# Convergence of a hybrid iterative method for finite families of generalized quasi-*ϕ*-asymptotically nonexpansive mappings

- Bashir Ali
^{1}Email author and - MS Minjibir
^{2}

**2012**:121

https://doi.org/10.1186/1687-1812-2012-121

© Ali and Minjibir; licensee Springer 2012

**Received: **10 January 2012

**Accepted: **4 July 2012

**Published: **23 July 2012

## Abstract

Strong convergence theorem for finite families of generalized quasi-*ϕ*-asymptotically nonexpansive mappings is proved in a real uniformly convex and uniformly smooth Banach space using a new modified hybrid iterative algorithm.

**MSC:**47H09, 47J25.

### Keywords

generalized quasi-*ϕ*-asymptotically nonexpansive mappings generalized projection map hybrid methods uniformly convex Banach space uniformly smooth Banach space

## 1 Introduction

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

*E*. The

*normalized duality mapping*$J:E\to {2}^{{E}^{\ast}}$ is defined by

*E*is said to be

*uniformly convex*if given $\u03f5\in (0,2]$, there exists $\delta >0$ such that for all $x,y\in E$ with $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $\parallel x-y\parallel \ge \u03f5$, we have $\parallel \frac{x+y}{2}\parallel \le 1-\delta $.

*E*is

*strictly convex*if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. The space

*E*is said to be

*smooth*if the limit

exists for all $x,y\in U$, where $U:=\{z\in E:\parallel z\parallel =1\}$. It is also *uniformly smooth* if the limit exists uniformly for $x,y\in U$. It is well known that if *E* is strictly convex, smooth and reflexive, then the duality map *J* is one-to-one, single-valued and onto. Also if *E* is uniformly smooth, then *J* is norm-to-norm uniformly continuous on bounded subsets of *E*.

Let *C* be a nonempty, closed, convex subset of *E*. Let $T:C\to C$ be a map, a point $x\in C$ is called a *fixed point* of *T* if $Tx=x$ and the set of all fixed points of *T* is denoted by $F(T)$. We recall that a point $p\in C$ is called an *asymptotic fixed point* of *T* if there exists a sequence $\{{x}_{n}\}\subset C$ which converges weakly to *p* and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$. The mapping *T* is called *Lipschitz* if there exists $L>0$ such that $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel $ for all $x,y\in C$, and if $L=1$, then *T* is called *nonexpansive*. *T* is *asymptotically nonexpansive* if there exists a sequence $\{{t}_{n}\}\subset [1,\mathrm{\infty})$ such that ${t}_{n}\to 1$ as $n\to \mathrm{\infty}$ and $\parallel {T}^{n}x-{T}^{n}y\parallel \le {t}_{n}\parallel x-y\parallel $ for all $n\in \mathbb{N}$ and for all $x,y\in C$. The map *T* is *quasi-nonexpansive* if $F(T)\ne \mathrm{\varnothing}$ and for all $x\in C$, $q\in F(T)$, $\parallel Tx-q\parallel \le \parallel x-q\parallel $ and is called *asymptotically quasi-nonexpansive* if $F(T)\ne \mathrm{\varnothing}$ and $\parallel {T}^{n}x-q\parallel \le {t}_{n}\parallel x-q\parallel $ for all $x\in C$, $q\in F(T)$ and the sequence $\{{t}_{n}\}\subset [1,\mathrm{\infty})$ satisfies ${t}_{n}\to 1$ as $n\to \mathrm{\infty}$. The mapping *T* is called *generalized asymptotically quasi-nonexpansive* if $F(T)\ne \mathrm{\varnothing}$, there exist sequences $\{{s}_{n}\}\subset [0,1]$, $\{{t}_{n}\}\subset [1,\mathrm{\infty})$ with ${s}_{n}\to 0$, ${t}_{n}\to 1$ as $n\to \mathrm{\infty}$ and $\parallel {T}^{n}x-q\parallel \le {t}_{n}\parallel x-q\parallel +{s}_{n}$ for all $x\in C$, $q\in F(T)$ and $n\in \mathbb{N}$.

*T*is said to be

- (i)
*asymptotically regular*on*C*if ${lim}_{n\to \mathrm{\infty}}\parallel {T}^{n+1}x-{T}^{n}x\parallel =0$ for all $x\in C$, - (ii)
*uniformly asymptotically regular*on*C*if ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}x\in K}\parallel {T}^{n+1}x-{T}^{n}x\parallel =0$ holds for any bounded subset*K*of*C*.

For a positive real number *L*, the map *T* is called *uniformly* *L-Lipschitzian* if $\parallel {T}^{n}x-{T}^{n}y\parallel \le L\parallel x-y\parallel $ for all $x,y\in C$ and $n\in \mathbb{N}$.

It is clear from these definitions that every nonexpansive mapping with a fixed point is quasi-nonexpansive and all asymptotically nonexpansive maps with fixed points are asymptotically quasi-nonexpansive. Recently, the class of generalized asymptotically quasi-nonexpansive mappings was introduced and studied by Shahzad and Zegeye [21]. They proved that every asymptotically quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping and the inclusion is proper. The class of quasi-nonexpansive mappings was introduced and studied first in 1967 by Diaz and Metcalf [7]. Goebel and Kirk [8] introduced the class of asymptotically nonexpansive mappings and proved that if *C* is a nonempty, closed, convex and bounded subset of a uniformly convex Banach space *E*, then an asymptotically nonexpansive mapping $T:C\to C$ has a fixed point.

Kirk [16], proved that if *E* is a reflexive Banach space with normal structure and *C* is a nonempty, closed, convex and bounded subset of *E*, a nonexpansive map $T:C\to C$ has a fixed point in *C*. This result was extended to a finite family of nonexpansive maps by Bellus and Kirk [3] and then to an infinite family of nonexpansive maps by Lim [17].

Let *H* be a real Hilbert space, *C* be a nonempty closed convex subset of *H*. Recall that for each $x\in H$ there exists a unique nearest point in *C* to *x* denoted by ${P}_{C}x$. That is, $\parallel x-{P}_{C}x\parallel \le \parallel x-y\parallel $ for all $y\in C$. ${P}_{C}$ is called a *metric projection* of *H* onto *C*.

It is well known that the metric projection is nonexpansive only in a Hilbert space. This fact actually characterizes Hilbert spaces. Alber [1], introduced a generalized projection map ${\prod}_{C}:E\to C$ in a Banach space which is an analogue of the metric projection in a Hilbert space.

Let *E* be a real normed linear space with single-valued normalized duality map. Consider the functional defined by $\varphi (x,y)={\parallel x\parallel}^{2}+{\parallel y\parallel}^{2}-2\u3008x,Jy\u3009$. We observe that in a Hilbert space, $\varphi (x,y)$ reduces to ${\parallel x-y\parallel}^{2}$. It is clear that for $x,y\in E$, the following inequality holds ${(\parallel x\parallel -\parallel y\parallel )}^{2}\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^{2}$. The generalized projection map ${\prod}_{C}:E\to C$ is a map that assigns to an arbitrary point $x\in E$, the minimum point of the functional $\varphi (x,\cdot )$ over *C*, that is, ${\prod}_{C}x={x}^{\ast}$ where $\varphi (x,{x}^{\ast})={min}_{y\in C}\varphi (x,y)$. Existence and uniqueness of the map ${\prod}_{C}$ follow from the properties of the functional *ϕ* and the strict monotonicity of *J* (see, for example, [2]).

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

*E*, a mapping $T:C\to C$ is said to be

- (i)
*relatively nonexpansive*if $F(T)=\tilde{F(T)}$ and $\varphi (q,Tx)\le \varphi (q,x)$ for all $x\in C$, $q\in F(T)$ where $\tilde{F(T)}$ denotes the set of asymptotic fixed points of*T*; - (ii)
*ϕ*-*nonexpansive*if $\varphi (Tx,Ty)\le \varphi (x,y)$ for all $x,y\in C$; - (iii)
*ϕ*-*asymptotically nonexpansive*if there exists a sequence $\{{t}_{n}\}\subset [1,\mathrm{\infty})$ satisfying ${t}_{n}\to \mathrm{\infty}$ as $n\to \mathrm{\infty}$ and $\varphi ({T}^{n}x,{T}^{n}y)\le {t}_{n}\varphi (x,y)$ for all $x,y\in C$, $n\in \mathbb{N}$; - (iv)
*quasi*-*ϕ*-*asymptotically nonexpansive*if $F(T)\ne \mathrm{\varnothing}$ and $\varphi (q,{T}^{n}x)\le {t}_{n}\varphi (q,x)$ for all $x\in C$, $q\in F(T)$, $n\in \mathbb{N}$, where $\{{t}_{n}\}$ is as in (iii) above.

We shall call the map *T* *generalized quasi*-*ϕ*-*asymptotically nonexpansive* in the light of [21], if $F(T)\ne \mathrm{\varnothing}$ and there exist sequences $\{{s}_{n}\}\subset [0,1]$, $\{{t}_{n}\}\subset [1,\mathrm{\infty})$ with ${s}_{n}\to 0$, ${t}_{n}\to 1$ as $n\to \mathrm{\infty}$ and $\varphi (q,{T}^{n}x)\le {t}_{n}\varphi (q,x)+{s}_{n}$ for all $x\in C$, $q\in F(T)$ and $n\in \mathbb{N}$.

Existence and approximations of fixed points of mappings of nonexpansive type and their generalizations were studied by numerous authors, see, for example, [3, 5, 7, 8, 10, 11, 14–17, 19, 21, 27] and the references therein.

where $\{{\alpha}_{n}\}$ and $\{{\beta}_{n}\}$ are sequences in $[a,b]$ such that $0<a<b<1$. They studied the scheme for two *quasi-nonexpansive maps* *S* and *T* and proved strong convergence of the sequence $\{{x}_{n}\}$ to a common fixed point of *S* and *T* in a real *strictly convex Banach space*. Takahashi and Tamura [25] proved strong and weak convergence of the sequence defined by (1.1) to a common fixed point of a pair of nonexpansive mappings *T* and *S* using a weaker condition on the maps.

Using a similar scheme, Wang [26] proved strong and weak convergence theorems for a pair of *nonself asymptotically nonexpansive mappings* in a uniformly convex Banach space.

Shahzad and Udomene [22] proved the necessary and sufficient conditions for the strong convergence of the scheme of type (1.1) to a common fixed point of two uniformly continuous asymptotically quasi-nonexpansive mappings in a real Banach space.

to a common fixed point of a finite family of nonself asymptotically nonexpansive mappings in a uniformly convex Banach space.

*et al.*[13] introduced and studied the following scheme:

for a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in a Banach space.

It is known that only weak convergence theorems were proved for nonexpansive maps even in Hilbert spaces using Mann and Ishikawa type schemes.

In 2000 Solodov and Svaiter [23] introduced a hybrid proximal point type iterative scheme and proved the strong convergence of the scheme to a zero of a maximal monotone operator.

In 2003 Nakajo and Takahashi [19] proposed a hybrid Mann scheme for nonexpansive mappings and nonexpansive semigroups and proved strong convergence theorems.

Kim and Xu [14] generalized the result of Nakajo and Takahashi by proving strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. Plubtieng and Ughchittrakool [20] introduced an Ishikawa type hybrid scheme for two asymptotically nonexpansive mappings and two asymptotically nonexpansive semigroups.

Takahashi *et al.* [24] studied a simpler hybrid scheme for nonexpansive mappings in Hilbert spaces. Inchan and Plubtieng [10], adopted this simpler scheme of Takahashi *et al.* with little modification for two nonexpansive maps and two nonexpansive semigroups. They proved the following theorem:

**Theorem 1.1** ([10])

*Let*

*H*

*be a real Hilbert space and let*

*C*

*be a nonempty*,

*closed*,

*convex*,

*and bounded subset of*

*H*.

*Let*$S,T:C\to C$

*be two asymptotically nonexpansive mappings with sequences*$\{{s}_{n}\}$

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

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

*Let*${x}_{0}\in C$.

*Then the sequence*$\{{x}_{n}\}$

*generated by*

*converges strongly to* ${z}_{0}={P}_{F}{x}_{0}$, *where* ${\theta}_{n}=(1-{\alpha}_{n})[({t}_{n}^{2}-1)+(1-{\beta}_{n}){t}_{n}^{2}({s}_{n}^{2}-1)]{(diamC)}^{2}\to 0$ *as* $n\to \mathrm{\infty}$ *and* $0\le {\alpha}_{n}\le a<1$, $0<b\le {\beta}_{n}\le c<1$ *for all* $n\in \mathbb{N}$.

Kimura and Takahashi [15] proved strong convergence theorem for the family of relatively nonexpansive mappings in strictly convex Banach spaces having Kadec-Klee property and Frechet differentiable norm.

Recently, Zhou *et al.* [28] have proved strong convergence theorem for the family ${T}_{i}:C\to C$, $i\in I$ of quasi-*ϕ*-asymptotically nonexpansive mappings, where *C* is a nonempty, closed, convex and bounded subset of a uniformly smooth and uniformly convex Banach space *E*.

More recently, Xu *et al.* [27] have studied a modified hybrid scheme for fixed point of families of quasi-*ϕ*-asymptotically nonexpansive mappings. They proved the following theorem:

**Theorem 1.2** ([27])

*Let*

*C*

*be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space*

*E*,

*and let*${T}_{i}:C\to C$, $i\in I$

*be a family of closed and quasi*-

*ϕ*-

*asymptotically nonexpansive mappings such that*$F:={\bigcap}_{i\in I}F({T}_{i})\ne \mathrm{\varnothing}$.

*Assume that every*${T}_{i}$, $i\in I$

*is asymptotically regular on*

*C*.

*Let*$\{{\alpha}_{n}\}$, $\{{\beta}_{n}\}$

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

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

*such that*${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\gamma}_{n}>0$.

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

*in*

*C*

*by*:

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

Motivated by these results, we have the purpose in this paper to study a new modified hybrid iterative scheme and prove a strong convergence theorem for a finite family of generalized quasi-*ϕ*-asymptotically nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space. Our theorems improve and unify several recent important results.

## 2 Preliminaries

Consider a sequence $\{{C}_{n}\}$ of nonempty closed and convex subsets of a reflexive Banach space *E*. Let $s-lim{C}_{n}$ denotes the set of all strong limits of sequences $\{{x}_{n}\}$ satisfying ${x}_{n}\in {C}_{n}$ for all $n\in \mathbb{N}$ and $w-lim{C}_{n}$ be the set of all weak limits of sequences $\{{y}_{i}\}$ satisfying ${y}_{i}\in {C}_{{n}_{i}}$ for all $i\in \mathbb{N}$ where $\{{C}_{{n}_{i}}\}$ is some subsequence of $\{{C}_{n}\}$. The sequence $\{{C}_{n}\}$ is said to converge to ${C}^{\ast}$ in the sense of Mosco [18] if $s-lim{C}_{n}=w-lim{C}_{n}={C}^{\ast}$. The Mosco limit of $\{{C}_{n}\}$ is denoted by $M-lim{C}_{n}$.

We shall make use of the following important results in the sequel.

**Lemma 2.1** (Kamimura and Takahashi [12])

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

**Lemma 2.2** (Ibaraki, Kimura and Takahashi [9])

*Let* *C* *be a nonempty closed convex subset of a real uniformly smooth and uniformly convex Banach space* *E*. *Let* $\{{C}_{n}\}$ *be a sequence of nonempty closed convex subsets of* *C*. *If* $M-lim{C}_{n}={C}^{\ast}$ *exists and is nonempty*, *then* $\{{\prod}_{{C}_{n}}x\}$ *converges strongly to* $\{{\prod}_{{C}^{\ast}}x\}$ *for each* $x\in E$.

The result in [9] is more general than the one presented here, but this is sufficient for our purpose.

**Lemma 2.3** *Let* *C* *be a nonempty closed convex subset of a real smooth Banach space and* $T:C\to C$ *be a closed generalized quasi*-*ϕ*-*asymptotically nonexpansive mapping*. *Then* $F(T)$ *is closed and convex*.

*Proof* By the closedness assumption on *T* and the definition of *ϕ*, the result follows immediately. □

## 3 Main results

**Theorem 3.1**

*Let*

*E*

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

*C*

*be a nonempty*,

*bounded*,

*closed and convex subset of*

*E*.

*Let*${T}_{k}:C\to C$, $k=1,2,3,\dots ,m$

*be a finite family of closed generalized quasi*-

*ϕ*-

*asymptotically nonexpansive maps with corresponding sequences*$\{{t}_{kn}\}$

*and*$\{{s}_{kn}\}$, $k=1,2,3,\dots ,m$

*such that*${t}_{kn}\to 1$

*and*${s}_{kn}\to 0$

*as*$n\to \mathrm{\infty}$.

*Let*$F={\bigcap}_{k=1}^{m}F({T}_{k})\ne \mathrm{\varnothing}$

*and let*${t}_{n}={max}_{1\le k\le m}{t}_{kn}$, $n\in \mathbb{N}$.

*Assume also that the maps*${T}_{k}$, $k=1,2,\dots ,m$

*are uniformly asymptotically regular*.

*Let*${x}_{0}\in C$

*be arbitrary and*${C}_{0}=C$

*and let*$M={sup}_{x,y\in C}\varphi (x,y)$.

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

*let*$\{{\beta}_{kn}\}$

*be sequences in*$(a,b)$

*for some*$a,b\in (0,1)$, $a<b$.

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

*be a sequence generated by*

*where* ${\gamma}_{kn}=({t}_{n}-1)(1-{\beta}_{kn})[1+{t}_{kn}(1-{\beta}_{(k-1)n})[1+{t}_{(k-1)n}(1-{\beta}_{(k-2)n})\times [1+{t}_{(k-2)n}(1-{\beta}_{(k-3)n})[\cdots [1+{t}_{2n}(1-{\beta}_{1n})]\cdots ]]]]M+{\sum}_{i=1}^{k}{s}_{in}{\prod}_{j=i}^{k}(1-{\beta}_{jn}){\prod}_{l=i+1}^{k}{t}_{ln}$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}={\prod}_{F}{x}_{0}$.

*Proof*We start by showing that $F\subset {C}_{n}$ $\mathrm{\forall}n\in \mathbb{N}\cup \{0\}$. We do this by induction. $F\subset {C}_{0}$ by definition. We suppose that $F\subset {C}_{N}$ for some $N\in \mathbb{N}\cup \{0\}$. We observe that for $v\in F$, using convexity of ${\parallel \cdot \parallel}^{2}$ and (3.1), we have

So $\varphi (v,{z}_{k(N+1)})\le \varphi (v,{x}_{N+1})+{\gamma}_{k(N+1)}$ for any $v\in F$ and $k\in \{1,2,\dots ,m\}$. This and the induction hypothesis give that $F\subset {C}_{k(N+1)}$ for all $k\in \{1,2,\dots ,m\}$. Therefore, $F\subset {C}_{N+1}$ and hence $F\subset {C}_{n}$ for all $n\in \mathbb{N}$.

Also by induction and using the fact that $\varphi (\cdot ,x)$ is continuous on *E* for any $x\in E$, it follows that ${C}_{kn}$ is closed for each $n\in \mathbb{N}$ and $k\in \{1,2,\dots ,m\}$, and consequently, ${C}_{n}$ is closed for each $n\in \mathbb{N}$.

We now prove that ${C}_{n}$ is convex for all $n\in \mathbb{N}$. We observe that $s\in {C}_{kn}$ is equivalent to $s\in {C}_{n-1}$ and ${\parallel {z}_{nk}\parallel}^{2}-{\parallel {x}_{n}\parallel}^{2}\le 2\u3008s,j{x}_{n}-j{z}_{nk}\u3009+{\gamma}_{kn}$. So the convexity of ${C}_{kn}$ for each $k\in \{1,2,\dots ,m\}$ and for each $n\in \mathbb{N}$ follows immediately by induction. Thus ${C}_{n}$ is convex for each $n\in \mathbb{N}$.

We now show that the sequence $\{{x}_{n}\}$ converges. Since $\{{C}_{n}\}$ is a decreasing sequence of closed, convex subsets of *E*, such that ${\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}\ne \mathrm{\varnothing}$, then the Mosco limit $M-{lim}_{n\to \mathrm{\infty}}{C}_{n}$ exists and $M-{lim}_{n\to \mathrm{\infty}}{C}_{n}={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}$. By Lemma 2.2, the sequence $\{{x}_{n}\}$ converges to ${x}^{\ast}:={\prod}_{{C}^{\ast}}{x}_{0}$, where ${C}^{\ast}={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}$.

*j*is norm-to-norm uniformly continuous on bounded subsets of

*E*, we get that, for each $k\in \{1,2,\dots ,m\}$, ${lim}_{n\to \mathrm{\infty}}\parallel j{x}_{n}-j{z}_{kn}\parallel =0$. Using (3.1) we obtain that

for $k=2,3,\dots ,m$. These imply ${T}_{1}({T}_{1}^{n}{x}_{n})\to {x}^{\ast}$ and ${T}_{k}({T}_{k}^{n}{z}_{(k-1)n})\to {x}^{\ast}$ as $n\to \mathrm{\infty}$, and for $k=2,3,\dots ,m$. By the closedness of each of the maps ${T}_{k}$, $k=1,2,\dots ,m$, we have that ${x}^{\ast}\in F$.

As *F* is a nonempty closed convex subset of ${C}^{\ast}:={\bigcap}_{n=1}^{\mathrm{\infty}}{C}_{n}$, we obtain that ${x}^{\ast}={\prod}_{F}{x}_{0}$. This completes the proof. □

The conditions of closedness and uniform asymptotic regularity on the maps ${\{{T}_{k}\}}_{k=1}^{m}$ can be replaced by the condition that each of the maps ${\{{T}_{k}\}}_{k=1}^{m}$ is uniformly Lipschitz. So we have the following theorem:

**Theorem 3.2** *Let* *E*, *C*, ${\{{T}_{k}\}}_{k=1}^{m}$, *F*, $\{{t}_{kn}\}$, $\{{s}_{kn}\}$, *and* $\{{x}_{n}\}$ *be as in Theorem * 3.1 *with the exception that* ${\{{T}_{k}\}}_{k=1}^{m}$ *are uniformly* ${L}_{k}$, $k=1,2,\dots ,m$, *Lipschitzian instead of uniformly asymptotically regular and closed*. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}={\prod}_{F}{x}_{0}$.

*Proof*The proof that $F\in {C}_{n}$, ${C}_{n}$ is closed, convex for each $n\in \mathbb{N}$ and ${lim}_{n\to \mathrm{\infty}}{x}_{n}={x}^{\ast}$ follows as in Theorem 3.1. Also relations (3.2), (3.3), (3.4) and (3.5) are obtainable as in Theorem 3.1. We only need to show that ${x}^{\ast}\in F$. Let $L:={max}_{1\le k\le m}{L}_{k}$, then using (3.4) and (3.5) we get

Finally, using these, the fact that ${x}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$, and the continuity of ${T}_{k}$ for each *k*, we obtain that ${x}^{\ast}\in F$ and this completes the proof. □

The following corollaries follow from Theorems 3.1 and 3.2.

**Corollary 3.3** *Let* *E* *be a real uniformly convex and uniformly smooth Banach space and* *C* *be a nonempty*, *bounded*, *closed and convex subset of* *E*. *Let* ${T}_{k}:C\to C$, $k=1,2,3,\dots ,m$ *be a finite family of quasi*-*ϕ*-*asymptptically nonexpansive maps with corresponding sequences* $\{{t}_{kn}\}$, $k=1,2,3,\dots ,m$, *such that* ${t}_{kn}\to 1$, *as* $n\to \mathrm{\infty}$. *Let* $F={\bigcap}_{k=1}^{m}F({T}_{k})\ne \mathrm{\varnothing}$ *and let* ${t}_{n}={max}_{1\le k\le m}{t}_{kn},n\in \mathbb{N}$. *Assume also that the maps* ${T}_{k}$, $k=1,2,\dots ,m$ *are either closed and uniformly asymptotically regular on* *C* *or uniformly Lipschitzian on* *C*. *Let* ${x}_{0}\in C$ *be arbitrary and* ${C}_{0}=C$. *For* $k=1,2,\dots ,m$, *let* $\{{\beta}_{kn}\}$ *be sequences in* $(a,b)$ *for some* $a,b\in (0,1)$, $a<b$. *Let* $\{{x}_{n}\}$ *be a sequence generated by* (3.1). *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}={\prod}_{F}{x}_{0}$.

**Corollary 3.4** *Let* *E* *be a real uniformly convex and uniformly smooth Banach space and* *C* *be a nonempty*, *bounded*, *closed and convex subset of* *E*. *Let* ${T}_{k}:C\to C$, $k=1,2,3,\dots ,m$ *be a finite family of* *ϕ*-*asymptotically nonexpansive maps with corresponding sequences* $\{{t}_{kn}\}$, $k=1,2,3,\dots ,m$, *such that* ${t}_{kn}\to 1$, *as* $n\to \mathrm{\infty}$. *Let* $F={\bigcap}_{k=1}^{m}F({T}_{k})\ne \mathrm{\varnothing}$ *and let* ${t}_{n}={max}_{1\le k\le m}{t}_{kn}$. *Assume also that the maps* ${T}_{k}$, $k=1,2,\dots ,m$ *are either closed and uniformly asymptotically regular on* *C* *or uniformly Lipschitzian on* *C*. *Let* ${x}_{0}\in C$ *be arbitrary and* ${C}_{0}=C$. *For* $k=1,2,\dots ,m$, *let* $\{{\beta}_{kn}\}$ *be sequences in* $(a,b)$ *for some* $a,b\in (0,1)$, $a<b$. *Let* $\{{x}_{n}\}$ *be a sequence generated by* (3.1). *Then the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}={\prod}_{F}{x}_{0}$.

**Corollary 3.5**

*Let*

*H*

*be a real Hilbert space*,

*C*

*be a nonempty*,

*bounded*,

*closed and convex subset of*

*H*.

*Let*${T}_{k}:C\to C$, $k=1,2,3,\dots ,m$

*be a finite family of generalized asymptotically quasi*-

*nonexpansive maps with corresponding sequences*$\{{t}_{kn}\}$

*and*$\{{s}_{kn}\}$, $k=1,2,3,\dots ,m$

*such that*${t}_{kn}\to 1$

*and*${s}_{kn}\to 0$

*as*$n\to \mathrm{\infty}$.

*Let*$F={\bigcap}_{k=1}^{m}F({T}_{k})\ne \mathrm{\varnothing}$

*and let*${t}_{n}={max}_{1\le k\le m}{t}_{kn}$.

*Assume also that the maps*${T}_{k}$, $k=1,2,\dots ,m$

*are either closed and uniformly asymptotically regular on*

*C*

*or uniformly*${L}_{k}$, $k=1,2,\dots ,m$

*Lipschitzian on*

*C*.

*Let*${x}_{0}\in C$

*be arbitrary and*${C}_{0}=C$.

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

*let*$\{{\beta}_{kn}\}$

*be sequences in*$(a,b)$

*for some*$a,b\in (0,1)$, $a<b$.

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

*be a sequence generated by*

*where* ${\gamma}_{kn}=({t}_{n}^{2}-1)(1-{\beta}_{kn})[1+{t}_{kn}^{2}(1-{\beta}_{k-1n})[1+{t}_{(k-1)n}^{2}(1-{\beta}_{(k-2)n})\times [1+{t}_{(k-2)n}^{2}(1-{\beta}_{(k-3)n})[\cdots [1+{t}_{2n}^{2}(1-{\beta}_{1n})]\cdots ]]]]{(diamC)}^{2}+{\sum}_{i=1}^{k}{s}_{in}{\prod}_{j=i}^{k}(1-{\beta}_{jn}){\prod}_{l=i+1}^{k}{t}_{ln}$. *Then*, *the sequence* $\{{x}_{n}\}$ *converges strongly to* ${x}^{\ast}={P}_{F}{x}_{0}$.

**Corollary 3.6** *Let* *H* *be a real Hilbert space*, *C* *be a nonempty*,*closed and convex subset of* *H*. *Let* ${T}_{k}:C\to C$, $k=1,2,3,\dots ,m$ *be a finite family of asymptotically nonexpansive maps with corresponding sequences* $\{{t}_{kn}\}$, $k=1,2,3,\dots ,m$, *such that* ${t}_{kn}\to 1$ *as* $n\to \mathrm{\infty}$. *Let* $F={\bigcap}_{k=1}^{m}F({T}_{k})\ne \mathrm{\varnothing}$ *and let* ${t}_{n}={max}_{1\le k\le m}{t}_{kn}$. *Let* ${x}_{0}\in C$ *be arbitrary and* ${C}_{1}=C$. *For* $k=1,2,\dots ,m$, *let* $\{{\beta}_{kn}\}$ *be sequences in* $(a,b)$ *for some* $a,b\in (0,1)$, $a<b$. *Let* $\{{x}_{n}\}$ *be a sequence generated by* (3.10). *Then the sequence* $\{{x}_{n}\}$ *converges to* ${P}_{F}{x}_{0}$.

**Remark 3.7** Theorem 3.1 and Corollary 3.5 extend and improve several important recent results. For instance, Corollary 3.5 is an improvement and generalization of Theorem 1.1 and Theorem 3.1 of [20].

**Remark 3.8** It is not clear whether Theorem 3.1 and Corollary 3.5 hold without the boundedness assumption on *C*.

## Declarations

### Acknowledgements

This work was conducted when the first author was visiting the Abdus Salam International Center for Theoretical Physics, Trieste, Italy, as an associate. He would like to thank the center for hospitality and financial support.

## Authors’ Affiliations

## References

- Alber Y: Metric and generalized projection operators in Banach spaces: properties and applications. 178. In
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*. Edited by: Karstsatos AG. Dekker, New York; 1996:15–50.Google Scholar - Alber Y, Guerre-Delabriere S: On the projection methods for fixed point problems.
*Analysis*2001, 21: 17–39.MathSciNetView ArticleGoogle Scholar - Belluce LP, Kirk WA: Fixed point theorem for families of contraction mappings.
*Pac. J. Math.*1966, 18: 213–217. 10.2140/pjm.1966.18.213MathSciNetView ArticleGoogle Scholar - Chidume CE, Ali B: Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces.
*J. Math. Anal. Appl.*2007, 326: 960–973. 10.1016/j.jmaa.2006.03.045MathSciNetView ArticleGoogle Scholar - Chidume CE, Ali B: Convergence theorems for finite families of asymptotically nonexpansive mappings.
*J. Inequal. Appl.*2007., 326: Article ID 68616. doi:10.1155/2007/68616Google Scholar - Das G, Debata JP: Fixed points of quasi-nonexpansive mappings.
*Indian J. Pure Appl. Math.*1986, 17: 1263–1269.MathSciNetGoogle Scholar - Diaz JB, Metcalf FB: On the structure of the set of subsequential limit points of successive approximations.
*Bull. Am. Math. Soc.*1967, 73: 516–519. 10.1090/S0002-9904-1967-11725-7MathSciNetView ArticleGoogle Scholar - Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings.
*Proc. Am. Math. Soc.*1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleGoogle Scholar - Ibaraki T, Kimura Y, Takahashi W: Convergence theorems for generalized projections and maximal monotone operators in Banach spaces.
*Abstr. Appl. Anal.*2003, 2003(10):621–629. 10.1155/S1085337503207065MathSciNetView ArticleGoogle Scholar - Inchan I, Plubtieng S: Strong convergence theorem of hybrid method for two asymptotically nonexpansive mappings in Hilbert spaces.
*Nonlinear Anal. Hybrid Syst.*2008, 2: 1125–1135. 10.1016/j.nahs.2008.09.006MathSciNetView ArticleGoogle Scholar - Ishikawa S: Fixed point theorems for asymptotically nonexpansive mappings.
*Proc. Am. Math. Soc.*1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleGoogle Scholar - Kamimura S, Takahashi W: Strong convergence of a proximal type algorithm in Banach space.
*SIAM J. Optim.*2002, 13: 938–945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar - Khan AR, Domlo AA, Fukhar-ud-din H: Common fixed point Noor iteration for finite family of asymptotically quasi-nonexpansive mappings in Banach spaces.
*J. Math. Anal. Appl.*2008, 341: 1–11. 10.1016/j.jmaa.2007.06.051MathSciNetView ArticleGoogle Scholar - Kim TH, Xu HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups.
*Nonlinear Anal.*2006, 24: 1140–1152.MathSciNetView ArticleGoogle Scholar - Kimura Y, Takahashi W: On a hybrid method for family of relatively nonexpansive mappings in a Banach space.
*J. Math. Anal. Appl.*2009, 357: 356–363. 10.1016/j.jmaa.2009.03.052MathSciNetView ArticleGoogle Scholar - Kirk WA: A fixed point theorem for mappings which do not increase distance.
*Am. Math. Mon.*1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleGoogle Scholar - Lim TC: A fixed point theorem for families of nonexpansive mappings.
*Pac. J. Math.*1974, 53: 487–493. 10.2140/pjm.1974.53.487View ArticleGoogle Scholar - Mosco U: Convergence of convex sets and solutions of variational inequalities.
*Adv. Math.*1969, 3: 510–585. 10.1016/0001-8708(69)90009-7MathSciNetView ArticleGoogle Scholar - Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.
*J. Math. Anal. Appl.*2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleGoogle Scholar - Plubtieng S, Ughchittrakool K: Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups.
*Nonlinear Anal.*2007, 67: 2306–2315. 10.1016/j.na.2006.09.023MathSciNetView ArticleGoogle Scholar - Shahzad N, Zegeye H: Strong convergence of an implicit iteration process for finite family of generalized asymptotically quasi-nonexpansive maps.
*Appl. Math. Comput.*2007, 189: 1058–1065. 10.1016/j.amc.2006.11.152MathSciNetView ArticleGoogle Scholar - Shahzad N, Udomene A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces.
*Fixed Point Theory Appl.*2006., 2006: Article ID 18909Google Scholar - Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space.
*Math. Program., Ser. A*2000, 87: 189–202.MathSciNetGoogle Scholar - Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert space.
*J. Math. Anal. Appl.*2007, 241: 276–289.MathSciNetGoogle Scholar - Takahashi W, Tamura T: Convergence theorems for pair of nonexpansive mappings.
*J. Convex Anal.*1998, 5: 45–56.MathSciNetGoogle Scholar - Wang L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings.
*J. Math. Anal. Appl.*2006, 323: 550–557. 10.1016/j.jmaa.2005.10.062MathSciNetView ArticleGoogle Scholar - Xu Y, Zhang X, Khang J, Su Y: Modified hybrid algorithm for family of quasi-
*ϕ*-asymptotically nonexpansive mappings.*Fixed Point Theory Appl.*2010., 2010: Article ID 170701. doi:10.1155/2010170701Google Scholar - Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for finite family of quasi-
*ϕ*-asymptotically nonexpansive mappings.*J. Appl. Math. Comput.*2010, 32: 453–464. 10.1007/s12190-009-0263-4MathSciNetView ArticleGoogle Scholar

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