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- Open Access

# A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces

- Shih-sen Chang
^{1}, - Jong Kyu Kim
^{2}Email author, - H W Joseph Lee
^{3}and - Chi Kin Chan
^{3}

**2012**:37

https://doi.org/10.1186/1029-242X-2012-37

© Chang et al; licensee Springer. 2012

**Received:**23 May 2011**Accepted:**16 February 2012**Published:**16 February 2012

## Abstract

The purpose of this article is to establish some new approximation theorems of common fixed points for a countable family of total asymptotically quasi-nonexpansive mappings in Banach spaces which generalize and improve the corresponding theorems of Chidume et al. and others.

**2000 AMS Subject Classification**: 47J05; 47H09; 49J25.

## Keywords

- total asymptotically nonexpansive mapping
- total asymptotically quasi-nonexpansive mapping
- asymptotically quasi-nonexpansive mapping
- asymptotically nonexpansive mapping
- asymptotically nonexpansive in the intermediate sense

## 1. Introduction

Throughout this article, we assume that *E* is a real Banach space, *C* is a nonempty closed convex subset of *E*. In the sequel, we use *F*(*T*) to denote the set of fixed points of a mapping *T*, and use $\Re $ and ${\Re}^{+}$ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively.

*T*:

*C*→

*C*is said to be

*nonexpansive*if,

*T*is called

*asymptotically nonexpansive*if, there exists a sequence {

*ν*

_{ n }} ⊂ (0, ∞) with lim

_{n→∞}

*ν*

_{ n }= 0 such that for all

*x*,

*y*∈

*C*

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. They proved that if *C* is a nonempty closed and convex bounded subset of a real uniformly convex Banach space and *T* : *C* → *C* is an asymptotically nonexpansive mapping, then *T* has a fixed point.

*T*:

*C*→

*C*is said to be

*asymptotically nonexpansive in the intermediate sense*[2], if it is continuous and the following inequality holds:

*F*(

*T*) ≠ ∅ and (1.1) holds for all

*x*∈

*C*,

*y*∈

*F*(

*T*), then

*T*is called

*asymptotically quasi-nonexpansive in the intermediate sense*. Observe that if we define

*σ*

_{ n }→ 0 as

*n*→ ∞ and (1.1) reduces to

The class of asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known [3] that if *C* is a nonempty closed and convex bounded subset of a uniformly convex Banach space *E* and *T* : *C* → *C* is asymptotically nonexpansive in the intermediate sense, then *T* has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.

Recently, Alber et al. [4] introduced the concept of *total asymptotically nonexpansive mappings* which is more general than asymptotically nonexpansive mappings and studied the approximation methods of fixed points for this kind of mappings.

**Definition 1.1**A mapping

*T*:

*C → C*is said to be

*total asymptotically nonexpansive*if, there exist nonnegative real sequences {

*ν*

_{ n }} and {

*μ*

_{ n }} with

*ν*

_{ n }→ 0,

*μ*

_{ n }→ 0 as

*n*→ ∞ and a strictly increasing continuous function $\zeta :{\Re}^{+}\to {\Re}^{+}$ with

*ζ*(0) = 0 such that for all

*x*,

*y*∈

*C*,

If *F*(*T*) ≠ ∅ and (1.4) holds for all *x* ∈ *C*, *y* ∈ *F*(*T*), then *T* is called *total asymptotically quasi-nonexpansive*.

**Remark 1.2**If

*ζ*(

*t*) =

*t*,

*t*≥ 0, then (1.4) reduces to

In addition, if *μ*_{
n
} = 0, ∀*n* ≥ 1, then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mappings. If *ν*_{
n
} = *μ*_{
n
} = 0, ∀*n* ≥ 1, then total asymptotically nonexpansive mappings coincide with nonexpansive mappings. If *ν*_{
n
} = 0 and *μ*_{
n
} = *σ*_{
n
} := max{0, *a*_{
n
}}, where *a*_{
n
} is defined by (1.2), then (1.4) reduces to (1.3) which has been studied as asymptotically nonexpansive mappings in the intermediate sense.

Within the past 30 years, research on iterative approximation of common fixed points of nonexpansive mappings, asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings have been considered by many authors (see, for example, [1–20] and the references therein).

Especially, recently Chidume and Ofoedu [13, 14] introduced the following iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces which extend and generalize the corresponding results of Kirk [3], Alber et al. [4], Quan et al. [5], Shahzad et al. [6], Chang et al. [9], Jung [10], Shioji et al. [11], Suzuki [12], and Schu [19].

**Theorem 1.3**[[13, 14]] Let

*E*be a real Banach space,

*C*be a nonempty closed convex subset of

*E*and

*T*

_{ i }:

*C*→

*C*,

*i*= 1, 2, ...,

*m*be

*m*total asymptotically nonexpansive mappings with sequences {

*ν*

_{ in }}, {

*μ*

_{ in }},

*i*= 1, 2, ...,

*m*, such that $\mathfrak{F}:={\bigcap}_{i=1}^{m}F\left({T}_{i}\right)\ne \varnothing $. Let {

*x*

_{ n }} be defined by

Suppose ${\sum}_{n=1}^{\infty}{\nu}_{in}<\infty $, ${\sum}_{n=1}^{\infty}{\mu}_{in}<\infty $, *i* = 1, 2, ..., *m* and suppose that there exist *M*_{
i
}, ${M}_{i}^{*}>0$ such that ${\zeta}_{i}\left({\lambda}_{i}\right)\le {M}_{i}^{*}{\lambda}_{i}$ for all *λ*_{
i
} ≥ *M*_{
i
}, *i* = 1, 2, ..., *m*. Then the sequence {*x*_{
n
}} is bounded and lim_{n→∞}||*x*_{
n
} - *p*|| exists, $p\in \mathfrak{F}$. Moreover, the sequence {*x*_{
n
}} converges strongly to a common fixed point of *T*_{
i
}, *i* = 1, 2, ..., *m* if and only if $\text{lim}{\text{inf}}_{n\to \infty}d\left({x}_{n},\mathfrak{F}\right)=0$, where $d\left({x}_{n},\mathfrak{F}\right)={\text{inf}}_{y\in \mathfrak{F}}\left|\right|{x}_{n}-y\left|\right|$, *n* ≥ 1.

**Theorem 1.4** [[13, 14]] Let *E* be a uniformly convex real space, *C* be a nonempty closed convex subset of *E*, and *T*_{
i
} : *C* → *C*, *i* = 1, 2, ..., *m* be *m* uniformly continuous total asymptotically nonexpansive mappings with sequences {*ν*_{
in
}}, {*μ*_{
in
}} ⊂ [0, ∞) such that ${\sum}_{n=1}^{\infty}{\nu}_{in}<\infty $, ${\sum}_{n=1}^{\infty}{\mu}_{in}<\infty $, *i* = 1, 2, ..., *m* and $\mathfrak{F}:={\bigcap}_{i=1}^{m}F\left({T}_{i}\right)\ne \varnothing $. Let {*α*_{
in
}} ⊂ [*ε*, 1 - *ε*] for some *ε* ∈ (0, 1). From arbitrary *x*_{1} ∈ *C*, define the sequence {*x*_{
n
}} by (1.5). Suppose that there exist *M*_{
i
}, ${M}_{i}^{*}>0$ such that ${\zeta}_{i}\left({\lambda}_{i}\right)\le {M}_{i}^{*}{\lambda}_{i}$ whenever *λ*_{
i
} ≥ *M*_{
i
}, *i* = 1, 2, ..., *m* and that one of *T*_{1}, *T*_{2}, ..., *T*_{
m
} is compact, then {*x*_{
n
}} converges strongly to some $p\in \mathfrak{F}$.

It is our purpose in this article to construct a new iterative sequence much simpler than (1.5) for approximation of common fixed points of a countable family of total asymptotically nonexpansive mappings and give necessary and sufficient conditions for the convergence of the scheme to common fixed points of the mappings in arbitrary real Banach spaces. As well as a sufficient condition for convergence of the iteration process to a common fixed point of mappings under the setting of uniformly convex Banach space is also established. The results presented in the article not only generalize and improve the corresponding results of Chidume et al. [13–15] but also unify, extend and generalize the corresponding result of [3–7, 9–12, 19].

## 2. Preliminaries

For the sake of convenience we first give the following lemmas which will be needed in proving our main results.

**Lemma 2.1**[[20]] Let

*E*be a uniformly convex Banach space,

*r >*0 be a positive number and

*B*

_{ r }(0) be a closed ball of

*E*. Then, for any sequence ${\left\{{x}_{i}\right\}}_{i=1}^{\infty}\subset {B}_{r}\left(0\right)$ and for any sequence ${\left\{{\lambda}_{i}\right\}}_{i=1}^{\infty}$ of positive numbers with ${\sum}_{n=1}^{\infty}{\lambda}_{n}=1$, there exists a continuous, strictly increasing and convex function

*g*: [0, 2

*r*) → [0, ∞),

*g*(0) = 0 such that for any positive integers

*i*,

*j*≥ 1,

*i*≠

*j*, the following holds:

**Lemma 2.2**Let {

*a*

_{ n }}, {

*b*

_{ n }}, and {

*λ*

_{ n }} be sequences of nonnegative real numbers such that

where *n*_{0} is some positive integer. If ${\sum}_{n=1}^{\infty}{\lambda}_{n}<\infty $ and ${\sum}_{n=1}^{\infty}{b}_{n}<\infty $, then {*a*_{
n
}} is bounded and lim_{n→∞}*a*_{
n
}exists. Moreover, if, in addition, lim inf_{n→∞}*a*_{
n
}= 0, then lim_{n→∞}*a*_{
n
} = 0.

## 3. Main results

**Definition 3.1**Let

*E*be a real Banach space,

*C*be a nonempty closed convex subset of

*E*.

- (1)Let {
*T*_{ i }} be a countable family of mappings from*C*into itself. {*T*_{ i }} is said to be*uniformly total asymptotically quasi-nonexpansive mappings*if, $\mathfrak{F}:={\bigcap}_{i=1}^{\infty}F\left({T}_{i}\right)\ne \varnothing $, and there exist nonnegative real sequences {*ν*_{ n }} and {*μ*_{ n }} with*ν*_{ n }→ 0,*μ*_{ n }→ 0 as*n*→ ∞ and a strictly increasing continuous function $\zeta :{\Re}^{+}\to {\Re}^{+}$ with*ζ*(0) = 0 such that for all*x*∈*C*, $p\in \mathfrak{F}$,$\left|\right|{T}_{i}^{n}x-p\left|\right|\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}\left|\right|x-p\left|\right|+\phantom{\rule{2.77695pt}{0ex}}{\nu}_{n}\zeta \left(\left|\right|x-p\left|\right|\right)\phantom{\rule{2.77695pt}{0ex}}\text{+}{\mu}_{n},\forall i\ge \text{1},n\ge \text{1}.$(3.1) - (2)A mapping
*T*:*C*→*C*is said to be*uniformly L-Lipschitz continuous*, if there exists a constant*L >*0 such that$\left|\right|{T}^{n}x-{T}^{n}y\left|\right|\phantom{\rule{2.77695pt}{0ex}}\le L\left|\right|x-y\left|\right|,\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in C,\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 1.$

*T*

_{ i }} be a countable family of uniformly total asymptotically quasi-nonexpansive mappings from

*C*into itself and for each

*i*≥ 1,

*T*

_{ i }is uniformly

*L*

_{ i }-Lipschitz continuous. For any given

*x*

_{0}∈

*C*, we define an iterative sequence {

*x*

_{ n }} by

**Theorem 3.2**Let

*E*be a real Banach space,

*C*be a nonempty closed convex subset of

*E*. Let ${\left\{{T}_{n}\right\}}_{n=1}^{\infty}$ be a countable family of uniformly total asymptotically quasi-nonexpansive mappings from

*C*into itself with nonnegative real sequences {

*ν*

_{ n }} and {

*μ*

_{ n }} and a strictly increasing continuous function $\zeta :{\Re}^{+}\to {\Re}^{+}$ such that

*ζ*(0) = 0, $\mathfrak{F}:={\bigcap}_{i=1}^{m}F\left({T}_{i}\right)\ne \varnothing $ and ${\sum}_{n=1}^{\infty}\left({\nu}_{n}\text{+}{\mu}_{n}\right)\infty $. Let {

*x*

_{ n }} be the sequence defined by (3.2), where {

*β*

_{n, i}},

*i*= 0, 1, 2, ... and {

*α*

_{ n }} are sequences in [0, 1] satisfying the following conditions:

- (a)
for each

*n*≥ 0, ${\sum}_{i=0}^{\infty}{\beta}_{n,i}=1$; If {*x*_{ n }} and $\mathfrak{F}$ are bounded, then the following conclusions hold: - (1)
for each $p\in \mathfrak{F}$, lim

_{n→∞}||*x*_{ n }-*p*|| exists; - (2)
the sequence {

*x*_{ n }} converges strongly to a common fixed point ${x}^{*}\in \mathfrak{F}$ if and only if $\text{lim}{\text{inf}}_{n\to \infty}d\left({x}_{n},\mathfrak{F}\right)=0$, where $d\left({x}_{n},\mathfrak{F}\right)={\text{inf}}_{y\in \mathfrak{F}}\left|\right|{x}_{n}-y\left|\right|$,*n*≥ 1.

**Proof**. (1) For any

*n*≥ 0 and for any given $p\in \mathfrak{F}$ we have

*γ*

_{ n }=

*ν*

_{ n }

*ζ*(

*M*) +

*μ*

_{ n }. By the assumption, ${\sum}_{n=1}^{\infty}{\gamma}_{n}<\infty $. Substituting (3.4) into (3.3) and simplifying we have

_{n→∞}||

*x*

_{ n }-

*p*|| exists. The conclusion (1) is proved.

- (2)From (3.5) we have that$d\left({x}_{n+1},\mathfrak{F}\right)\le d\left({x}_{n},\mathfrak{F}\right)+{\gamma}_{n},for\phantom{\rule{2.77695pt}{0ex}}each\phantom{\rule{2.77695pt}{0ex}}n\ge 1.$

Then Lemma 2.2 implies that ${\text{lim}}_{n\to \infty}d\left({x}_{n},\mathfrak{F}\right)$ exists. By the assumption that $\text{lim}{\text{inf}}_{n\to \infty}d\left({x}_{n},\mathfrak{F}\right)=0$, therefore we have ${\text{lim}}_{n\to \infty}d\left({x}_{n},\mathfrak{F}\right)=0$.

This completes the proof of Theorem 3.2.

**Theorem 3.3** Let $E,C,{\left\{{T}_{n}\right\}}_{n=1}^{\infty}$, $\mathfrak{F}$ be the same as in Theorem 3.2. If there exist constants *K*, *K** *>* 0 such that *ζ*(*t*) ≤ *K***t*, ∀*t* ≥ *K*, then the sequence {*x*_{
n
}} defined by (3.2) is bounded and so the conclusions of Theorem 3.2 still hold.

**Proof**In fact, from (3.2) for any given $p\in \mathfrak{F}$, we have that

*ξ*

_{ n }=

*ν*

_{ n }

*ζ*(

*K*) +

*μ*

_{ n }. Substituting (3.6) into (3.3) and simplifying, we have that

By Lemma 2.2, for each $p\in \mathfrak{F}$, lim_{n→∞}||*x*_{
n
} - *p*|| exists, and so {*x*_{
n
}} is bounded. The conclusions of Theorem 3.3 can be obtained from Theorem 3.2 immediately.

From Theorem 3.3 we can obtain the following result:

**Corollary 3.4**Let

*E*,

*C*be as in Theorem 3.2. Let

*T*

_{ i }:

*C*→

*C*,

*i*= 1, 2, ...,

*m*be

*m*total asymptotically quasi-nonexpansive mappings with nonnegative real sequences {

*ν*

_{ in }}, {

*μ*

_{ in }} and a strictly increasing continuous function ${\zeta}_{i}:{\Re}^{+}\to {\Re}^{+}$ with

*ζ*

_{ i }(0) = 0 such that $\mathfrak{F}\phantom{\rule{1em}{0ex}}\text{:}={\bigcap}_{i=1}^{m}F\left({T}_{i}\right)$ is nonempty and bounded, ${\sum}_{n=1}^{\infty}\left({\nu}_{in}\text{+}\phantom{\rule{2.77695pt}{0ex}}{\mu}_{in}\right)\infty $,

*i*= 1, 2, ...,

*m*. Let {

*x*

_{ n }} be the sequence defined by:

*β*

_{n, i}},

*i*= 0, 1, 2, ...,

*m*, and {

*α*

_{ n }} are sequences in [0, 1] satisfying the following conditions:

- (a)
for each

*n*≥ 0, ${\sum}_{i=0}^{m}{\beta}_{n,i}=1$;

If there exist constants *K*, *K** *>* 0 such that for each *i* = 1, 2, ..., *m*, *ζ*_{
i
}(*t*) ≤ *K***t*, ∀*t* ≥ *K*, then {*x*_{
n
}} is bounded and the conclusions of Theorem 3.2 still hold.

**Proof** Let *ν*_{
n
} = max_{1≤i≤m}*ν*_{
in
}, *μ*_{
n
} = max_{1≤i≤m}*μ*_{
in
} and *ζ* = max_{1≤i≤m}*ζ*_{
i
}, then ${\sum}_{n=1}^{\infty}\left\{{\nu}_{n}\text{+}{\mu}_{n}\right\}\infty $ and $\zeta :{\Re}^{+}\to {\Re}^{+}$ is a strictly increasing continuous function with *ζ*(0) = 0 and there exist constants *K*, *K** *>* 0 such that *ζ*(*t*) ≤ *K***t*, ∀*t* ≥ *K*. Therefore all conditions in Theorem 3.3 are satisfied. The conclusions of Corollary 3.4 can be obtained from Theorem 3.3 immediately.

If the space *E* is uniformly convex, and one of {*T*_{
i
}} is compact, then we can obtain the following more better result.

**Theorem 3.5**Let

*E*be a uniformly convex real Banach space,

*C*be a nonempty closed convex subset of

*E*and ${\left\{{T}_{i}\right\}}_{i=1}^{\infty}$ be a countable family of uniformly total asymptotically quasi-nonexpansive mappings from

*C*into itself with nonnegative real sequences {

*ν*

_{ n }} and {

*μ*

_{ n }} and a strictly increasing continuous function $\zeta :{\Re}^{+}\to {\Re}^{+}$ such that

*ζ*(0) = 0, $\mathfrak{F}:={\bigcap}_{i=1}^{\infty}F\left({T}_{i}\right)\ne \varnothing $ and ${\sum}_{n=1}^{\infty}\left({\nu}_{n}\text{+}{\mu}_{n}\right)\infty $, and for each

*i*≥ 1,

*T*

_{ i }is uniformly

*Li*-Lipschitzian continuous. Let {

*x*

_{ n }} be the sequence defined by (3.2), where {

*β*

_{n, i}},

*i*= 0, 1, 2, ... and {

*α*

_{ n }} are sequences in [0, 1] satisfying the conditions (a), (b) in Theorem 3.2 and lim inf

*β*

_{n,0}

*β*

_{n; i}> 0 for any

*i*≥ 1. If {

*x*

_{ n }} and $\mathfrak{F}$ both are bounded and one of {

*T*

_{ i }} is compact, then the following conclusions hold:

- (1)
${\text{lim}}_{n\to \infty}\left|\right|{x}_{n}-{T}_{j}^{n}{x}_{n}\left|\right|\phantom{\rule{2.77695pt}{0ex}}=0$ uniformly in

*j*≥ 1, - (2)
the sequence {

*x*_{ n }} converges strongly to some point $p\in \mathfrak{F}$.

**Proof**(1) Since {

*x*

_{ n }} and $\mathfrak{F}$ both are bounded and the norm ||·||

^{2}is convex, for any given $p\in \mathfrak{F}$, it follows from (3.2) that

*j*≥ 1 we have

*j*≥ 1

*β*

_{n,0}

*β*

_{n, i}> 0 for any

*i*≥ 1, this implies that

*g*, we have that

- (2)From (3.2) we have$\left|\right|{x}_{n+1}-{x}_{n}\left|\right|\phantom{\rule{2.77695pt}{0ex}}=\left(1-{\alpha}_{n}\right)\left|\right|{z}_{n}-{x}_{n}\left|\right|\phantom{\rule{2.77695pt}{0ex}}\le \sum _{i=1}^{\infty}{\beta}_{n,i}\left|\right|{T}_{i}^{n}{x}_{n}-{x}_{n}\left|\right|.$(3.13)

*ε >*0, from (3.12) there exists a positive integer

*n*

_{0}such that

*T*

_{ i }} which is compact. Without loss of generality, we can assume that

*T*

_{1}is compact. Thus, there exists a subsequence $\left\{{T}_{1}^{{n}_{k}}{x}_{{n}_{k}}\right\}$ of $\left\{{T}_{1}^{n}{x}_{n}\right\}$ such that ${T}_{1}^{{n}_{k}}{x}_{{n}_{k}}\to {x}^{*}$ (as

*k*→ ∞) for some point

*x** ∈

*C*. Since

*T*

_{1}is

*L*

_{1}-Lipschitzian, it is continuous. Thus we have ${T}_{1}{T}_{1}^{{n}_{k}}{x}_{{n}_{k}}\to {T}_{1}{x}^{*}$ (as

*k*→ ∞). From (3.12), we have that ${\text{lim}}_{k\to \infty}{x}_{{n}_{k}}={x}^{*}$ Also from (3.12) for each

*i*≥ 1, ${\text{lim}}_{k\to \infty}{T}_{i}^{{n}_{k}}{x}_{{n}_{k}}={x}^{*}$. Thus for each

*i*≥ 1, ${\text{lim}}_{k\to \infty}{T}_{i}{T}_{i}^{{n}_{k}}{x}_{{n}_{k}}={T}_{i}{x}^{*}$. By (3.14), ${\text{lim}}_{k\to \infty}\left|\right|{x}_{{n}_{k}+1}-{x}_{{n}_{k}}\left|\right|\phantom{\rule{2.77695pt}{0ex}}=0$, it follows that ${\text{lim}}_{k\to \infty}{x}_{{n}_{k}+1}={x}^{*}$. Next, we prove that ${x}^{*}\in \mathfrak{F}$. In fact, for each

*i*≥ 1, since

*T*

_{ i }is uniformly

*L*

_{ i }-Lipschitz continuous, we have

Therefore, we have *x** = *T*_{
i
}*x**, for each *i* ≥ 1. This implies that ${x}^{*}\in \mathfrak{F}$. But by Theorem 3.2, for each $p\in \mathfrak{F}$, lim_{n→∞}||*x*_{
n
} - *p*|| exists. Hence {*x*_{
n
}} converges strongly to ${x}^{*}\in \mathfrak{F}$. This completes the proof of Theorem 3.5.

**Remark 3.6** By the same way as given in the proof of Theorem 3.3, we can prove that if the condition " {*x*_{
n
}} is bounded" in Theorem 3.5 is replaced by the condition "if there exist constants *K*, *K** *>* 0 such that *ζ*(*t*) ≤ *K***t*, ∀*t* ≥ *K*", then the conclusions of Theorem 3.5 still hold.

**Definition 3.7**Let {

*T*

_{ i }} be a family of mappings from

*C*into itself.

- (1){
*T*_{ i }} is said to be*a family of uniformly asymptotically nonexpansive mappings*if, there exists a sequence of nonnegative real numbers {*ν*_{ n }} with*ν*_{ n }→ 0 (*as n*→ ∞) such that for any*x*,*y*∈*C*and for any*i*≥ 1$\left|\right|{T}_{i}^{n}x-{T}_{i}^{n}y\left|\right|\phantom{\rule{2.77695pt}{0ex}}\le \left(1+{\nu}_{n}\right)\left|\right|x-y\left|\right|,\phantom{\rule{1em}{0ex}}\forall n\ge 1.$(3.16) - (2){
*T*_{ i }} is said to be*a family of uniformly asymptotically nonexpansive in the intermediate sense*if, for each*i*≥ 1,*T*_{ i }is continuous and there exists a sequence {*σ*_{ n }} of nonnegative real numbers with*σ*_{ n }→ 0 (*as n*→ ∞) such that for any*x*,*y*∈*C*and for any*i*≥ 1,$\left|\right|{T}_{i}^{n}x-{T}_{i}^{n}y\left|\right|\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}\left|\right|x-y\left|\right|+\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{n},\forall n\ge 1.$(3.17)

From Theorem 3.5 and Remark 3.6 we can obtain the following

**Theorem 3.8** Let *E*, *C* be the same as in Theorem 3.5. Let {*T*_{
i
}} be a countable family of uniformly asymptotically nonexpansive mappings from *C* into itself with nonnegative real sequences {*ν*_{
n
}} such that $\mathfrak{F}:={\bigcap}_{i=1}^{\infty}F\left({T}_{i}\right)$ is nonempty and bounded and ${\sum}_{n=1}^{\infty}{\nu}_{n}<\infty $. Let {*x*_{
n
}} be the sequence defined by (3.2), where {*β*_{n, i}}, *i* = 0, 1, 2, ... and {*α*_{
n
}} are sequences in [0, 1] satisfying the conditions (a), (b) in Theorem 3.2 and lim inf *β*_{n, 0}*β*_{n, i}> 0 for any *i* ≥ 1. If one of {*T*_{
i
}} is compact, then the conclusions in Theorem 3.5 still hold:

**Proof**. Letting

*μ*

_{ n }= 0, ∀

*n*≥ 1,

*ζ*(

*t*) =

*t*,

*t*≥ 0,

*K*= 0, and

*K** = 1, therefore we have ${\sum}_{n=1}^{\infty}\left({\nu}_{n}\phantom{\rule{2.77695pt}{0ex}}\text{+}{\mu}_{n}\right)\infty $ and

*ζ*(

*t*) =

*K**

*t*, ∀

*t*≥ 0. Again since

*ν*

_{ n }→ 0, {

*ν*

_{ n }} is bounded. Setting

*L*= 1 + sup

_{n≥1}

*ν*

_{ n }, it follows from (3.16) that

i.e., for each *i* ≥ 1, *T*_{
i
} is uniformly *L*-Lipschitz continuous. Therefore all conditions in Theorem 3.5 and Remark 3.6 are satisfied. The conclusions of Theorem 3.8 can be obtained from Theorem 3.5 and Remark 3.6 immediately.

**Theorem 3.9** Let *E*, *C* be the same as in Theorem 3.5. Let {*T*_{
i
}} be a countable family of uniformly asymptotically nonexpansive in the intermediate sense from *C* into itself with a nonnegative real sequence {*σ*_{
n
}} such that $\mathfrak{F}:={\bigcap}_{i=1}^{\infty}F\left({T}_{i}\right)$ is nonempty and bounded and ${\sum}_{n=1}^{\infty}{\sigma}_{n}<\infty $. Let {*x*_{
n
}} be the sequence defined by (3.2), where {*β*_{
n
}, _{
i
}}, *i* = 0, 1, 2, ... and {*α*_{
n
}} are sequences in [0, 1] satisfying the conditions (a), (b) in Theorem 3.2 and lim inf *β*_{n, 0}*β*_{n, i}> 0 for any *i* ≥ 1. If for each *i* ≥ 1, *T*_{
i
} is uniformly *L*_{
i
}-Lipschitzian continuous and one of {*T*_{
i
}} is compact, then the conclusions in Theorem 3.5 still hold.

## Declarations

### Acknowledgements

The authors would like to express their thanks to the referees for their helpful comments and suggestions. This work was supported by the National Research Foundation of Korean Grant funded by the Korean Government (2011-0002581).

## Authors’ Affiliations

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