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# Uniformly closed replaced AKTT or ^{∗}AKTT condition to get strong convergence theorems for a countable family of relatively quasi-nonexpansive mappings and systems of equilibrium problems

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

**2014**:103

https://doi.org/10.1186/1687-1812-2014-103

© Zhang et al.; licensee Springer. 2014

**Received:**9 February 2014**Accepted:**31 March 2014**Published:**6 May 2014

## Abstract

The purpose of this paper is to construct a new iterative scheme and to get a strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized *f*-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but does not satisfy condition AKTT and ^{∗}AKTT. Our results can be applied to solve a convex minimization problem. In addition, this paper clarifies an ambiguity in a useful lemma. The results of this paper modify and improve many other important recent results.

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

## Keywords

- relatively quasi-nonexpansive mapping
- equilibrium problems
- generalized
*f*-projection operator - hybrid algorithm
- uniformly closed mappings

## 1 Introduction and preliminaries

*E*be a real Banach space and

*C*be a nonempty closed convex subset of

*E*. A mapping $T:C\to C$ is called nonexpansive if

*E*be a real Banach space and

*C*be a nonempty closed convex subset of

*E*. A point $p\in C$ is said to be an asymptotic fixed point of

*T*if there exists a sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset C$ such that ${x}_{n}\rightharpoonup p$ and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$. The set of asymptotic fixed point is denoted by $\stackrel{\u02c6}{F}(T)$. We say that a mapping

*T*is relatively nonexpansive (see [1–4]) if the following conditions are satisfied:

- (I)
$F(T)\ne \mathrm{\varnothing}$;

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

- (III)
$F(T)=\stackrel{\u02c6}{F}(T)$.

If *T* satisfies (I) and (II), then *T* is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings.

*E*be a real Banach space. The modulus of smoothness of

*E*is the function ${\rho}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by

*E*is uniformly smooth if and only if

Let $dimE\ge 2$. The modulus of convexity of *E* is the function ${\delta}_{E}(\u03f5):=inf\{1-\parallel \frac{x+y}{2}\parallel :\parallel x\parallel =\parallel y\parallel =1;\u03f5=\parallel x-y\parallel \}$. *E* is uniformly convex if for any $\u03f5\in (0,2]$, there exists $\delta =\delta (\u03f5)>0$ such that if $x,y\in E$ with $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $\parallel x-y\parallel \ge \u03f5$, then $\parallel \frac{1}{2}(x+y)\parallel \le 1-\delta $. Equivalently, *E* is uniformly convex if and only if ${\delta}_{E}(\u03f5)>0$ for all $\u03f5\in (0,2]$. A normed space *E* is called strictly convex if for all $x,y\in E$, $x\ne y$, $\parallel x\parallel =\parallel y\parallel =1$, we have $\parallel \lambda x+(1-\lambda )y\parallel <1$, $\mathrm{\forall}\lambda \in (0,1)$.

*E*. We denote by

*J*the normalized duality mapping from

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

*J*are well known (see [5–7] for more details):

- (1)
If

*E*is uniformly smooth, then*J*is norm-to-norm uniformly continuous on each bounded subset of*E*. - (2)
If

*E*is reflexive, then*J*is a mapping from*E*onto ${E}^{\ast}$. - (3)
If

*E*is smooth, then*J*is single valued.

*ϕ*the functional on $E\times E$ defined by

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

*C*be a nonempty closed convex subset of

*E*. Following Alber [8], the generalized projection ${\mathrm{\Pi}}_{C}$ from

*E*onto

*C*is defined by

*J*. It is obvious that

*f*-projection operator and its properties. Let $G:C\times {E}^{\ast}\to R\cup \{+\mathrm{\infty}\}$ be a functional defined as follows:

*ρ*is a positive number and $f:C\to R\cup \{+\mathrm{\infty}\}$ is proper, convex, and lower semi-continuous. From the definitions of

*G*and

*f*, it is easy to see the following properties:

- (i)
$G(\xi ,\phi )$ is convex and continuous with respect to

*φ*when*ξ*is fixed; - (ii)
$G(\xi ,\phi )$ is convex and lower semi-continuous with respect to

*ξ*when*φ*is fixed.

**Definition 1.1** [9]

*E*be a real Banach space with its dual ${E}^{\ast}$. Let

*C*be a nonempty, closed, and convex subset of

*E*. We say that ${\mathrm{\Pi}}_{C}^{f}:{E}^{\ast}\to {2}^{C}$ is

*a generalized*

*f-projection operator*if

For the generalized *f*-projection operator, Wu and Huang [9] proved in the following theorem some basic properties.

**Lemma 1.2** [9]

*Let*

*E*

*be a real reflexive Banach space with its dual*${E}^{\ast}$.

*Let*

*C*

*be a nonempty*,

*closed*,

*and convex subset of*

*E*.

*Then the following statements hold*:

- (i)
${\mathrm{\Pi}}_{C}^{f}$

*is a nonempty closed convex subset of**C**for all*$\phi \in {E}^{\ast}$. - (ii)
*If**E**is smooth*,*then for all*$\phi \in {E}^{\ast}$, $x\in {\mathrm{\Pi}}_{C}^{f}\phi $*if and only if*$\u3008x-y,\phi -Jx\u3009+\rho f(y)-\rho f(x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C.$ - (iii)
*If**E**is strictly convex and*$f:C\to R\cup \{+\mathrm{\infty}\}$*is positive homogeneous*(*i*.*e*., $f(tx)=tf(x)$*for all*$t>0$*such that*$tx\in C$*where*$x\in C$),*then*${\mathrm{\Pi}}_{C}^{f}$*is a single*-*valued mapping*.

Fan *et al.* [10] showed that the condition *f* is positive homogeneous which appeared in Lemma 1.2 can be removed.

**Lemma 1.3** [10]

*Let* *E* *be a real reflexive Banach space with its dual* ${E}^{\ast}$ *and* *C* *a nonempty*, *closed*, *and convex subset of* *E*. *Then if* *E* *is strictly convex*, *then* ${\mathrm{\Pi}}_{C}^{f}$ *is a single*-*valued mapping*.

*J*is a single-valued mapping when

*E*is a smooth Banach space. There exists a unique element $\phi \in {E}^{\ast}$ such that $\phi =Jx$ for each $x\in E$. This substitution in (1.3) gives

Now, we consider the second generalized *f*-projection operator in a Banach space.

**Definition 1.4** [11]

*E*be a real Banach space and

*C*a nonempty, closed, and convex subset of

*E*. We say that ${\mathrm{\Pi}}_{C}^{f}:E\to {2}^{C}$ is

*a generalized*

*f-projection operator*if

*T*is equivalent to

- (1)
$F(T)\ne \mathrm{\varnothing}$;

- (2)
$G(p,JTx)\le G(p,Jx)$, $\mathrm{\forall}x\in C$, $p\in F(T)$.

**Lemma 1.5** [12]

*Let*

*E*

*be a Banach space and*$f:E\to R\cup \{+\mathrm{\infty}\}$

*be a lower semi*-

*continuous convex functional*.

*Then there exist*$x\in {E}^{\ast}$

*and*$\alpha \in R$

*such that*

We know that the following lemmas hold for operator ${\mathrm{\Pi}}_{C}^{f}$.

**Lemma 1.6** [13]

*Let*

*C*

*be a nonempty*,

*closed*,

*and convex subset of a smooth and reflexive Banach space*

*E*.

*Then the following statements hold*:

- (i)
${\mathrm{\Pi}}_{C}^{f}$

*is a nonempty*,*closed*,*and convex subset of**C**for all*$x\in E$; - (ii)
*for all*$x\in E$, $\stackrel{\u02c6}{x}\in {\mathrm{\Pi}}_{C}^{f}x$*if and only if*$\u3008\stackrel{\u02c6}{x}-y,Jx-J\stackrel{\u02c6}{x}\u3009+\rho f(y)-\rho f(x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C;$ - (iii)
*if**E**is strictly convex*,*then*${\mathrm{\Pi}}_{C}^{f}x$*is a single*-*valued mapping*.

**Lemma 1.7** [13]

*Let*

*C*

*be a nonempty*,

*closed*,

*and convex subset of a smooth and reflexive Banach space*

*E*.

*Let*$x\in E$

*and*$\stackrel{\u02c6}{x}\in {\mathrm{\Pi}}_{C}^{f}x$.

*Then*

The fixed points set $F(T)$ of a relatively quasi-nonexpansive mapping is closed convex as given in the following lemma.

*Let* *C* *be a nonempty closed convex subset of a smooth*, *uniformly convex Banach space* *E*. *Let* *T* *be a closed relatively quasi*-*nonexpansive mapping of* *C* *into itself*. *Then* $F(T)$ *is closed and convex*.

Also, this following lemma will be used in the sequel.

**Lemma 1.9** [16]

*Let* *C* *be a nonempty closed convex subset of a smooth*, *uniformly convex Banach space* *E*. *Let* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *and* ${\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *be sequences in* *E* *such that either* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *or* ${\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *is bounded*. *If* ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{y}_{n})=0$, *then* ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{y}_{n}\parallel =0$.

**Lemma 1.10** [17]

*Let*$p>1$

*and*$r>0$

*be two fixed real numbers*.

*Then a Banach space*

*X*

*is uniformly convex if and only if there is a continuous*,

*strictly increasing and convex function*$g:{R}^{+}\to {R}^{+}$, $g(0)=0$,

*such that*

*for all* $x,y\in {B}_{r}$ *and* $0\le \lambda \le 1$, *where* ${W}_{p}(\lambda )=\lambda {(1-\lambda )}^{p}+{\lambda}^{p}(1-\lambda )$.

**Remark** We can see from the Lemma 1.10 that the function *g* has no relation with the selection of *x*, *y* and *λ*. However, the key point above, in the process of generalization and application about this lemma, has been ambiguous gradually. For instance, the following lemma states that the function *g* has something to do with *λ*, which always leads to failure in the proof.

**Lemma** (stated in [[11], Lemma 2.10])

*Let*

*E*

*be a uniformly convex real Banach space*.

*For arbitrary*$r>0$,

*let*${B}_{r}(0):=\{x\in E:\parallel x\parallel \le r\}$

*and*$\lambda \in [0,1]$.

*Then there exists a continuous strictly increasing convex function*

*such that for every*$x,y\in {B}_{r}(0)$,

*the following inequality holds*:

Let *F* be a bifunction of $C\times C$ into *R*. The equilibrium problem is to find ${x}^{\ast}\in C$ such that $F({x}^{\ast},y)\ge 0$, for all $y\in C$. We shall denote the solutions set of the equilibrium problem by $EP(F)$. Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases.

For solving the equilibrium problem for a bifunction $F:C\times C\to R$, let us assume that *F* satisfies the following conditions:

(A1) $F(x,x)=0$ for all $x\in C$;

(A2) *F* is monotone, *i.e.*, $F(x,y)+F(y,x)\le 0$ for all $x,y\in C$;

(A3) for each $x,y\in C$, ${lim}_{t\to 0}F(tz+(1-t)x,y)\le F(x,y)$;

(A4) for each $x\in C$, $y\mapsto F(x,y)$ is convex and lower semi-continuous.

**Lemma 1.11** [18]

*Let*

*C*

*be a nonempty closed convex subset of a smooth*,

*strictly convex and reflexive Banach space*

*E*

*and let*

*F*

*be a bifunction of*$C\times C$

*into*

*R*

*satisfying*(A1)-(A4).

*Let*$r>0$

*and*$x\in E$.

*Then there exists*$z\in C$

*such that*

**Lemma 1.12** [19]

*Let*

*C*

*be a nonempty closed convex subset of a smooth*,

*strictly convex and reflexive Banach space*

*E*.

*Assume that*$F:C\times C\to R$

*satisfies*(A1)-(A4).

*For*$r>0$

*and*$x\in E$,

*define a mapping*${T}_{r}^{F}:E\to C$

*as follows*:

*for all*$z\in E$.

*Then the following hold*:

- (1)
${T}_{r}^{F}$

*is single valued*; - (2)${T}_{r}^{F}$
*is a firmly nonexpansive*-*type mapping*,*i*.*e*.,*for any*$x,y\in E$,$\u3008{T}_{r}^{F}x-{T}_{r}^{F}y,J{T}_{r}^{F}x-J{T}_{r}^{F}y\u3009\le \u3008{T}_{r}^{F}x-{T}_{r}^{F}y,Jx-Jy\u3009;$ - (3)
$F({T}_{r}^{F})=EP(F)$;

- (4)
$EP(F)$

*is closed and convex*.

**Lemma 1.13** [19]

*Let*

*C*

*be a nonempty closed convex subset of a smooth*,

*strictly convex and reflexive Banach space*

*E*.

*Assume that*$F:C\times C\to R$

*satisfies*(A1)-(A4)

*and let*$r>0$.

*Then for each*$x\in E$

*and*$q\in F({T}_{r}^{F})$,

*C*into

*E*, where

*C*is a nonempty closed convex subset of a real Banach space

*E*. For a subset

*B*of

*C*, we say that

- (i)$(\{{T}_{n}\},B)$ satisfies condition AKTT (see [15]) if$\sum _{n=1}^{\mathrm{\infty}}sup\{\parallel {T}_{n+1}x-{T}_{n}x\parallel :x\in B\}<\mathrm{\infty};$
- (ii)$(\{{T}_{n}\},B)$ satisfies condition
^{∗}AKTT (see [15]) if$\sum _{n=1}^{\mathrm{\infty}}sup\{\parallel J{T}_{n+1}x-J{T}_{n}x\parallel :x\in B\}<\mathrm{\infty}.$

Recently, Shehu [11] proved strong convergence theorems for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized *f*-projection operator. The author obtained the following theorem.

**Theorem 1.14** [11]

*Let*

*E*

*be a uniformly convex real Banach space which is also uniformly smooth*.

*Let*

*C*

*be a nonempty closed convex subset of*

*E*.

*For each*$k=1,2,\dots ,m$,

*let*${F}_{k}$

*be a bifunction from*$C\times C$

*satisfying*(A1)-(A4)

*and let*${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*be an infinite family of relatively quasi*-

*nonexpansive mappings of*

*C*

*into itself such that*$F:=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\cap ({\bigcap}_{k=1}^{m}EP({F}_{k}))\ne \mathrm{\varnothing}$.

*Let*$f:E\to R$

*be a convex and lower semi*-

*continuous mapping with*$C\subset int(D(f))$

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

*is iteratively generated by*${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,

*where* *J* *is the duality mapping on* *E*. *Suppose* ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *is a sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$ ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) *satisfying* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). *Suppose that for each bounded subset* *B* *of* *C*, *the ordered pair* $(\{{T}_{n}\},B)$ *satisfies either condition AKTT or condition* ^{∗}
*AKTT*. *Let* *T* *be the mapping from* *C* *into* *E* *defined by* $Tx={lim}_{n\to \mathrm{\infty}}{T}_{n}x$ *for all* $x\in C$ *and suppose that* *T* *is closed and* $F(T)={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$. *Then* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *converges strongly to* ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$.

In this paper we will construct a new iterative scheme and will get strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized *f*-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but not satisfy condition AKTT and ^{∗}AKTT.

## 2 Main results

Now, we shall first introduce the notion of uniformly closed mappings and give an example which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of *G*. Another example shall be given which is uniformly closed but not satisfy condition AKTT and ^{∗}AKTT.

**Definition 2.1** Let *E* be a Banach space, *C* be a nonempty closed convex subset of *E*. Let ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}:C\to E$ be a sequence of mappings of *C* into *E* such that ${\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$ is nonempty. ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is said to be *uniformly closed*, if $p\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$, whenever $\{{x}_{n}\}\subset C$ converges strongly to *p* and $\parallel {x}_{n}-{T}_{n}{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$.

**Example 1**Let $E={l}^{2}$, where

for all $n\ge 1$.

for all $n\ge 0$.

**Conclusion 2.2** ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *has a unique fixed point* 0, *that is*, $F({T}_{n})=\{0\}\ne \mathrm{\varnothing}$, $\mathrm{\forall}n\ge 0$.

*Proof* The conclusion is obvious. □

*G*, where

is said to be the asymptotic fixed point set of ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$.

**Conclusion 2.3** ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *is a countable family of relatively quasi*-*nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional G*.

*Proof*By Conclusion 2.2, we only need to show that $G(0,J{T}_{n}x)\le G(0,Jx)$, $\mathrm{\forall}x\in E$. Note that $E={l}^{2}$ is a Hilbert space, for any $n\ge 0$ we can derive

as $n\to \mathrm{\infty}$, so ${x}_{0}$ is an asymptotic fixed point of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$. Joining with Conclusion 2.2, we can obtain ${\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})\ne \stackrel{\u02c6}{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$.

Thus, ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of *G*. □

**Conclusion 2.4** ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *is a countable family of uniformly closed relatively quasi*-*nonexpansive mappings in the sense of functional* *G*.

*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 $n\to \mathrm{\infty}$, there exists a sufficiently large nature number *N*, such that ${z}_{n}\ne {x}_{m}$ for any $n,m>N$ (since ${x}_{n}$ is not a Cauchy sequence it cannot converge to any element in *E*). Then ${T}_{n}{z}_{n}=-{z}_{n}$ for $n>N$, it follows from $\parallel {z}_{n}-{T}_{n}{z}_{n}\parallel \to 0$ that $2{z}_{n}\to 0$ and hence ${z}_{n}\to {z}_{0}=0$.

Therefore, ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional *G*. □

Now, we give an example which is a countable family of uniformly closed quasi-nonexpansive mappings but not satisfied condition AKTT and ^{∗}AKTT.

**Example 2**Let $X={\mathrm{\Re}}^{2}$. For any complex number $x=r{e}^{i\theta}\in X$, define a countable family of quasi-nonexpansive mappings as follows:

*Proof*It is easy to see that ${\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})=\{0\}$. We first prove that $\{{T}_{n}\}$ is uniformly closed. In fact, for any strong convergent sequence $\{{x}_{n}\}\subset X$ such that ${x}_{n}\to {x}_{0}$ and $\parallel {x}_{n}-{T}_{n}{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, there must be ${x}_{0}=0\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$. Otherwise, if ${x}_{n}\to {x}_{0}\ne 0$, and

This is a contradiction. Therefore, $\{{T}_{n}\}$ is uniformly closed.

*n*by the definition of ${T}_{n}$, we have

That is to say, $\{{T}_{n}\}$ does not satisfied condition AKTT and ^{∗}AKTT. □

Now we are in a position to present our main theorems.

**Theorem 2.5** *Let* ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *be a countable family of uniformly closed relatively quasi*-*nonexpansive mappings of* *C* *into itself and other conditions are the same as Theorem* 1.14 *except for condition AKTT*, ^{∗}
*AKTT and condition ‘Let* *T* *be the mapping from* *C* *into* *E* *defined by* $Tx={lim}_{n\to \mathrm{\infty}}{T}_{n}x$ *for all* $x\in C$ *and suppose that* *T* *is closed and* $F(T)={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$
*’*. *Then the sequence* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *generated by* (1.5) *converges strongly to* ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$.

*Proof*We first show that ${C}_{n}$, $\mathrm{\forall}n\ge 1$, is closed and convex. It is obvious that ${C}_{1}=C$ is closed and convex. Suppose that ${C}_{n}$ is closed convex for some $n>1$. From the definition of ${C}_{n+1}$, we have $z\in {C}_{n+1}$ implies $G(z,J{u}_{n})\le G(z,J{x}_{n})$. This is equivalent to

This implies that ${C}_{n+1}$ is closed convex for the same $n>1$. Hence, ${C}_{n}$ is closed and convex for all $n\ge 1$. This shows that ${\mathrm{\Pi}}_{{C}_{n+1}}^{f}{x}_{0}$ is well defined for all $n\ge 0$.

By taking ${\theta}_{n}^{k}={T}_{{r}_{k,n}}^{{F}_{k}}{T}_{{r}_{k-1,n}}^{{F}_{k-1}}\cdots {T}_{{r}_{2,n}}^{{F}_{2}}{T}_{{r}_{1,n}}^{{F}_{1}}$, $k=1,2,\dots ,m$ and ${\theta}_{n}^{0}=I$ for all $n\ge 1$, we obtain ${u}_{n}={\theta}_{n}^{m}{y}_{n}$.

So, ${x}^{\ast}\in {C}_{n}$. This implies that $F\subset {C}_{n}$, $\mathrm{\forall}n\ge 1$ and the sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ generated by (1.5) is well defined.

and so ${\{G({x}_{n},J{x}_{0})\}}_{n=0}^{\mathrm{\infty}}$ is nondecreasing. It follows that the limit of ${\{G({x}_{n},J{x}_{0})\}}_{n=0}^{\mathrm{\infty}}$ exists.

It then follows from Lemma 1.9 that $\parallel {x}_{m}-{x}_{n}\parallel \to 0$ as $m,n\to \mathrm{\infty}$. Hence, ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a Cauchy sequence. Since *E* is a Banach space and *C* is closed and convex, there exists $p\in C$ such that ${x}_{n}\to p$ as $n\to \mathrm{\infty}$.

*E*is uniformly convex and smooth, we have from Lemma 1.9

*J*is uniformly norm-to-norm continuous on bounded sets and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{u}_{n}\parallel =0$, we obtain

*E*is uniformly smooth, we know that ${E}^{\ast}$ is uniformly convex. Then from Lemma 1.10, we have

*g*, we have ${lim}_{n\to \mathrm{\infty}}\parallel J{x}_{n}-J{T}_{n}{x}_{n}\parallel =0$. Since ${J}^{-1}$ is also uniformly norm-to-norm continuous on bounded sets, we have

Since ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are uniformly closed, and ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a Cauchy sequence. Then $p\in F(T)={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$.

This implies that $p\in EP({F}_{k})$, $k=1,2,\dots ,m$. Thus, $p\in {\bigcap}_{k=1}^{m}EP({F}_{k})$. Hence, we have $p\in F={\bigcap}_{k=1}^{m}EP({F}_{k})\cap ({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))$.

*ξ*when

*φ*is fixed. This implies that

From the definition of ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$ and $p\in F$, we see that $p=w$. This completes the proof. □

**Corollary 2.6**

*Let*

*E*

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

*and let*

*C*

*be a nonempty closed convex subset of*

*E*.

*For each*$k=1,2,\dots ,m$,

*let*${F}_{k}$

*be a bifunction from*$C\times C$

*satisfying*(A1)-(A4)

*and let*${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*be a countable family of uniformly closed relatively quasi*-

*nonexpansive mappings of*

*C*

*into itself such that*$F:=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\cap ({\bigcap}_{k=1}^{m}EP({F}_{k}))\ne \mathrm{\varnothing}$.

*Suppose*${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$

*is iteratively generated by*${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,

*where* *J* *is the duality mapping on* *E*. *Suppose* ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *is a sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$, *and* ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) *satisfying* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). *Then* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *converges strongly to* ${\mathrm{\Pi}}_{F}{x}_{0}$.

*Proof* Take $f(x)=0$ for all $x\in E$ in Theorem 2.5, then $G(\xi ,Jx)=\varphi (\xi ,x)$ and ${\mathrm{\Pi}}_{C}^{f}{x}_{0}={\mathrm{\Pi}}_{C}{x}_{0}$. Then Corollary 2.6 holds. □

Take ${F}_{k}\equiv 0$ ($k=1,2,\dots ,m$), it is obvious that the following holds.

**Corollary 2.7**

*Let*

*E*

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

*and let*

*C*

*be a nonempty closed convex subset of*

*E*.

*Let*${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*be a countable family of uniformly closed relatively quasi*-

*nonexpansive mappings of*

*C*

*into itself such that*$F=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\ne \mathrm{\varnothing}$.

*Let*$f:E\to R$

*be a convex and lower semi*-

*continuous mapping with*$C\subset int(D(f))$

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

*is iteratively generated by*${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,

*where* *J* *is the duality mapping on* *E*. *Suppose* ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *is a sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$, *and* ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) *satisfying* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). *Then* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *converges strongly to* ${\mathrm{\Pi}}_{F}{x}_{0}$.

## 3 Applications

**Theorem 3.1**

*Let*

*E*

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

*and let*

*C*

*be a nonempty closed convex subset of*

*E*.

*For each*$k=1,2,\dots ,m$,

*let*${\phi}_{k}$

*be a bifunction from*$C\times C$

*satisfying*(A1)-(A4)

*and let*${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*be a countable family of uniformly closed relatively quasi*-

*nonexpansive mappings of*

*C*

*into itself such that*$F:=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\cap ({\bigcap}_{k=1}^{m}CMP({\phi}_{k}))\ne \mathrm{\varnothing}$.

*Let*$f:E\to R$

*be a convex and lower semi*-

*continuous mapping with*$C\subset int(D(f))$

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

*is iteratively generated by*${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,

*where* *J* *is the duality mapping on* *E*. *Suppose* ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *is a sequence in* $(0,1)$ *such that* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$ *and* ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) *satisfying* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). *Then* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *converges strongly to* ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$.

*Proof* Define ${F}_{k}(x,y)={\phi}_{k}(y)-{\phi}_{k}(x)$, $x,y\in C$ and $k=1,2,\dots ,m$. Then $F({T}_{{r}_{k}}^{{F}_{k}})=EP({F}_{k})=CMP({\phi}_{k})=F({T}_{{r}_{k}}^{{\phi}_{k}})$ for each $k=1,2,\dots ,m$, and therefore ${\{{F}_{k}\}}_{k=1}^{m}$ satisfies conditions (A1) and (A2). Furthermore, one can easily show that ${\{{F}_{k}\}}_{k=1}^{m}$ satisfies (A3) and (A4). Therefore, from Theorem 2.5, we obtain Theorem 3.1. □

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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