# Convergence theorems for mixed type asymptotically nonexpansive mappings

- Weiping Guo
^{1}, - Yeol Je Cho
^{2}Email author and - Wei Guo
^{3}

**2012**:224

https://doi.org/10.1186/1687-1812-2012-224

© Guo et al.; licensee Springer. 2012

**Received: **27 April 2012

**Accepted: **16 November 2012

**Published: **11 December 2012

## Abstract

In this paper, we introduce a new two-step iterative scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong and weak convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

## Keywords

## 1 Introduction

*K*be a nonempty subset of a real normed linear space

*E*. A mapping $T:K\to K$ is said to be

*asymptotically nonexpansive*if there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=1$ such that

for all $x,y\in K$ and $n\ge 1$.

In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self-mappings, and proved that if *K* is a nonempty closed convex subset of a real uniformly convex Banach space *E* and *T* is an asymptotically nonexpansive self-mapping of *K*, then *T* has a fixed point.

Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive self-mappings in Banach spaces (see [2–16]), which extend and improve the result of Goebel and Kirk in several ways.

Recently, Chidume *et al.* [10] introduced the concept of asymptotically nonexpansive nonself-mappings, which is a generalization of an asymptotically nonexpansive self-mapping, as follows.

**Definition 1.1** [10]

*K*be a nonempty subset of a real normed linear space

*E*. Let $P:E\to K$ be a nonexpansive retraction of

*E*onto

*K*. A nonself-mapping $T:K\to E$ is said to be

*asymptotically nonexpansive*if there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that

for all $x,y\in K$ and $n\ge 1$.

Let *K* be a nonempty closed convex subset of a real uniformly convex Banach space *E*.

*et al.*[10] studied the following iteration scheme:

for each $n\ge 1$, where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$ and *P* is a nonexpansive retraction of *E* onto *K*, and proved some strong and weak convergence theorems for an asymptotically nonexpansive nonself-mapping.

for each $n\ge 1$, where ${T}_{1},{T}_{2}:K\to E$ are two asymptotically nonexpansive nonself-mappings and $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ are real sequences in $[0,1)$, and proved some strong and weak convergence theorems for two asymptotically nonexpansive nonself-mappings. Recently, Guo and Guo [12] proved some new weak convergence theorems for the iteration process (1.4).

The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and to prove some strong and weak convergence theorems for the new iteration scheme in uniformly convex Banach spaces.

## 2 Preliminaries

*E*be a real Banach space,

*K*be a nonempty closed convex subset of

*E*and $P:E\to K$ be a nonexpansive retraction of

*E*onto

*K*. Let ${S}_{1},{S}_{2}:K\to K$ be two asymptotically nonexpansive self-mappings and ${T}_{1},{T}_{2}:K\to E$ be two asymptotically nonexpansive nonself-mappings. Then we define the new iteration scheme of mixed type as follows:

for each $n\ge 1$, where $\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$ are two sequences in $[0,1)$.

If ${S}_{1}$ and ${S}_{2}$ are the identity mappings, then the iterative scheme (2.1) reduces to the sequence (1.4).

We denote the set of common fixed points of ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ and ${T}_{2}$ by $F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$ and denote the distance between a point *z* and a set *A* in *E* by $d(z,A)={inf}_{x\in A}\parallel z-x\parallel $.

Now, we recall some well-known concepts and results.

*E*be a real Banach space, ${E}^{\ast}$ be the dual space of

*E*and $J:E\to {2}^{{E}^{\ast}}$ be the

*normalized duality mapping*defined by

for all $x\in E$, where $\u3008\cdot ,\cdot \u3009$ denotes duality pairing between *E* and ${E}^{\ast}$. A single-valued normalized duality mapping is denoted by *j*.

A subset *K* of a real Banach space *E* is called a *retract* of *E* [10] if there exists a continuous mapping $P:E\to K$ such that $Px=x$ for all $x\in K$. Every closed convex subset of a uniformly convex Banach space is a retract. A mapping $P:E\to E$ is called a *retraction* if ${P}^{2}=P$. It follows that if a mapping *P* is a retraction, then $Py=y$ for all *y* in the range of *P*.

*E*is said to satisfy

*Opial’s condition*[17] if, for any sequence $\{{x}_{n}\}$ of

*E*, ${x}_{n}\to x$ weakly as $n\to \mathrm{\infty}$ implies that

for all $y\in E$ with $y\ne x$.

*E*is said to have a

*Fréchet differentiable norm*[18] if, for all $x\in U=\{x\in E:\parallel x\parallel =1\}$,

exists and is attained uniformly in $y\in U$.

A Banach space *E* is said to have the *Kadec-Klee property* [19] if for every sequence $\{{x}_{n}\}$ in *E*, ${x}_{n}\to x$ weakly and $\parallel {x}_{n}\parallel \to \parallel x\parallel $, it follows that ${x}_{n}\to x$ strongly.

Let *K* be a nonempty closed subset of a real Banach space *E*. A nonself-mapping $T:K\to E$ is said to be *semi-compact* [11] if, for any sequence $\{{x}_{n}\}$ in *K* such that $\parallel {x}_{n}-T{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, there exists a subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ such that $\{{x}_{{n}_{j}}\}$ converges strongly to some ${x}^{\ast}\in K$.

**Lemma 2.1** [15]

*Let*$\{{a}_{n}\}$, $\{{b}_{n}\}$

*and*$\{{c}_{n}\}$

*be three nonnegative sequences satisfying the following condition*:

*for each* $n\ge {n}_{0}$, *where* ${n}_{0}$ *is some nonnegative integer*, ${\sum}_{n={n}_{0}}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}$ *and* ${\sum}_{n={n}_{0}}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}$. *Then* ${lim}_{n\to \mathrm{\infty}}{a}_{n}$ *exists*.

**Lemma 2.2** [8]

*Let*

*E*

*be a real uniformly convex Banach space and*$0<p\le {t}_{n}\le q<1$

*for each*$n\ge 1$.

*Also*,

*suppose that*$\{{x}_{n}\}$

*and*$\{{y}_{n}\}$

*are two sequences of*

*E*

*such that*

*hold for some* $r\ge 0$. *Then* ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$.

**Lemma 2.3** [10]

*Let* *E* *be a real uniformly convex Banach space*, *K* *be a nonempty closed convex subset of* *E* *and* $T:K\to E$ *be an asymptotically nonexpansive mapping with a sequence* $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ *and* ${k}_{n}\to 1$ *as* $n\to \mathrm{\infty}$. *Then* $I-T$ *is demiclosed at zero*, *i*.*e*., *if* ${x}_{n}\to x$ *weakly and* ${x}_{n}-T{x}_{n}\to 0$ *strongly*, *then* $x\in F(T)$, *where* $F(T)$ *is the set of fixed points of* *T*.

**Lemma 2.4** [16]

*Let*

*X*

*be a uniformly convex Banach space and*

*C*

*be a convex subset of*

*X*.

*Then there exists a strictly increasing continuous convex function*$\gamma :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*with*$\gamma (0)=0$

*such that*,

*for each mapping*$S:C\to C$

*with a Lipschitz constant*$L>0$,

*for all* $x,y\in C$ *and* $0<\alpha <1$.

**Lemma 2.5** [16]

*Let*

*X*

*be a uniformly convex Banach space such that its dual space*${X}^{\ast}$

*has the Kadec*-

*Klee property*.

*Suppose*$\{{x}_{n}\}$

*is a bounded sequence and*${f}_{1},{f}_{2}\in {W}_{w}(\{{x}_{n}\})$

*such that*

*exists for all* $\alpha \in [0,1]$, *where* ${W}_{w}(\{{x}_{n}\})$ *denotes the set of all weak subsequential limits of *$\{{x}_{n}\}$. *Then* ${f}_{1}={f}_{2}$.

## 3 Strong convergence theorems

In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.

**Lemma 3.1**

*Let*

*E*

*be a real uniformly convex Banach space and*

*K*

*be a nonempty closed convex subset of*

*E*.

*Let*${S}_{1},{S}_{2}:K\to K$

*be two asymptotically nonexpansive self*-

*mappings with*$\{{k}_{n}^{(1)}\},\{{k}_{n}^{(2)}\}\subset [1,\mathrm{\infty})$

*and*${T}_{1},{T}_{2}:K\to E$

*be two asymptotically nonexpansive nonself*-

*mappings with*$\{{l}_{n}^{(1)}\},\{{l}_{n}^{(2)}\}\subset [1,\mathrm{\infty})$

*such that*${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}^{(i)}-1)<\mathrm{\infty}$

*and*${\sum}_{n=1}^{\mathrm{\infty}}({l}_{n}^{(i)}-1)<\mathrm{\infty}$

*for*$i=1,2$,

*respectively*,

*and*$F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})\ne \mathrm{\varnothing}$.

*Let*$\{{x}_{n}\}$

*be the sequence defined by*(2.1),

*where*$\{{\alpha}_{n}\}$

*and*$\{{\beta}_{n}\}$

*are two real sequences in*$[0,1)$.

*Then*

- (1)
${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-q\parallel $

*exists for any*$q\in F$; - (2)
${lim}_{n\to \mathrm{\infty}}d({x}_{n},F)$

*exists*.

*Proof*(1) Set ${h}_{n}=max\{{k}_{n}^{(1)},{k}_{n}^{(2)},{l}_{n}^{(1)},{l}_{n}^{(2)}\}$. For any $q\in F$, it follows from (2.1) that

- (2)Taking the infimum over all $q\in F$ in (3.2), we have$d({x}_{n+1},F)\le [1+({h}_{n}^{2}-1)]d({x}_{n},F)$

for each $n\ge 1$. It follows from ${\sum}_{n=1}^{\mathrm{\infty}}({h}_{n}^{2}-1)<\mathrm{\infty}$ and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □

**Lemma 3.2**

*Let*

*E*

*be a real uniformly convex Banach space and*

*K*

*be a nonempty closed convex subset of*

*E*.

*Let*${S}_{1},{S}_{2}:K\to K$

*be two asymptotically nonexpansive self*-

*mappings with*$\{{k}_{n}^{(1)}\},\{{k}_{n}^{(2)}\}\subset [1,\mathrm{\infty})$

*and*${T}_{1},{T}_{2}:K\to E$

*be two asymptotically nonexpansive nonself*-

*mappings with*$\{{l}_{n}^{(1)}\},\{{l}_{n}^{(2)}\}\subset [1,\mathrm{\infty})$

*such that*${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}^{(i)}-1)<\mathrm{\infty}$

*and*${\sum}_{n=1}^{\mathrm{\infty}}({l}_{n}^{(i)}-1)<\mathrm{\infty}$

*for*$i=1,2$,

*respectively*,

*and*$F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})\ne \mathrm{\varnothing}$.

*Let*$\{{x}_{n}\}$

*be the sequence defined by*(2.1)

*and the following conditions hold*:

- (a)
$\{{\alpha}_{n}\}$

*and*$\{{\beta}_{n}\}$*are two real sequences in*$[\u03f5,1-\u03f5]$*for some*$\u03f5\in (0,1)$; - (b)
$\parallel x-{T}_{i}y\parallel \le \parallel {S}_{i}x-{T}_{i}y\parallel $

*for all*$x,y\in K$*and*$i=1,2$.

*Then* ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{S}_{i}{x}_{n}\parallel ={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}_{i}{x}_{n}\parallel =0$ *for* $i=1,2$.

*Proof*Set ${h}_{n}=max\{{k}_{n}^{(1)},{k}_{n}^{(2)},{l}_{n}^{(1)},{l}_{n}^{(2)}\}$. For any given $q\in F$, ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-q\parallel $ exists by Lemma 3.1. Now, we assume that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-q\parallel =c$. It follows from (3.2) and ${\sum}_{n=1}^{\mathrm{\infty}}({h}_{n}^{2}-1)<\mathrm{\infty}$ that

*E*onto

*K*, we have

This completes the proof. □

Now, we find two mappings, ${S}_{1}={S}_{2}=S$ and ${T}_{1}={T}_{2}=T$, satisfying the condition (b) in Lemma 3.2 as follows.

**Example 3.1** [20]

*T*is nonexpansive. In fact, if $x,y\in [0,1]$ or $x,y\in [-1,0)$, then we have

This implies that *T* is nonexpansive and so *T* is an asymptotically nonexpansive mapping with ${k}_{n}=1$ for each $n\ge 1$. Similarly, we can show that *S* is an asymptotically nonexpansive mapping with ${l}_{n}=1$ for each $n\ge 1$.

Next, we show that two mappings *S*, *T* satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:

Therefore, the condition (b) in Lemma 3.2 is satisfied.

**Theorem 3.1** *Under the assumptions of Lemma * 3.2, *if one of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$ *is completely continuous*, *then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) *converges strongly to a common fixed point of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$.

*Proof*Without loss of generality, we can assume that ${S}_{1}$ is completely continuous. Since $\{{x}_{n}\}$ is bounded by Lemma 3.1, there exists a subsequence $\{{S}_{1}{x}_{{n}_{j}}\}$ of $\{{S}_{1}{x}_{n}\}$ such that $\{{S}_{1}{x}_{{n}_{j}}\}$ converges strongly to some ${q}^{\ast}$. Moreover, we know that

for $i=1,2$. Thus it follows that ${q}^{\ast}\in F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$. Furthermore, since ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{q}^{\ast}\parallel $ exists by Lemma 3.1, we have ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{q}^{\ast}\parallel =0$. This completes the proof. □

**Theorem 3.2** *Under the assumptions of Lemma * 3.2, *if one of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$ *is semi*-*compact*, *then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) *converges strongly to a common fixed point of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$.

*Proof* Since ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{S}_{i}{x}_{n}\parallel ={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}_{i}{x}_{n}\parallel =0$ for $i=1,2$ by Lemma 3.2 and one of ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ and ${T}_{2}$ is semi-compact, there exists a subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ such that $\{{x}_{{n}_{j}}\}$ converges strongly to some ${q}^{\ast}\in K$. Moreover, by the continuity of ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ and ${T}_{2}$, we have $\parallel {q}^{\ast}-{S}_{i}{q}^{\ast}\parallel ={lim}_{j\to \mathrm{\infty}}\parallel {x}_{{n}_{j}}-{S}_{i}{x}_{{n}_{j}}\parallel =0$ and $\parallel {q}^{\ast}-{T}_{i}{q}^{\ast}\parallel ={lim}_{j\to \mathrm{\infty}}\parallel {x}_{{n}_{j}}-{T}_{i}{x}_{{n}_{j}}\parallel =0$ for $i=1,2$. Thus it follows that ${q}^{\ast}\in F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$. Since ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{q}^{\ast}\parallel $ exists by Lemma 3.1, we have ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{q}^{\ast}\parallel =0$. This completes the proof. □

**Theorem 3.3**

*Under the assumptions of Lemma*3.2,

*if there exists a nondecreasing function*$f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*with*$f(0)=0$

*and*$f(r)>0$

*for all*$r\in (0,\mathrm{\infty})$

*such that*

*for all* $x\in K$, *where* $F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$, *then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) *converges strongly to a common fixed point of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$.

*Proof* Since ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{S}_{i}{x}_{n}\parallel ={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{T}_{i}{x}_{n}\parallel =0$ for $i=1,2$ by Lemma 3.2, we have ${lim}_{n\to \mathrm{\infty}}f(d({x}_{n},F))=0$. Since $f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is a nondecreasing function satisfying $f(0)=0$, $f(r)>0$ for all $r\in (0,\mathrm{\infty})$ and ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F)$ exists by Lemma 3.1, we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F)=0$.

*K*. In fact, from (3.2), we have

*m*,

*n*, $m>n\ge 1$, we have

Thus it follows from ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F)=0$ that $\{{x}_{n}\}$ is a Cauchy sequence. Since *K* is a closed subset of *E*, the sequence $\{{x}_{n}\}$ converges strongly to some ${q}^{\ast}\in K$. It is easy to prove that $F({S}_{1})$, $F({S}_{2})$, $F({T}_{1})$ and $F({T}_{2})$ are all closed and so *F* is a closed subset of *K*. Since ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F)=0$, ${q}^{\ast}\in F$, the sequence $\{{x}_{n}\}$ converges strongly to a common fixed point of ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ and ${T}_{2}$. This completes the proof. □

## 4 Weak convergence theorems

In this section, we prove weak convergence theorems for the iterative scheme defined by (2.1) in uniformly convex Banach spaces.

**Lemma 4.1**

*Under the assumptions of Lemma*3.1,

*for all*${q}_{1},{q}_{2}\in F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$,

*the limit*

*exists for all* $t\in [0,1]$, *where* $\{{x}_{n}\}$ *is the sequence defined by* (2.1).

*Proof* Set ${a}_{n}(t)=\parallel t{x}_{n}+(1-t){q}_{1}-{q}_{2}\parallel $. Then ${lim}_{n\to \mathrm{\infty}}{a}_{n}(0)=\parallel {q}_{1}-{q}_{2}\parallel $ and, from Lemma 3.1, ${lim}_{n\to \mathrm{\infty}}{a}_{n}(1)={lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{q}_{2}\parallel $ exists. Thus it remains to prove Lemma 4.1 for any $t\in (0,1)$.

*m*. Observe that

Thus we have ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{a}_{n}(t)\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{a}_{n}(t)$, that is, ${lim}_{n\to \mathrm{\infty}}\parallel t{x}_{n}+(1-t){q}_{1}-{q}_{2}\parallel $ exists for all $t\in (0,1)$. This completes the proof. □

**Lemma 4.2**

*Under the assumptions of Lemma*3.1,

*if*

*E*

*has a Fréchet differentiable norm*,

*then*,

*for all*${q}_{1},{q}_{2}\in F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$,

*the limit*

*exists*, *where* $\{{x}_{n}\}$ *is the sequence defined by* (2.1). *Furthermore*, *if* ${W}_{w}(\{{x}_{n}\})$ *denotes the set of all weak subsequential limits of* $\{{x}_{n}\}$, *then* $\u3008{x}^{\ast}-{y}^{\ast},j({q}_{1}-{q}_{2})\u3009=0$ *for all* ${q}_{1},{q}_{2}\in F$ *and* ${x}^{\ast},{y}^{\ast}\in {W}_{w}(\{{x}_{n}\})$.

*Proof* This follows basically as in the proof of Lemma 3.2 of [12] using Lemma 4.1 instead of Lemma 3.1 of [12]. □

**Theorem 4.1** *Under the assumptions of Lemma * 3.2, *if* *E* *has a Fréchet differentiable norm*, *then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) *converges weakly to a common fixed point of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$.

*Proof*Since

*E*is a uniformly convex Banach space and the sequence $\{{x}_{n}\}$ is bounded by Lemma 3.1, there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ which converges weakly to some $q\in K$. By Lemma 3.2, we have

for $i=1,2$. It follows from Lemma 2.3 that $q\in F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$.

*q*. Suppose that there exists a subsequence $\{{x}_{{m}_{j}}\}$ of $\{{x}_{n}\}$ such that $\{{x}_{{m}_{j}}\}$ converges weakly to some ${q}_{1}\in K$. Then, by the same method given above, we can also prove that ${q}_{1}\in F$. So, $q,{q}_{1}\in F\cap {W}_{w}(\{{x}_{n}\})$. It follows from Lemma 4.2 that

Therefore, ${q}_{1}=q$, which shows that the sequence $\{{x}_{n}\}$ converges weakly to *q*. This completes the proof. □

**Theorem 4.2** *Under the assumptions of Lemma * 3.2, *if the dual space* ${E}^{\ast}$ *of* *E* *has the Kadec*-*Klee property*, *then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) *converges weakly to a common fixed point of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$.

*Proof* Using the same method given in Theorem 4.1, we can prove that there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ which converges weakly to some $q\in F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$.

*q*. Suppose that there exists a subsequence $\{{x}_{{m}_{j}}\}$ of $\{{x}_{n}\}$ such that $\{{x}_{{m}_{j}}\}$ converges weakly to some ${q}^{\ast}\in K$. Then, as for

*q*, we have ${q}^{\ast}\in F$. It follows from Lemma 4.1 that the limit

exists for all $t\in [0,1]$. Again, since $q,{q}^{\ast}\in {W}_{w}(\{{x}_{n}\})$, ${q}^{\ast}=q$ by Lemma 2.5. This shows that the sequence $\{{x}_{n}\}$ converges weakly to *q*. This completes the proof. □

**Theorem 4.3** *Under the assumptions of Lemma * 3.2, *if* *E* *satisfies Opial’s condition*, *then the sequence* $\{{x}_{n}\}$ *defined by* (2.1) *converges weakly to a common fixed point of* ${S}_{1}$, ${S}_{2}$, ${T}_{1}$ *and* ${T}_{2}$.

*Proof* Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ which converges weakly to some $q\in F=F({S}_{1})\cap F({S}_{2})\cap F({T}_{1})\cap F({T}_{2})$.

*q*. Suppose that there exists a subsequence $\{{x}_{{m}_{j}}\}$ of $\{{x}_{n}\}$ such that $\{{x}_{{m}_{j}}\}$ converges weakly to some $\overline{q}\in K$ and $\overline{q}\ne q$. Then, as for

*q*, we have $\overline{q}\in F$. Using Lemma 3.1, we have the following two limits exist:

which is a contradiction and so $q=\overline{q}$. This shows that the sequence $\{{x}_{n}\}$ converges weakly to *q*. This completes the proof. □

## Declarations

### Acknowledgements

The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

## Authors’ Affiliations

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