$ in *E* such that ${w}_{n}\to q$ as $n\to \mathrm{\infty}$ with ${w}_{n}\ne q$. Since ${A}_{i}$ is uniformly continuous, so is ${S}_{i}$. Furthermore, we obtain that ${S}_{i}{w}_{n}\to {S}_{i}q$ as $n\to \mathrm{\infty}$, then $\{{w}_{n}-{S}_{i}{w}_{n}\}$ is the bounded sequence for $i=1,2,\dots ,N$. Hence, there exists the common natural number ${n}_{0}$ such that $\parallel {w}_{n}-{S}_{i}{w}_{n}\parallel \cdot \parallel {w}_{n}-q\parallel <\frac{{r}_{1}}{2}$ for $n\ge {n}_{0}$ and $i=1,2,\dots ,N$, then we redefine ${x}_{0}={w}_{{n}_{0}}$ and $\parallel {x}_{0}-{S}_{i}{x}_{0}\parallel \cdot \parallel {x}_{0}-q\parallel <\frac{{r}_{1}}{2}$. Thus, ${max}_{1\le i\le N}\{\parallel {x}_{0}-{S}_{i}{x}_{0}\parallel \cdot \parallel {x}_{0}-q\parallel \}\in R(\varphi )$. It is to ensure that ${\mathrm{\Phi}}^{-1}({r}_{0})$ is defined well.

Step I. We show that $\{{x}_{n}\}$ is a bounded sequence.

*n*,

*i.e.*, ${x}_{n}\in {B}_{1}$. We prove that ${x}_{n+1}\in {B}_{1}$. Suppose it is not the case, then $\parallel {x}_{n+1}-q\parallel >R>\frac{R}{2}$. Since ${S}_{i}$ is uniformly continuous for $i=1,2,\dots ,N$, then for ${\u03f5}_{0}=\frac{\mathrm{\Phi}(\frac{R}{2})}{8R}$, there exists common $\delta >0$ such that $\parallel {S}_{i}x-{S}_{i}y\parallel <{\u03f5}_{0}$ when $\parallel x-y\parallel <\delta $. Denote

which is a contradiction. So, ${x}_{n+1}\in {B}_{1}$, *i.e.*, $\{{x}_{n}\}$ is a bounded sequence, from which it follows that $\{{x}_{n}^{1}\},\{{x}_{n}^{2}\},\dots ,\{{x}_{n}^{N-1}\}$ are all bounded sequences as well.

Step II. We want to prove $\parallel {x}_{n}-q\parallel \to 0$ as $n\to \mathrm{\infty}$.

By Lemma 2.3, we obtain ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-q\parallel =0$. This completes the proof. □

**Remark 3.2**Theorem 3.1 generalizes Theorem 3.1 of [1] and Theorem 2 of [2] in the following cases:

- (a)
- (b)
The condition of $\{{c}_{n}^{i}\}$ is weakened to ${c}_{n}^{N}=o({b}_{n}^{N})$ from ${lim}_{n\to \mathrm{\infty}}\frac{{c}_{n}^{i}}{{b}_{n}^{i}+{c}_{n}^{i}}=0$ ($i=1,2,\dots ,N$).

- (c)

**Theorem 3.3**

*Let*

*E*, $\{{u}_{n}^{i}\}$, $\{{a}_{n}^{i}\}$, $\{{b}_{n}^{i}\}$, $\{{c}_{n}^{i}\}$ ($i=1,2,\dots ,N$)

*be as in Theorem*3.1

*and let*${\{{T}_{i}\}}_{i=1}^{N}:E\to E$

*be*

*N*

*uniformly continuous*

*ϕ*-

*strongly pseudocontractive mappings*.

*Then*,

*for some*${x}_{0}\in E$,

*the multi*-

*step iterative sequence with errors*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*defined by*

*converges strongly to the unique common fixed point of* ${\{{T}_{i}\}}_{i=1}^{N}$.

*Proof* See [1]. □

## Declarations

### Acknowledgements

The authors are very grateful to Professor Yeol-Je Cho for good suggestions which helped to improve the manuscript. This work is supported by the Hebei Natural Science Foundation No. A2011210033.

## Authors’ Affiliations

## References

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