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A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces
Journal of Inequalities and Applications volume 2012, Article number: 37 (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 quasinonexpansive mappings in Banach spaces which generalize and improve the corresponding theorems of Chidume et al. and others.
2000 AMS Subject Classification: 47J05; 47H09; 49J25.
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.
Recall that a mapping 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.
A mapping T : C → C is said to be asymptotically nonexpansive in the intermediate sense [2], if it is continuous and the following inequality holds:
If F(T) ≠ ∅ and (1.1) holds for all x ∈ C, y ∈ F(T), then T is called asymptotically quasinonexpansive in the intermediate sense. Observe that if we define
then 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 quasinonexpansive.
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 quasinonexpansive 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, 2r) → [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 quasinonexpansive 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}xp\left\right\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}\left\rightxp\left\right+\phantom{\rule{2.77695pt}{0ex}}{\nu}_{n}\zeta \left(\left\rightxp\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 LLipschitz 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\rightxy\left\right,\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in C,\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 1.
Let {T_{ i }} be a countable family of uniformly total asymptotically quasinonexpansive 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 quasinonexpansive 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
Denoting by M={\text{sup}}_{n\ge 0,p\in \mathfrak{F}}\left\{\left\right{x}_{n}\left\right+\left\right{x}_{n}p\left\right\right\}<\infty, from (3.2) we have
where 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
It follows from Lemma 2.2 that lim_{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
By the assumption, it is easy to see that
This implies that
Therefore, we have
where 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 quasinonexpansive 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:
where {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 quasinonexpansive 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 LiLipschitzian 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
By Lemma 2.1, for any positive integer j ≥ 1 we have
Since
where
First substituting (3.11) into (3.10), then substituting (3.10) into (3.9) and simplifying we have that
This together with Theorem 3.2 (1) shows that for each j ≥ 1
By conditions (b) and lim inf β_{n,0}β_{n, i}> 0 for any i ≥ 1, this implies that
By the property of g, we have that
The conclusion (1) is proved.

(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)
For any given ε > 0, from (3.12) there exists a positive integer n_{0} such that
This together with (3.13) yields that
By the assumption that, there exists a mapping in {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\rightxy\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\rightxy\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 LLipschitz 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.
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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 (20110002581).
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SsC and JKK conceived the study and participated in its design and coordination. JKK and HWJL suggested many good ideas that are useful for achievement this paper and made the revision. JKK and CKC prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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Chang, Ss., Kim, J.K., Lee, H.W.J. et al. A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces. J Inequal Appl 2012, 37 (2012). https://doi.org/10.1186/1029242X201237
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DOI: https://doi.org/10.1186/1029242X201237
Keywords
 total asymptotically nonexpansive mapping
 total asymptotically quasinonexpansive mapping
 asymptotically quasinonexpansive mapping
 asymptotically nonexpansive mapping
 asymptotically nonexpansive in the intermediate sense