# Simple projection algorithm for a countable family of weak relatively nonexpansive mappings and applications

- Jingling Zhang
^{1}, - Yongfu Su
^{1}Email author and - Qingqing Cheng
^{1}

**2012**:205

https://doi.org/10.1186/1687-1812-2012-205

© Zhang et al.; licensee Springer 2012

**Received: **14 April 2012

**Accepted: **29 October 2012

**Published: **13 November 2012

## Abstract

Let *E* be a uniformly convex and uniformly smooth Banach space, let *C* be a nonempty closed convex subset of *E*, let $\{{T}_{n}\}:C\to C$ be a countable family of weak relatively nonexpansive mappings such that $F={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})\ne \mathrm{\varnothing}$. For any given gauss ${x}_{0}\in C$, define a sequence $\{{x}_{n}\}$ in *C* by the following algorithm:

Then $\{{x}_{n}\}$ converges strongly to $q={\mathrm{\Pi}}_{F}{x}_{0}$.

**MSC:**47H05, 47H09, 47H10.

## Keywords

## 1 Introduction

*E*be a real Banach space with the dual ${E}^{\ast}$. We denote by

*J*the normalized duality mapping from

*E*to ${2}^{{E}^{\ast}}$ defined by

where $\u3008\cdot ,\cdot \u3009$ denotes the generalized duality pairing. The duality mapping *J* has the following properties: (1) if *E* is smooth, then *J* is single-valued; (2) if *E* is strictly convex, then *J* is one-to-one; (3) if *E* is reflexive, then *J* is surjective; (4) if *E* is uniformly smooth, then *J* is uniformly norm-to-norm continuous on each bounded subset of *E*; (5) if ${E}^{\ast}$ is uniformly convex, then *J* is uniformly continuous on bounded subsets of *E* and *J* is singe-valued and also one-to-one (see [1–4]).

*E*be a smooth Banach space with the dual ${E}^{\ast}$. The functional $\varphi :E\times E\to R$ is defined by

for all $x,y\in E$.

Let *C* be a closed convex subset of *E*, and let *T* be a mapping from *C* into itself. We denote by $F(T)$ the set of fixed points of *T*. A point *p* in *C* is said to be an asymptotic fixed point of *T* [5] if *C* contains a sequence $\{{x}_{n}\}$ which converges weakly to *p* such that the strong ${lim}_{n\to \mathrm{\infty}}({x}_{n}-T{x}_{n})=0$. The set of asymptotic fixed points of *T* will be denoted by $\stackrel{\u02c6}{F}(T)$. A mapping *T* from *C* into itself is called nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ for all $x,y\in C$ and relatively nonexpansive if $F(T)=\stackrel{\u02c6}{F}(T)$ and $\varphi (p,Tx)\le \varphi (p,x)$ for all $x\in C$ and $p\in F(T)$. The asymptotic behavior of a relatively nonexpansive mapping was studied in [1, 6–9].

where $\{{\alpha}_{n}\}$ and $\{{\beta}_{n}\}$ are sequences in the interval $[0,1]$.

- (i)
${\alpha}_{n}\to 0$;

- (ii)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$;

- (iii)
${\sum}_{n=0}^{\mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|<\mathrm{\infty}$ or ${lim}_{n\to \mathrm{\infty}}\frac{{\alpha}_{n}}{{\alpha}_{n+1}}=1$.

By the restriction of condition (ii), it is widely believed that Halpern’s iteration process (1.2) has slow convergence though the rate of convergence has not been determined. Halpern [11] proved that conditions (i) and (ii) are necessary in the strong convergence of (1.2) for a nonexpansive mapping *T* on a closed convex subset *C* of a Hilbert space *H*. Moreover, Wittmann [13] showed that (1.2) converges strongly to ${P}_{F(T)}u$ when $\{{\alpha}_{n}\}$ satisfies (i), (ii) and (iii), where ${P}_{F(T)}(\cdot )$ is the metric projection onto $F(T)$.

Both iteration processes (1.1) and (1.3) have only weak convergence in a general Banach space (see [17] for more details). As a matter of fact, the process (1.1) may fail to converge, while the process (1.3) can still converge for a Lipschitz pseudo-contractive mapping in a Hilbert space [18]. For example, Reich [19] proved that if *E* is a uniformly convex Banach space with the Fréchet differentiable norm and if $\{{\alpha}_{n}\}$ is chosen such that ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})=\mathrm{\infty}$, then the sequence $\{{x}_{n}\}$ defined by (1.1) converges weakly to a fixed point of *T*. However, we note that Mann’s iteration process (1.1) has only weak convergence even in a Hilbert space [17].

*T*in a Hilbert space

*H*:

where *C* is a closed convex subset of *H*, ${P}_{K}$ denotes the metric projection from *H* onto a closed convex subset *K* of *H*. They proved that if the sequence $\{{\alpha}_{n}\}$ is bounded above from one, then the sequence $\{{x}_{n}\}$ generated by (1.4) converges strongly to ${P}_{F(T)}({x}_{0})$, where $F(T)$ denotes the fixed point set of *T*.

*T*in a Banach space

*E*:

They proved the following convergence theorem.

**Theorem MT** *Let* *E* *be a uniformly convex and uniformly smooth Banach space*, *let* *C* *be a nonempty closed convex subset of* *E*, *let* *T* *be a relatively nonexpansive mapping from* *C* *into itself*, *and let* $\{{\alpha}_{n}\}$ *be a sequence of real numbers such that* $0\le {\alpha}_{n}<1$ *and* ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$. *Suppose that* $\{{x}_{n}\}$ *is given by* (1.6), *where* *J* *is the duality mapping on* *E*. *If* $F(T)$ *is nonempty*, *then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F(T)}{x}_{0}$, *where* ${\mathrm{\Pi}}_{F(T)}(\cdot )$ *is the generalized projection from* *C* *onto* $F(T)$.

In 2007, Plubtieng and Ungchittrakool [21] proposed the following hybrid algorithms for two relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.

**Theorem SK1**

*Let*

*E*

*be a uniformly convex and uniformly smooth real Banach space*,

*let*

*C*

*be a nonempty closed convex subset of*

*E*,

*let*

*T*,

*S*

*be two relatively nonexpansive mappings from*

*C*

*into itself with*$F:=F(T)\cap F(S)$

*is nonempty*.

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

*be defined by*

*with the following restrictions*:

- (i)
$0\le {\alpha}_{n}<1$, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$;

- (ii)
$0\le {\beta}_{n}^{(1)},{\beta}_{n}^{(1)},{\beta}_{n}^{(3)}\le 1$, ${lim}_{n\to \mathrm{\infty}}{\beta}_{n}^{(1)}=0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}^{(2)}{\beta}_{n}^{(3)}>0$.

*Then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F}{x}_{0}$, *where* ${\mathrm{\Pi}}_{F}$ *is the generalized projection from* *C* *onto F*.

**Theorem SK2**

*Let*

*E*

*be a uniformly convex and uniformly smooth Banach space*,

*let*

*C*

*be a nonempty closed convex subset of*

*E*,

*let*

*T*,

*S*

*be two relatively nonexpansive mappings from*

*C*

*into itself with*$F:=F(T)\cap F(S)$

*is nonempty*.

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

*be defined by*

*with the following restrictions*:

- (i)
$0<{\alpha}_{n}<1$, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1$;

- (ii)
$0\le {\beta}_{n}^{(1)},{\beta}_{n}^{(1)},{\beta}_{n}^{(3)}\le 1$, ${lim}_{n\to \mathrm{\infty}}{\beta}_{n}^{(1)}=0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}^{(2)}{\beta}_{n}^{(3)}>0$.

*Then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F}{x}_{0}$, *where* ${\mathrm{\Pi}}_{F}$ *is the generalized projection from* *C* *onto F*.

In 2010, Su, Xu and Zhang [22] proposed the following hybrid algorithms for two countable families of weak relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.

**Theorem SKZ**

*Let*

*E*

*be a uniformly convex and uniformly smooth real Banach space*,

*let*

*C*

*be a nonempty closed convex subset of*

*E*,

*let*$\{{T}_{n}\}$, $\{{S}_{n}\}$

*be two countable families of weak relatively nonexpansive mappings from*

*C*

*into itself such that*$F:=({\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n}))\cap ({\bigcap}_{n=0}^{\mathrm{\infty}}F({S}_{n}))\ne \mathrm{\varnothing}$.

*Define a sequence*$\{{x}_{n}\}$

*in*

*C*

*by the following algorithm*:

*with the conditions*

- (i)
${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}^{(1)}{\beta}_{n}^{(2)}>0$;

- (ii)
${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}^{(1)}{\beta}_{n}^{(3)}>0$;

- (iii)
$0\le {\alpha}_{n}\le \alpha <1$

*for some*$\alpha \in (0,1)$.

*Then* $\{{x}_{n}\}$ *converges strongly to* ${\mathrm{\Pi}}_{F}{x}_{0}$, *where* ${\mathrm{\Pi}}_{F}$ *is the generalized projection from* *C* *onto F*.

Unfortunately, in recent years, many hybrid algorithms have been very complex, so these complex algorithms are not applicable or are very difficult in applications. Naturally, we hope to obtain some simple and practical algorithms. The purpose of this article is to present a simple projection algorithm for a countable family of weak relatively nonexpansive mappings and to prove strong convergence theorems in Banach spaces.

In addition, we shall give an example which is a countable family of weak relatively nonexpansive mappings, but not a countable family of relatively nonexpansive mappings.

## 2 Preliminaries

*E*be a smooth Banach space with the dual ${E}^{\ast}$. The functional $\varphi :E\times E\to R$ is defined by

for all $x,y\in E$. Observe that in a Hilbert space *H*, (2.1) reduces to $\varphi (x,y)={\parallel x-y\parallel}^{2}$, $x,y\in H$.

*C*is a nonempty, closed and convex subset of a Hilbert space

*H*and ${P}_{C}:H\to C$ is the metric projection of

*H*onto

*C*, then ${P}_{C}$ is nonexpansive. This is true only when

*H*is a real Hilbert space. In this connection, Alber [23] has recently introduced a generalized projection operator ${\mathrm{\Pi}}_{C}$ in a Banach space

*E*which is an analogue of the metric projection in Hilbert spaces. The generalized projection ${\mathrm{\Pi}}_{C}:E\to C$ is a map that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi (y,x)$, that is, ${\mathrm{\Pi}}_{C}x=\overline{x}$, where $\overline{x}$ is the solution to the minimization problem

*J*. In a Hilbert space, ${\mathrm{\Pi}}_{C}={P}_{C}$. It is obvious from the definition of the functional

*ϕ*that

for all $x,y\in E$. See [24] for more details.

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

**Remark 2.1** If *E* is a reflexive strictly convex and smooth Banach space, then for $x,y\in E$, $\varphi (x,y)=0$ if and only if $x=y$. It is sufficient to show that if $\varphi (x,y)=0$ then $x=y$. From (2.3), we have $\parallel x\parallel =\parallel y\parallel $. This implies $\u3008x,Jy\u3009={\parallel x\parallel}^{2}={\parallel Jy\parallel}^{2}$. From the definition of *J*, we have $Jx=Jy$. Since *J* is one-to-one, then we have $x=y$; see [13, 16, 25] for more details.

In this paper, we give the definitions of a countable family of relatively nonexpansive mappings and a countable family of weak relatively nonexpansive mappings which are generalizations of a relatively nonexpansive mapping and a weak relatively nonexpansive mapping respectively. We also give an example which is a countable family of weak relatively nonexpansive mappings, but not a countable family of relatively nonexpansive mappings.

Let *C* be a closed convex subset of *E*, and let ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ be a countable family of mappings from *C* into itself. We denote by *F* the set of common fixed points of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$. That is $F={\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})$, where $F({T}_{n})$ denotes the set of fixed points of ${T}_{n}$ for all $n\ge 0$. A point *p* in *C* is said to be an *asymptotic fixed point* of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ if *C* contains a sequence $\{{x}_{n}\}$ which converges weakly to *p* such that ${lim}_{n\to \mathrm{\infty}}\parallel {T}_{n}{x}_{n}-{x}_{n}\parallel =0$. The set of asymptotic fixed points of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ will be denoted by $\stackrel{\u02c6}{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$. A point *p* in *C* is said to be a *strong asymptotic fixed point* of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ if *C* contains a sequence $\{{x}_{n}\}$ which converges strongly to *p* such that ${lim}_{n\to \mathrm{\infty}}\parallel {T}_{n}{x}_{n}-{x}_{n}\parallel =0$. The set of strong asymptotic fixed points of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ will be denoted by $\tilde{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$.

**Definition 2.2**The countable family of mappings ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is said to be

*a countable family of relatively nonexpansive mappings*if the following conditions are satisfied:

- (1)
$F({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$ is nonempty;

- (2)
$\varphi (u,{T}_{n}x)\le \varphi (u,x)$, $\mathrm{\forall}u\in F({T}_{n})$, $x\in C$, $n\ge 0$;

- (3)
$\stackrel{\u02c6}{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})={\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})$.

**Definition 2.3**The countable family of mappings ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is said to be

*a countable family of weak relatively nonexpansive mappings*if the following conditions are satisfied:

- (1)
$F({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$ is nonempty;

- (2)
$\varphi (u,{T}_{n}x)\le \varphi (u,x)$, $\mathrm{\forall}u\in F({T}_{n})$, $x\in C$, $n\ge 0$;

- (3)
$\tilde{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})={\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})$.

**Definition 2.4** [21]

*T*is said to be

*a relatively nonexpansive mapping*if the following conditions are satisfied:

- (1)
$F(T)$ is nonempty;

- (2)
$\varphi (u,Tx)\le \varphi (u,x)$, $\mathrm{\forall}u\in F(T)$, $x\in C$;

- (3)
$\tilde{F}(T)=F(T)$.

**Definition 2.5**The mapping

*T*is said to be

*a weak relatively nonexpansive mapping*if the following conditions are satisfied:

- (1)
$F(T)$ is nonempty;

- (2)
$\varphi (u,Tx)\le \varphi (u,x)$, $\mathrm{\forall}u\in F(T)$, $x\in C$;

- (3)
$\tilde{F}(T)=F(T)$.

Definition 2.4 (Definition 2.5) is a special form of Definition 2.2 (Definition 2.3) as ${T}_{n}\equiv T$ for all $n\ge 0$.

The hybrid algorithms for a fixed point of relatively nonexpansive mappings and applications have been studied by many authors; see, for example, [1, 6, 7, 18, 26, 27]. In recent years, the definition of a weak relatively nonexpansive mapping has been presented and studied by many authors [7, 18, 25, 27], but they have not given an example of a mapping which is weak relatively nonexpansive, but not relatively nonexpansive.

In the next section, we shall give an example which is a countable family of weak relatively nonexpansive mappings, but not a countable family of relatively nonexpansive mappings.

We need the following lemmas for the proof of our main results.

**Lemma 2.6** [24]

*Let* *E* *be a uniformly convex and smooth real Banach space and let* $\{{x}_{n}\}$, $\{{y}_{n}\}$ *be two sequences of* *E*. *If* $\varphi ({x}_{n},{y}_{n})\to 0$ *and either* $\{{x}_{n}\}$ *or* $\{{y}_{n}\}$ *is bounded*, *then* $\parallel {x}_{n}-{y}_{n}\parallel \to 0$.

*Let*

*C*

*be a nonempty closed convex subset of a smooth real Banach space*

*E*

*and*$x\in E$.

*Then*, ${x}_{0}={\mathrm{\Pi}}_{C}x$

*if and only if*

*Let*

*E*

*be a reflexive*,

*strictly convex and smooth real Banach space*,

*let*

*C*

*be a nonempty closed convex subset of*

*E*

*and let*$x\in E$.

*Then*

**Lemma 2.9** [27]

*Let*

*E*

*be a uniformly convex Banach space and*${B}_{r}(0)=\{x\in E:\parallel x\parallel \le r\}$

*be a closed ball of*

*E*.

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

*with*$g(0)=0$

*such that*

*for all* $x,y,z\in {B}_{r}(0)$ *and* $\lambda ,\mu ,\gamma \in [0,1]$ *with* $\lambda +\mu +\gamma =1$.

It is easy to prove the following result.

**Lemma 2.10** *Let* *E* *be a strictly convex and smooth real Banach space*, *let* *C* *be a closed convex subset of* *E*, *and let* *T* *be a weak relatively nonexpansive mapping from* *C* *into itself*. *Then* $F(T)$ *is closed and convex*.

## 3 Main results

Firstly, we give an example which is a countable family of weak relatively nonexpansive mappings, but not a countable family of relatively nonexpansive mappings in the Banach space ${l}^{2}$.

for all $n\ge 1$.

for all $n\ge 0$.

**Conclusion 3.1** $\{{x}_{n}\}$ *converges weakly to* ${x}_{0}$.

*Proof*For any $f=({\zeta}_{1},{\zeta}_{2},{\zeta}_{3},\dots ,{\zeta}_{k},\dots )\in {l}^{2}={({l}^{2})}^{\ast}$, we have

as $n\to \mathrm{\infty}$. That is, $\{{x}_{n}\}$ converges weakly to ${x}_{0}$. □

**Conclusion 3.2** $\{{x}_{n}\}$ *is not a Cauchy sequence*, *so*, *it does not converge strongly to any element of* ${l}^{2}$.

*Proof* In fact, we have $\parallel {x}_{n}-{x}_{m}\parallel =\sqrt{2}$ for any $n\ne m$. Then $\{{x}_{n}\}$ is not a Cauchy sequence. □

**Conclusion 3.3** ${T}_{n}$ *has a unique fixed point* 0, *that is*, $F({T}_{n})=\{0\}$ *for all* $n\ge 0$.

*Proof* The conclusion is obvious. □

**Conclusion 3.4** ${x}_{0}$ *is an asymptotic fixed point of* ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$.

*Proof*Since $\{{x}_{n}\}$ converges weakly to ${x}_{0}$ and

as $n\to \mathrm{\infty}$, so, ${x}_{0}$ is an asymptotic fixed point of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$. □

**Conclusion 3.5** ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *has a unique strong asymptotic fixed point* 0, *so*, ${\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})=\tilde{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$.

*Proof* In fact, for any strong convergent sequence $\{{z}_{n}\}\subset E$ such that ${z}_{n}\to {z}_{0}$ and $\parallel {z}_{n}-{T}_{n}{z}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, from Conclusion 3.2, there exists a sufficiently large natural number *N* such that ${z}_{n}\ne {x}_{m}$ for any $n,m>N$. Then $T{z}_{n}=-{z}_{n}$ for $n>N$, it follows from $\parallel {z}_{n}-{T}_{n}{z}_{n}\parallel \to 0$ that $2{z}_{n}\to 0$ and hence ${z}_{n}\to {z}_{0}=0$. □

**Conclusion 3.6** ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *is a countable family of weak relatively nonexpansive mappings*.

*Proof*Since $E={L}^{2}$ is a Hilbert space, for any $n\ge 0$, we have

From Conclusion 3.5, we have ${\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})=\tilde{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$, then ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of weak relatively nonexpansive mappings. □

**Conclusion 3.7** ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *is not a countable family of relatively nonexpansive mappings*.

*Proof* From Conclusions 3.3 and 3.4, we have ${\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})\ne \stackrel{\u02c6}{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$, so, ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is not a countable family of relatively nonexpansive mappings. □

Secondly, we give another example which is a weak relatively nonexpansive mapping, but not a relatively nonexpansive mapping in the Banach space ${l}^{2}$.

**Conclusion 3.8** $\{{x}_{n}\}$ *converges weakly to* ${x}_{0}$.

*Proof*For any $f=({\zeta}_{1},{\zeta}_{2},{\zeta}_{3},\dots ,{\zeta}_{k},\dots )\in {l}^{2}={({l}^{2})}^{\ast}$, we have

as $n\to \mathrm{\infty}$. That is, $\{{x}_{n}\}$ converges weakly to ${x}_{0}$. □

**Conclusion 3.9** $\{{x}_{n}\}$ *is not a Cauchy sequence*, *so*, *it does not converge strongly to any element of* ${l}^{2}$.

*Proof* In fact, we have $\parallel {x}_{n}-{x}_{m}\parallel =\sqrt{2}$ for any $n\ne m$. Then $\{{x}_{n}\}$ is not a Cauchy sequence. □

**Conclusion 3.10** *T* *has a unique fixed point* 0, *that is*, $F(T)=\{0\}$.

*Proof* The conclusion is obvious. □

**Conclusion 3.11** ${x}_{0}$ *is an asymptotic fixed point of* *T*.

*Proof*Since $\{{x}_{n}\}$ converges weakly to ${x}_{0}$ and

as $n\to \mathrm{\infty}$, then ${x}_{0}$ is an asymptotic fixed point of *T*. □

**Conclusion 3.12** *T* *has a unique strong asymptotic fixed point* 0, *so*, $F(T)=\tilde{F}(T)$.

*Proof* In fact, for any strong convergent sequence $\{{z}_{n}\}\subset E$ such that ${z}_{n}\to {z}_{0}$ and $\parallel {z}_{n}-T{z}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, from Conclusion 3.9, there exists a sufficiently large natural number *N* such that ${z}_{n}\ne {x}_{m}$, for any $n,m>N$. Then $T{z}_{n}=-{z}_{n}$ for $n>N$, it follows from $\parallel {z}_{n}-T{z}_{n}\parallel \to 0$ that $2{z}_{n}\to 0$ and hence ${z}_{n}\to {z}_{0}=0$. □

**Conclusion 3.13** *T* *is a weak relatively nonexpansive mapping*.

*Proof*Since $E={L}^{2}$ is a Hilbert space, we have

From Conclusion 3.12, we have $F(T)=\tilde{F}(T)$, then *T* is a weak relatively nonexpansive mapping. □

**Conclusion 3.14** *T* *is not a relatively nonexpansive mapping*.

*Proof* From Conclusions 3.10 and 3.11, we have $F(T)\ne \stackrel{\u02c6}{F}(T)$, so, *T* is not a relatively nonexpansive mapping. □

Next, we prove our convergence theorems as follows.

**Theorem 3.15**

*Let*

*E*

*be a uniformly convex and uniformly smooth Banach space*,

*let*

*C*

*be a nonempty closed convex subset of*

*E*,

*let*$\{{T}_{n}\}:C\to C$

*be a countable family of weak relatively nonexpansive mappings such that*$F={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})\ne \mathrm{\varnothing}$.

*For any given gauss*${x}_{0}\in C$,

*define a sequence*$\{{x}_{n}\}$

*in*

*C*

*by the following algorithm*:

*Then* $\{{x}_{n}\}$ *converges strongly to* $q={\mathrm{\Pi}}_{F}{x}_{0}$.

*Proof* Firstly, ${C}_{n}$ is closed and convex. Since *T* is a closed hemi-relatively nonexpansive mapping, then $F(T)\subseteq {C}_{n}$, $n=0,1,2,3,\dots $ .

*m*, that

*m*. By using Lemma 2.6, we get that

holds, uniformly for all *m*. Then $\{{x}_{n}\}$ is a Cauchy sequence. Therefore, there exists a point $p\in C$ such that ${x}_{n}\to p$.

By using Lemma 2.6, we have $\parallel {x}_{n}-{T}_{n}{x}_{n}\parallel \to 0$; therefore, $p\in F(T)$.

*n*, also from Lemma 2.8, we have

Combining (3.4) and (3.5), we know that $\varphi (p,{x}_{0})=\varphi ({\mathrm{\Pi}}_{F}{x}_{0},{x}_{0})$. Therefore, it follows from the uniqueness of ${\mathrm{\Pi}}_{F}{x}_{0}$ that $p={\mathrm{\Pi}}_{F}{x}_{0}$. This completes the proof. □

**Theorem 3.16**

*Let*

*E*

*be a uniformly convex and uniformly smooth Banach space*,

*let*

*C*

*be a nonempty closed convex subset of*

*E*,

*let*$\{{T}_{n}\}:C\to C$

*be a countable family of weak relatively nonexpansive mappings such that*$F={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})\ne \mathrm{\varnothing}$.

*For any given gauss*${x}_{0}\in C$,

*define a sequence*$\{{x}_{n}\}$

*in*

*C*

*by the following algorithm*:

*Then* $\{{x}_{n}\}$ *converges strongly to* $q={\mathrm{\Pi}}_{F}{x}_{0}$.

*Proof*Let $\{{x}_{n}\}$ be defined by (3.1). We claim that

This is a contradiction to ${x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}$ and $z({t}_{0})\in {C}_{n+1}$. This completes the proof. □

## 4 Applications

Now, we apply Theorem 3.15 to prove a strong convergence theorem concerning maximal monotone operators in a Banach space *E*.

*A*be a multi-valued operator from

*E*to ${E}^{\ast}$ with the domain $D(A)=\{z\in E:Az\ne \mathrm{\varnothing}\}$ and range $R(A)=\{z\in E:z\in D(A)\}$. An operator A is said to be monotone if

for each ${x}_{1},{x}_{2}\in D(A)$ and ${y}_{1}\in A{x}_{1}$, ${y}_{2}\in A{x}_{2}$. A monotone operator A is said to be maximal if its graph $G(A)=\{(x,y):y\in Ax\}$ is not properly contained in the graph of any other monotone operator. We know that if *A* is a maximal monotone operator, then ${A}^{-1}0$ is closed and convex. The following result is also well known.

**Theorem 4.1** (Rockafellar [28])

*Let* *E* *be a reflexive*, *strictly convex and smooth Banach space*, *and let* *A* *be a monotone operator from* *E* *to* ${E}^{\ast}$. *Then* *A* *is maximal if and only if* $R(J+rA)={E}^{\ast}$ *for all* $r>0$.

*E*be a reflexive, strictly convex and smooth Banach space, and let

*A*be a maximal monotone operator from

*E*to ${E}^{\ast}$. Using Theorem 4.1 and strict convexity of

*E*, we obtain that for every $r>0$ and $x\in E$, there exists a unique ${x}_{r}$ such that

Then we can define a single valued mapping ${J}_{r}:E\to D(A)$ by ${J}_{r}={(J+rA)}^{-1}J$ and such a ${J}_{r}$ is called the resolvent of *A*. We know that ${A}^{-1}=F({J}_{r})$ for all $r>0$, see [4, 15] for more details. Using Theorem 3.15, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space. Such a problem has been also studied in [4, 5, 15, 20, 22, 24, 29–36].

**Theorem 4.2**

*Let*

*E*

*be a uniformly convex and uniformly smooth real Banach space*,

*let*

*A*

*be a maximal monotone operators from*

*E*

*to*${E}^{\ast}$

*such that*${A}^{-1}0\ne \mathrm{\varnothing}$,

*let*${J}_{r}$

*be the resolvent of A*,

*where*$r>0$.

*For any given gauss*${x}_{0}\in {C}_{0}=C$,

*define a sequence*$\{{x}_{n}\}$

*in*

*C*

*by the following algorithm*:

*with the condition*, ${r}_{n}>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{n}>0$. *Then* $\{{x}_{n}\}$ *converges strongly to* $q={\mathrm{\Pi}}_{{A}^{-1}0}{x}_{0}$.

*Proof* We only need to prove that ${\{{J}_{{r}_{n}}\}}_{n=0}^{\mathrm{\infty}}$, is a countable family of weak relatively nonexpansive mappings.

*A*, we have

for all $n\ge 0$. Thirdly, we prove the set of strong asymptotic fixed points $\tilde{F}({\{{J}_{{r}_{n}}\}}_{n=0}^{\mathrm{\infty}})={\bigcap}_{n=0}^{\mathrm{\infty}}F({J}_{{r}_{n}})={A}^{-1}0$.

*J*is uniformly norm-to-norm continuous on bounded sets, we obtain

*A*that

for all $w\in D(A)$ and ${w}^{\ast}\in Aw$. Letting $n\to \mathrm{\infty}$, we have $\u3008w-p,{w}^{\ast}\u3009\ge 0$ for all $w\in D(A)$ and ${w}^{\ast}\in Aw$. Therefore, from the maximality of *A*, we obtain $p\in {A}^{-1}0$. On the other hand, we know that $F({J}_{{r}_{n}})={A}^{-1}0$, $F({J}_{{r}_{n}})\subset \tilde{F}({J}_{{r}_{n}})$ for all $n\ge 0$; therefore, ${A}^{-1}0={\bigcap}_{n=0}^{\mathrm{\infty}}F({J}_{{r}_{n}})=\tilde{F}({\bigcap}_{n=0}^{\mathrm{\infty}}{J}_{{r}_{n}})$. From above three conclusions, we have proved ${\{{J}_{{r}_{n}}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of weak relatively nonexpansive mappings. By using Theorem 3.16, we can conclude that $\{{x}_{n}\}$ converges strongly to ${\mathrm{\Pi}}_{{A}^{-1}0}{x}_{0}$. This completes the proof. □

## Declarations

### Acknowledgements

This project is supported by the National Natural Science Foundation of China under Grant (11071279).

## Authors’ Affiliations

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