- Research Article
- Open access
- Published:
Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 632137 (2010)
Abstract
We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fréchet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.
1. Introduction
Let and
be a real Banach space and the dual space of
, respectively. Let
be a nonempty subset of
. Let
denote the normalized duality mapping from
into
given by
, for all
, where
denotes the duality pairing between
and
. If
is smooth or
is strictly convex, then
is single-valued.
Throughout this paper, we denote the single valued duality mapping by and denote the set of fixed points of a nonlinear mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ1_HTML.gif)
Definition 1.1.
A mapping with domain
and range
in
is called
(i)pseudocontractive [1] if, for all , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ2_HTML.gif)
(ii)-strictly pseudocontractive [2] if for all
, there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ3_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ4_HTML.gif)
(iii)L-Lipschitzian if, for all , there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ5_HTML.gif)
Remark 1.2.
It is obvious by the definition that
(1)every strictly pseudocontractive mapping is pseudocontractive,
(2)every -strictly pseudocontractive mapping is
-Lipschitzian; see [3].
Remark 1.3.
Let be a nonempty subset of a real Hilbert space and
a mapping. Then
is said to be
-strictly pseudocontractive [2] if, for all
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ6_HTML.gif)
It is well know that (1.6) is equivalent to the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ7_HTML.gif)
It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. Moreover, we know from [4] that the class of pseudocontractions also includes properly the class of strict pseudocontractions. A mapping is called accretive if, for all
, there exists
such that
. It is also known that
is accretive if and only if
is pseudocontractive. Hence, a solution of the equation
is a solution of the fixed point of
. Note that if
, then
is
-strictly accretive if and only if
is
-strictly pseudocontractive.
In 1953, Mann [5] introduced the iteration as follows: a sequence defined by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ8_HTML.gif)
where . If
is a nonexpansive mapping with a fixed point and the control sequence
is chosen so that
, then the sequence
defined by (1.8) converges weakly to a fixed point of
( this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [6] ). However, if
is a Lipschitzian pseudocontractive mapping, then Mann iteration defined by (1.8) may fail to converge in a Hilbert space; see [4].
In 1967, Browder-Petryshyn [2] introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann's iteration (1.8) with a constant sequence for all
. Respectively, Marino-Xu [7] and Zhou [8] extended the results of Browder-Petryshyn [2] to Mann's iteration process (1.8). To be more precise, they proved the following theorem.
Theorem 1.4 (see [7]).
Let be a closed convex subset of a real Hilbert space
. Let
be a
-strict pseudocontraction for some
, and assume that
admits a fixed point in
. Let a sequence
be the sequence generated by Mann's algorithm (1.8). Assume that the control sequence
is chosen so that
for all
and
. Then
converges weakly to a fixed point of
.
Meanwhile, Marino, and Xu raised the open question: whether Theorem 1.4 can be extended to Banach spaces which are uniformly convex and have a Fréchet differentiable norm. Later, Zhou [9] and Zhang-Su [10], respectively, extended the result above to -uniformly smooth and
-uniformly smooth Banach spaces which are uniformly convex or satisfy Opial's condition.
In 2001, Osilike-Udomene [11] proved the convergence theorems of the Mann [5] and Ishikawa [12] iteration methods in the framework of -uniformly smooth and uniformly convex Banach spaces. They also obtained that a sequence
defined by (1.8) converges weakly to a fixed point of
under suitable control conditions. However, the sequence
excluded the canonical choice
. This was a motivation for Zhang-Guo [13] to improve the results in the same space. Observe that the results of Osilike-Udomene [11] and Zhang-Guo [13] hold under the assumption that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ9_HTML.gif)
for some and
is a constant depending on the geometry of the space.
Let be a uniformly smooth real Banach space. Then there exists a nondecreasing continuous function
with
and
for
such that, for all
, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ10_HTML.gif)
Recently, Chidume-Shahzad [17] extended the results of Osilike-Udomene [11] and Zhang-Guo [13] by using Reich's inequality (1.10) to the much more general real Banach spaces which are uniformly smooth and uniformly convex. Under the assumption that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ11_HTML.gif)
for some , they proved the following theorem.
Theorem 1.6 (see [17]).
Let be a uniformly smooth real Banach space which is also uniformly convex and
a nonempty closed convex subset of
. Let
be a
-strict pseudocontraction for some
with
. For a fixed
, define a sequence
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ12_HTML.gif)
where is a real sequence in
satisfying the following conditions:
(i);
(ii).
Then, converges weakly to a fixed point of
.
However, we would like to point out that the results of Chidume-Shahzad [17] do not hold in real Hilbert spaces. Indeed, we know from Chidume [14] that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ13_HTML.gif)
If is a real Hilbert space, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ14_HTML.gif)
On the other hand, by assumption (1.11), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ15_HTML.gif)
which is a contradiction.
It is known that one can extend his result from a single strict pseudocontraction to a finite family of strict pseudocontractions by replacing the convex combination of these mappings in the iteration under suitable conditions. The construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors; see also [18–22] and the references therein.
Our motivation in this paper is the following:
(1)to modify the normal Mann iteration process for finding common fixed points of an infinitely countable family of strict pseudocontractions,
(2)to improve and extend the results of Chidume-Shahzad [17] from a real uniformly smooth and uniformly convex Banach space to a real uniformly convex Banach space which has the Fréchet differentiable norm.
Motivated and inspired by Marino-Xu [7], Osilike-Udomene [11], Zhou [8], Zhang-Guo [13], and Chidume-Shahzad [17], we consider the following Mann-type iteration: and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ16_HTML.gif)
where is a real sequence in
and
is a countable family of strict pseudocontractions on a closed and convex subset
of a real Banach space
.
In this paper, we prove the weak convergence of a Mann-type iteration process (1.16) in a uniformly convex Banach space which has the Fréchet differentiable norm for a countable family of strict pseudocontractions under some appropriate conditions. The results obtained in this paper improve and extend the results of Chidume-Shahzad [17], Marino-Xu [7], Osilike-Udomene [11], Zhou [8], and Zhang-Guo [13] in some aspects.
We will use the following notation:
(i) for weak convergence and
for strong convergence.
(ii) denotes the weak
-limit set of
.
2. Preliminaries
A Banach space is said to be strictly convex if
for all
with
and
. A Banach space
is called uniformly convex if for each
there is a
such that, for
with
, and
holds. The modulus of convexity of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ17_HTML.gif)
for all .
is uniformly convex if
, and
for all
. It is known that every uniformly convex Banach space is strictly convex and reflexive. Let
. Then the norm of
is said to be Gâteaux differentiable if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ18_HTML.gif)
exists for each . In this case
is called smooth. The norm of
is said to be Fréchet differentiable or
is Fréchet smooth if, for each
, the limit is attained uniformly for
. In other words, there exists a function
with
as
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ19_HTML.gif)
for all . In this case the norm is Gâteaux differentiable and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ20_HTML.gif)
for all . On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ21_HTML.gif)
for all , where
is a function defined on
such that
. The norm of
is called uniformly Fréchet differentiable if the limit is attained uniformly for
.
Let be the modulus of smoothness of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ22_HTML.gif)
A Banach space is said to be uniformly smooth if
as
. Let
, then
is said to be
-uniformly smooth if there exists
such that
. It is easy to see that if
is
-uniformly smooth, then
is uniformly smooth. It is well known that
is uniformly smooth if and only if the norm of
is uniformly Fréchet differentiable, and hence the norm of
is Fréchet differentiable, and it is also known that if
is Fréchet smooth, then
is smooth. Moreover, every uniformly smooth Banach space is reflexive. For more details, we refer the reader to [14, 23]. A Banach space
is said to satisfy Opial's condition [24] if
and
; then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ23_HTML.gif)
In the sequel, we will need the following lemmas.
Lemma 2.1 (see [23]).
Let be a Banach space and
the duality mapping. Then one has the following:
(i) for all
, where
;
(ii) for all
, where
.
Lemma 2.2 (see [25]).
Let be a real uniformly convex Banach space,
a nonempty, closed, and convex subset of
, and
a continuous pseudocontractive mapping. Then,
is demiclosed at zero, that is, for all sequence
with
and
it follows that
.
Lemma 2.3 (see [25]).
Let be a real reflexive Banach space which satisfies Opial's condition,
a nonempty, closed and convex subset of
and
a continuous pseudocontractive mapping. Then,
is demiclosed at zero.
Lemma 2.4 (see [26]).
Let be a real uniformly convex Banach space with a Fréchet differentiable norm. Let
be a closed and convex subset of
and
a family of
-Lipschitzian self-mappings on
such that
and
. For arbitrary
, define
for all
. Then for every
,
exists, in particular, for all
and
,
.
Let and
, be sequences of nonnegative real numbers satisfying the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ24_HTML.gif)
If and
, then
exists. If, in addition,
has a subsequence converging to 0, then
.
To deal with a family of mappings, the following conditions are introduced. Let be a subset of a real Banach space
, and let
be a family of mappings of
such that
. Then
is said to satisfy the AKTT-condition [28] if for each bounded subset
of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ25_HTML.gif)
Lemma 2.6 (see [28]).
Let be a nonempty and closed subset of a Banach space
, and let
be a family of mappings of
into itself which satisfies the AKTT-condition, then the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ26_HTML.gif)
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ27_HTML.gif)
for each bounded subset of
.
So we have the following results proved by Boonchari-Saejung [29, 30].
Let be a closed and convex subset of a smooth Banach space
. Suppose that
is a family of
-strictly pseudocontractive mappings from
into
with
and
is a real sequence in
such that
. Then the following conclusions hold:
(1) is a
-strictly pseudocontractive mapping;
(2).
Lemma 2.8 (see [30]).
Let be a closed and convex subset of a smooth Banach space
. Suppose that
is a countable family of
-strictly pseudocontractive mappings of
into itself with
. For each
, define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ28_HTML.gif)
where is a family of nonnegative numbers satisfying
(i) for all
;
(ii) for all
;
(iii).
Then
(1)each is a
-strictly pseudocontractive mapping;
(2) satisfies AKTT-condition;
(3)If is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ29_HTML.gif)
then and
.
For convenience, we will write that satisfies the AKTT-condition if
satisfies the AKTT-condition and
is defined by Lemma 2.6 with
.
3. Main Results
Lemma 3.1.
Let be a real Banach space, and let
be a nonempty, closed, and convex subset of
. Let
be a family of
-strict pseudocontractions for some
such that
. Define a sequence
by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ30_HTML.gif)
where satisfying
and
. If
satisfies the AKTT-condition, then
(i) exists for all
;
(ii).
Proof.
Let , and put
. First, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ31_HTML.gif)
Since is a
-strict pseudocontraction, there exists
. By Lemma 2.1 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ32_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ33_HTML.gif)
Hence, by , we have from Lemma 2.5 that
exists; consequently,
is bounded. Moreover, by (3.3), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ34_HTML.gif)
where . It follows that
. Since
is bounded,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ35_HTML.gif)
Since satisfies the AKTT-condition, it follows that
. This completes the proof of (i) and (ii).
Lemma 3.2.
Let be a real Banach space with the Fréchet differentiable norm. For
, let
be defined for
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ36_HTML.gif)
Then, , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ37_HTML.gif)
for all .
Proof.
Let . Since
has the Fréchet differentiable norm, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ38_HTML.gif)
Then , and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ39_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ40_HTML.gif)
Suppose that . Put
and
. By (3.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ41_HTML.gif)
This completes the proof.
Remark 3.3.
In a real Hilbert space, we see that for
.
In our more general setting, throughout this paper we will assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ42_HTML.gif)
where is a function appearing in (3.8).
So we obtain the following results.
Lemma 3.4.
Let be a real Banach space with the Fréchet differentiable norm, and let
be a nonempty, closed, and convex subset of
. Let
be a family of
-strict pseudocontractions for some
such that
. Define a sequence
by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ43_HTML.gif)
where satisfying
and
. If
satisfies the AKTT-condition, then
.
Proof.
Let , and put
. Then by (3.8) and (3.13) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ44_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ45_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ46_HTML.gif)
By (3.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ47_HTML.gif)
Since ,
, and
, it follows from Lemma 2.5 that
exists. Hence, by Lemma 3.1(ii), we can conclude that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ48_HTML.gif)
it follows from Lemma 2.6 that . This completes the proof.
Now, we prove our main result.
Theorem 3.5.
Let be a real uniformly convex Banach space with the Fréchet differentiable norm, and let
be a nonempty, closed, and convex subset of
. Let
be a family of
-strict pseudocontractions for some
such that
. Define a sequence
by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ49_HTML.gif)
where satisfying
and
. If
satisfies the AKTT-condition, then
converges weakly to a common fixed point of
.
Proof.
Let , and define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ50_HTML.gif)
Then . By (3.8), we have for bounded
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ51_HTML.gif)
This implies that is nonexpansive. By Lemma 3.1(i), we know that
is bounded. By Lemma 3.4, we also know that
. Applying Lemma 2.2, we also have
.
Finally, we will show that is a singleton. Suppose that
. Hence
. By Lemma 2.4,
exists. Suppose that
and
are subsequences of
such that
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ52_HTML.gif)
Hence ; consequently,
as
. This completes the proof.
As a direct consequence of Theorem 3.5, Lemmas 2.7 and 2.8 we also obtain the following results.
Theorem 3.6.
Let be a real uniformly convex Banach space with the Fréchet differentiable norm, and let
be a nonempty, closed, and convex subset of
. Let
be a sequence of
-strict pseudocontractions of
into itself such that
and
. Define a sequence
by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ53_HTML.gif)
where satisfying
and
and
satisfies conditions (i)–(iii) of Lemma 2.8. Then,
converges weakly to a common fixed point of
.
Remark 3.7.
-
(i)
Theorems 3.5 and 3.6 extend and improve Theorems
and
of Chidume-Shahzad [17] in the following senses:
(i)from real uniformly smooth and uniformly convex Banach spaces to real uniformly convex Banach spaces with Fréchet differentiable norms;
(ii)from finite strict pseudocontractions to infinite strict pseudocontractions.
Using Opial's condition, we also obtain the following results in a real reflexive Banach space.
Theorem 3.8.
Let be a real Fréchet smooth and reflexive Banach space which satisfies Opial's condition, and let
be a nonempty, closed, and convex subset of
. Let
be a family of
-strict pseudocontractions for some
such that
. Define a sequence
by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ54_HTML.gif)
where satisfying
and
. If
satisfies the AKTT-condition, then
converges weakly to a common fixed point of
.
Proof.
Let . By Lemma 3.1(i), we know that
exists. Since
has the Fréchet differentiable norm, by Lemma 3.4, we know that
. It follows from Lemma 2.3 that
. Finally, we show that
is a singleton. Let
, and let
and
be subsequences of
chosen so that
and
. If
, then Opial's condition of
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ55_HTML.gif)
This is a contradiction, and thus the proof is complete.
Theorem 3.9.
Let be a real Fréchet smooth and reflexive Banach space which satisfies Opial's condition, and let
be a nonempty, closed, and convex subset of
. Let
be a sequence of
-strict pseudocontractions of
into itself such that
and
. Define a sequence
by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F632137/MediaObjects/13663_2010_Article_1316_Equ56_HTML.gif)
where satisfying
and
and
satisfies conditions (i)–(iii) of Lemma 2.8. Then,
converges weakly to a common fixed point of
.
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Acknowledgments
The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund, Thailand. The first author is supported by the Royal Golden Jubilee Grant PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.
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Cholamjiak, P., Suantai, S. Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces. Fixed Point Theory Appl 2010, 632137 (2010). https://doi.org/10.1155/2010/632137
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DOI: https://doi.org/10.1155/2010/632137