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

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:

$$\{\begin{array}{c}{C}_0=C,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,T_n{x}_n)=\varphi (z,x_n)\},n=0,1,2,3,\dots ,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0.\hfill \end{array}$$

Then $\{x_n\}$ converges strongly to $q={\mathrm{\Pi }}_F{x}_0$.

MSC:47H05, 47H09, 47H10.

1 Introduction

Let 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

$$Jx=\{f\in E^{\ast }:〈x,f〉={\parallel x\parallel }^2={\parallel f\parallel }^2\},$$

where $〈\cdot ,\cdot 〉$ 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]).

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

$$\varphi (x,y)={\parallel x\parallel }^2-2〈x,Jy〉+{\parallel y\parallel }^2,$$

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 $\hat{F}(T)$. 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 $F(T)=\hat{F}(T)$ 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].

Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one was introduced in 1953 by Mann [10] and is well known as Mann’s iteration process defined as follows:

$$\{\begin{array}{c}{x}_0\text{chosen arbitrarily},\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,n\ge 0,\hfill \end{array}$$

(1.1)

where the sequence $\{{\alpha }_n\}$ is chosen in $[0,1]$. Fourteen years later, Halpern [11] proposed the new innovation iteration process which resembled Mann’s iteration (1.1). It is defined by

$$\{\begin{array}{c}{x}_0\text{chosen arbitrarily},\hfill \\ x_{n+1}={\alpha }_{n}u+(1-{\alpha }_n)T{x}_n,n\ge 0,\hfill \end{array}$$

(1.2)

where the element $u\in C$ is fixed. Seven years later, Ishikawa [2] enlarged and improved Mann’s iteration (1.1) to the new iteration method, which is often cited as Ishikawa’s iteration process and defined recursively by

$$\{\begin{array}{c}{x}_0\text{chosen arbitrarily},\hfill \\ y_n={\beta }_n{x}_n+(1-{\beta }_n)T{x}_n,\hfill \\ x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{y}_n,n\ge 0,\hfill \end{array}$$

(1.3)

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

In both Hilbert space [11–13] and uniformly smooth Banach space [14–16] the iteration process (1.2) has been proved to be strongly convergent if the sequence $\{{\alpha }_n\}$ satisfies the following conditions:

• (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].

Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [20] proposed the following modification of the Mann iteration method for a single nonexpansive mapping T in a Hilbert space H:

$$\{\begin{array}{c}{x}_0\in C\text{chosen arbitrarily},\hfill \\ y_n={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,\hfill \\ C_n=\{z\in C:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\hfill \\ Q_n=\{z\in C:〈x_n-z,x_0-x_n〉\ge 0\},\hfill \\ x_{n+1}=P_{C_n\cap Q_n}(x_0),\hfill \end{array}$$

(1.4)

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.

The ideas to generalize the process (1.4) from a Hilbert space to a Banach space have recently been made. By using available properties on a uniformly convex and uniformly smooth Banach space, Matsushita and Takahashi [9] presented their ideas as the following method for a single relatively nonexpansive mapping T in a Banach space E:

$$\{\begin{array}{c}{x}_0\in C\text{chosen arbitrarily},\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_0+(1-{\alpha }_n)JT{x}_n),\hfill \\ C_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ Q_n=\{z\in C:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_n\cap Q_n}(x_0).\hfill \end{array}$$

(1.5)

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

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{z}_n),\hfill \\ z_n=J^{-1}({\beta }_n^{(1)}J{x}_n+{\beta }_n^{(2)}JT{x}_n+{\beta }_n^{(3)}JS{x}_n),\hfill \\ H_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ W_n=\{z\in C:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{H_n\cap W_n}(x_0)\hfill \end{array}$$

(1.6)

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

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_0+(1-{\alpha }_n)J{z}_n),\hfill \\ z_n=J^{-1}({\beta }_n^{(1)}J{x}_n+{\beta }_n^{(2)}JT{x}_n+{\beta }_n^{(3)}JS{x}_n),\hfill \\ H_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)+{\alpha }_n({\parallel x_0\parallel }^2+2〈z,J{x}_n-J{x}_0〉)\},\hfill \\ W_n=\{z\in C:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{H_n\cap W_n}(x_0)\hfill \end{array}$$

(1.7)

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:

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}},\hfill \\ z_n=J^{-1}({\beta }_n^{(1)}J{x}_n+{\beta }_n^{(2)}J{T}_n{x}_n+{\beta }_n^{(3)}J{S}_n{x}_n),\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{z}_n),\hfill \\ C_n=\{z\in C_{n-1}\cap Q_{n-1}:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ C_0=\{z\in C:\varphi (z,y_0)\le \varphi (z,x_0)\},\hfill \\ Q_n=\{z\in C_{n-1}\cap Q_{n-1}:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ Q_0=C,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_n\cap Q_n}(x_0)\hfill \end{array}$$

(1.8)

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

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

$$\varphi (x,y)={\parallel x\parallel }^2-2〈x,Jy〉+{\parallel y\parallel }^2,$$

(2.1)

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

Recall that if 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

$$\varphi (\overline{x},x)=\underset{y\in C}{min}\varphi (y,x).$$

(2.2)

The existence and uniqueness of the operator ${\mathrm{\Pi }}_C$ follow from the properties of the functional $\varphi (y,x)$ and strict monotonicity of the mapping J. In a Hilbert space, ${\mathrm{\Pi }}_C=P_C$. It is obvious from the definition of the functional ϕ that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (y,x)\le ({\parallel y\parallel }^2+{\parallel x\parallel }^2)$$

(2.3)

and

$$\varphi (x,y)=\varphi (x,z)+\varphi (z,y)-2〈x-z,Jz-Jy〉$$

(2.4)

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 $〈x,Jy〉={\parallel x\parallel }^2={\parallel Jy\parallel }^2$. 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 $\hat{F}({\{T_n\}}_{n=0}^{\mathrm{\infty }})$. 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) $\hat{F}({\{T_n\}}_{n=0}^{\mathrm{\infty }})={\bigcap }_{n=0}^{\mathrm{\infty }}F(T_n)$.

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]

The mapping 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 $\parallel x_n-y_n\parallel \to 0$.

Lemma 2.7 [23, 24, 26]

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

$$〈x_0-y,Jx-J{x}_0〉\ge 0\mathit{\text{for all}}y\in C.$$

Lemma 2.8 [23, 24, 26]

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

$$\varphi (y,{\mathrm{\Pi }}_{c}x)+\varphi ({\mathrm{\Pi }}_{c}x,x)\le \varphi (y,x)\mathit{\text{for all}}y\in C.$$

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

$${\parallel \lambda x+\mu y+\gamma z\parallel }^2\le \lambda {\parallel x\parallel }^2+\mu {\parallel y\parallel }^2+\gamma {\parallel z\parallel }^2-\lambda \mu g(\parallel x-y\parallel )$$

(2.5)

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 $l^2$.

Example 1 Let $E=l^2$, where

It is well known that $l^2$ is a Hilbert space, so ${(l^2)}^{\ast }=l^2$. Let $\{x_n\}\subset E$ be a sequence defined by

where

$${\xi }_{n,k}=\{\begin{array}{cc}1\hfill & \text{if }k=1,n+1,\hfill \\ 0\hfill & \text{if }k\ne 1,k\ne n+1,\hfill \end{array}$$

for all $n\ge 1$.

Define a countable family of mappings $T_n:E\to E$ as follows:

$$T_n(x)=\{\begin{array}{cc}\frac{n}{n+1}{x}_n\hfill & \text{if }x=x_n,\hfill \\ -x\hfill & \text{if }x\ne x_n,\hfill \end{array}$$

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

$$f(x_n-x_0)=〈f,x_n-x_0〉=\sum _{k=2}^{\mathrm{\infty }}{\zeta }_k{\xi }_{n,k}={\zeta }_{n+1}\to 0,$$

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

$$\parallel T_n{x}_n-x_n\parallel =\parallel \frac{n}{n+1}{x}_n-x_n\parallel =\frac{1}{n+1}\parallel x_n\parallel \to 0$$

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 $2z_n\to 0$ 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

$$\begin{array}{rl}\varphi (0,T_{n}x)& ={\parallel 0-T_{n}x\parallel }^2={\parallel T_{n}x\parallel }^2\\ \le {\parallel x\parallel }^2={\parallel x-0\parallel }^2=\varphi (0,x),\mathrm{\forall }x\in E.\end{array}$$

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 ${\bigcap }_{n=0}^{\mathrm{\infty }}F(T_n)\ne \hat{F}({\{T_n\}}_{n=0}^{\mathrm{\infty }})$, 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 $l^2$.

Example 2 Let $E=l^2$, where

It is well known that $l^2$ is a Hilbert space, so ${(l^2)}^{\ast }=l^2$. Let $\{x_n\}\subset E$ be a sequence defined by

where

$${\xi }_{n,k}=\{\begin{array}{cc}1\hfill & \text{if }k=1,n+1,\hfill \\ 0\hfill & \text{if }k\ne 1,k\ne n+1,\hfill \end{array}$$

for all $n\ge 1$. Define the mapping $T:E\to E$ as follows

$$T(x)=\{\begin{array}{cc}\frac{n}{n+1}{x}_n\hfill & \text{if }x=x_n(\mathrm{\exists }n\ge 1),\hfill \\ -x\hfill & \text{if }x\ne x_n(\mathrm{\forall }n\ge 1).\hfill \end{array}$$

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

$$f(x_n-x_0)=〈f,x_n-x_0〉=\sum _{k=2}^{\mathrm{\infty }}{\zeta }_k{\xi }_{n,k}={\zeta }_{n+1}\to 0,$$

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

$$\parallel T{x}_n-x_n\parallel =\parallel \frac{n}{n+1}{x}_n-x_n\parallel =\frac{1}{n+1}\parallel x_n\parallel \to 0$$

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 $2z_n\to 0$ 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

$$\begin{array}{rl}\varphi (0,Tx)& ={\parallel 0-Tx\parallel }^2={\parallel Tx\parallel }^2\\ \le {\parallel x\parallel }^2={\parallel x-0\parallel }^2=\varphi (0,x),\mathrm{\forall }x\in E.\end{array}$$

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 $F(T)\ne \hat{F}(T)$, 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:

$$\{\begin{array}{c}{C}_0=C,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,T_n{x}_n)\le \varphi (z,x_n)\},n=0,1,2,3,\dots ,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0.\hfill \end{array}$$

(3.1)

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 $F(T)\subseteq C_n$, $n=0,1,2,3,\dots$ .

Since $x_n={\mathrm{\Pi }}_{C_n}{x}_0$ and $C_n\subset C_{n-1}$, then we get

$$\varphi (x_n,x_0)\le \varphi (x_{n+1},x_0),\text{for all }n\ge 0.$$

(3.2)

Therefore, $\{\varphi (x_n,x_0)\}$ is nondecreasing. On the other hand, by Lemma 2.8 we have

$$\begin{array}{rl}\varphi (x_n,x_0)& =\varphi ({\mathrm{\Pi }}_{C_n}{x}_0,x_0)\\ \le \varphi (p,x_0)-\varphi (p,x_n)\le \varphi (p,x_0),\end{array}$$

for all $p\in F(T)\subset C_n$ and for all $n\ge 0$. Therefore, $\varphi (x_n,x_0)$ is also bounded. This together with (3.2) implies that the limit of $\{\varphi (x_n,x_0)\}$ exists. Put

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,x_0)=d.$$

(3.3)

From Lemma 2.8, we have, for any positive integer m, that

$$\begin{array}{rl}\varphi (x_{n+m},x_n)& =\varphi (x_{n+m},{\mathrm{\Pi }}_{C_n}{x}_0)\\ \le \varphi (x_{n+m},x_0)-\varphi ({\mathrm{\Pi }}_{C_n}{x}_0,x_0)\\ =\varphi (x_{n+m},x_0)-\varphi (x_{n+1},x_0),\end{array}$$

for all $n\ge 0$. This together with (3.3) implies that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+m},x_n)=0$$

holds, uniformly for all m. By using Lemma 2.6, we get that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+m}-x_n\parallel =0$$

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

Since $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0\subset C_{n+1}\subset C_n$, then

$$\varphi (x_{n+1},T_n{x}_n)\le \varphi (x_{n+1},x_n),n=0,1,2,3,\dots .$$

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

Finally, we prove that $p={\mathrm{\Pi }}_F{x}_0$. From Lemma 2.8, we have

$$\varphi (p,{\mathrm{\Pi }}_F{x}_0)+\varphi ({\mathrm{\Pi }}_F{x}_0,x_0)\le \varphi (p,x_0).$$

On the other hand, since $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0$ and $F\subset C_n$ for all n, also from Lemma 2.8, we have

$$\varphi ({\mathrm{\Pi }}_F{x}_0,x_{n+1})+\varphi (x_{n+1},x_0)\le \varphi ({\mathrm{\Pi }}_F{x}_0,x_0).$$

(3.4)

By the definition of $\varphi (x,y)$, we know that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},x_0)=\varphi (p,x_0).$$

(3.5)

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:

$$\{\begin{array}{c}{C}_0=C,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,T_n{x}_n)=\varphi (z,x_n)\},n=0,1,2,3,\dots ,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0.\hfill \end{array}$$

(3.6)

Then $\{x_n\}$ converges strongly to $q={\mathrm{\Pi }}_F{x}_0$.

Proof Let $\{x_n\}$ be defined by (3.1). We claim that

$$x_{n+1}\in \{z\in C_n:\varphi (z,T_n{x}_n)=\varphi (z,x_n)\},n=0,1,2,3,\dots .$$

Therefore,

$$x_{n+1}={\mathrm{\Pi }}_{\{z\in C_n:\varphi (z,T_n{x}_n)=\varphi (z,x_n)\}}{x}_0.$$

If not, there exists $x_{n+1}$ such that

$$\varphi (x_{n+1},T_n{x}_n)<\varphi (x_{n+1},x_n).$$

We define

$$z(t)=(1-t)x_{n+1}+t{x}_0\in C,t\in [0,1].$$

Observe that $z(0)=x_{n+1}$. Since $\varphi (\cdot ,T_n{x}_n)$, $\varphi (\cdot ,x_n)$ are continuous, then there exists $t_0\in (0,1)$ such that

$$\varphi (z(t_0),T_n{x}_n)<\varphi (z(t_0),x_n),$$

that is, $z(t_0)\in C_{n+1}$. On the other hand, we have

$$\begin{array}{rcl}\varphi (z(t_0),x_0)& =& \varphi ((1-t_0)x_{n+1}+t_0{x}_0,x_0)\\ =& {\parallel (1-t_0)x_{n+1}+t_0{x}_0\parallel }^2\\ -2〈(1-t_0)x_{n+1}+t_0{x}_0,J{x}_0〉+{\parallel x_0\parallel }^2\\ \le & (1-t_0){\parallel x_{n+1}\parallel }^2+t_0{\parallel x_0\parallel }^2\\ -2(1-t_0)〈x_{n+1},J{x}_0〉-2t_0〈x_0,J{x}_0〉+{\parallel x_0\parallel }^2\\ =& (1-t_0){\parallel x_{n+1}\parallel }^2+t_0{\parallel x_0\parallel }^2\\ -2(1-t_0)〈x_{n+1},J{x}_0〉-2t_0{\parallel x_0\parallel }^2+{\parallel x_0\parallel }^2\\ =& (1-t_0){\parallel x_{n+1}\parallel }^2+(1-t_0){\parallel x_0\parallel }^2\\ -2(1-t_0)〈x_{n+1},J{x}_0〉\\ =& (1-t_0)\varphi (x_{n+1},x_0)<\varphi (x_{n+1},x_0).\end{array}$$

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.

Let 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

$$〈x_1-x_2,y_1-y_2〉\ge 0$$

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 $r>0$.

Let 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

$$Jx\in J{x}_r+rA{x}_r.$$

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:

$$\{\begin{array}{c}{C}_0=C,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,J_{r_n}{x}_n)=\varphi (z,x_n)\},n=0,1,2,3,\dots ,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,\hfill \end{array}$$

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.

Firstly, we have ${\bigcap }_{n=0}^{\mathrm{\infty }}F(J_{r_n})=A^{-1}0\ne \mathrm{\varnothing }$. Secondly, from the monotonicity of A, we have

$$\begin{array}{rcl}\varphi (p,J_{r_n}w)& =& {\parallel p\parallel }^2-2〈p,J{J}_{r_n}w〉+{\parallel J_{r_n}w\parallel }^2\\ =& {\parallel p\parallel }^2+2〈p,Jw-J{J}_{r_n}w-Jw〉+{\parallel J_{r_n}w\parallel }^2\\ =& {\parallel p\parallel }^2+2〈p,Jw-J{J}_{r_n}w〉-2〈p,Jw〉+{\parallel J_{r_n}w\parallel }^2\\ =& {\parallel p\parallel }^2-2〈J_{r_n}w-p-J_{r_n}w,Jw-J{J}_{r_n}w-Jw〉\\ -2〈p,Jw〉+{\parallel J_{r_n}w\parallel }^2\\ =& {\parallel p\parallel }^2-2〈J_{r_n}w-p,Jw-J{J}_{r_n}w-Jw〉\\ +2〈J_{r_n}w,Jw-J{J}_{r_n}w〉-2〈p,Jw〉+{\parallel J_{r_n}w\parallel }^2\\ \le & {\parallel p\parallel }^2+2〈J_{r_n}w,Jw-J{J}_{r_n}w〉-2〈p,Jw〉+{\parallel J_{r_n}w\parallel }^2\\ =& {\parallel p\parallel }^2-2〈p,Jw〉+{\parallel w\parallel }^2-{\parallel J_{r_n}w\parallel }^2\\ +2〈J_{r_n}w,Jw〉-{\parallel w\parallel }^2\\ =& \varphi (p,w)-\varphi (J_{r_n}w,w)\\ \le & \varphi (p,w)\end{array}$$

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

We first show that $\tilde{F}({\{J_{r_n}\}}_{n=0}^{\mathrm{\infty }})\subset A^{-1}0$. Let $p\in \tilde{F}({\{J_{r_n}\}}_{n=0}^{\mathrm{\infty }})$, then there exists $\{z_n\}\subset E$ such that $z_n\to p$ and ${lim}_{n\to \mathrm{\infty }}\parallel z_n-J_{r_n}{z}_n\parallel =0$. Since J is uniformly norm-to-norm continuous on bounded sets, we obtain

$$\frac{1}{r_n}(J{z}_n-J{J}_{r_n}{z}_n)\to 0.$$

It follows from

$$\frac{1}{r_n}(J{z}_n-J{J}_{r_n}{z}_n)\in A{J}_{r_n}{z}_n$$

and the monotonicity of A that

$$〈w-J_{r_n}{z}_n,w^{\ast }-\frac{1}{r_n}(J{z}_n-J{J}_{r_n}{z}_n)〉\ge 0$$

for all $w\in D(A)$ and $w^{\ast }\in Aw$. Letting $n\to \mathrm{\infty }$, we have $〈w-p,w^{\ast }〉\ge 0$ 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. □