- Research Article
- Open access
- Published:
Strong Convergence of a New Iteration for a Finite Family of Accretive Operators
Fixed Point Theory and Applications volume 2009, Article number: 491583 (2009)
Abstract
The viscosity approximation methods are employed to establish strong convergence of the modified Mann iteration scheme to a common zero of a finite family of accretive operators on a strictly convex Banach space with uniformly Gâteaux differentiable norm. Our work improves and extends various results existing in the current literature.
1. Introduction
Let be a Banach space with dual space of
, and let
a nonempty closed convex subset
. Let
be a positive integer, and let
. We denote by
the normalized duality map from E to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ1_HTML.gif)
A mapping is said to be nonexpansive if
, for all
. A mapping
is called
-contraction if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ2_HTML.gif)
In the last ten years, many papers have been written on the approximation of fixed point for nonlinear mappings by using some iterative processes (see, e.g.,[1–20]).
An operator is said to be accretive if
, for all
,
and
. If
is accretive and
is identity mapping, then we define, for each
, a nonexpansive single-valued mapping
by
, which is called the resolvent of
. we also know that for an accretive operator
,
, where
and
. An accretive operator
is said to be
-accretive, if
for all
. If
is a Hilbert space, then accretive operator is monotone operator. There are many papers throughout literature dealing with the solution of
(
) by utilizing certain iterative sequence (see [1–3, 8–10, 13, 16, 20]).
In 2005, Kim and Xu [10] introduced the following Halpern type iterative sequence for -accretive operator
: Let  
  be a nonempty closed convex subset of  
. For any  
, the sequence 
 is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ3_HTML.gif)
where  and   
, for some  
, satisfy the following conditions:
(C1),
(C2),
(C3), and
(C4).
They proved that the iterative sequence converges strongly to a zero of
.
Recently, Zegeye and Shahzad [20] proved a strong convergence theorem for a finite family of accretive operators by using the Halpern type iteration: Let    be a nonempty closed convex subset of  
. For any  
, the sequence  
  is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ4_HTML.gif)
where     with  
,
,  for  
,
, and  
  satisfies the conditions: (C1), (C2), (C3), or (
).
).
More recently, Hu and Liu [8] proposed a generalized Halpern type iteration: Let  be a nonempty closed convex subset of
. For any  
, the sequence  
  is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ5_HTML.gif)
where    with  
, for  
,
  and  
. Assume
,
,
,   and   
satisfy the following conditions: (C1), (C2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ6_HTML.gif)
They proved that the sequence converges strongly to a common zero of
.
In this paper, we introduce and study a new iterative sequence: Let    be a nonempty closed convex subset of  
  and  
  a  
-contraction. For any  
, the sequence  
  is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ7_HTML.gif)
where    with  
,  for  
,
  and  
,
  and  
. The iterative sequence (1.7) is a natural generalization of all the above mentioned iterative sequences.
(i)In contrast to the iterations (1.3)–(1.5), the convex composition of the iteration (1.7) deals with only instead of
and
.
(ii)If we take , for all
, in (1.7), then (1.7) reduces to Mann iteration. In 2000, Kamimura and Takahashi [9] proved that if
is a Hilbert space and
and
are chosen such that
,
and
, then the Mann iterative sequence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ8_HTML.gif)
converges weakly to a zero of . However, the Mann iteration scheme has only weak convergence for nonexpansive mappings even in a Hilbert space (see [4]).
Our main purpose is to prove strong convergence theorems for a finite family of accretive operators on a strictly convex Banach space with uniformly Gteaux differentiable norm by using viscosity approximation methods. Our theorems extend the comparable results in the following three aspects.
(1)In contrast to weak convergence results on a Hilbert Space in [9], strong convergence of the iterative sequence is obtained in the general setup of a Banach space.
(2)The restrictions (C3), (), and (C4) on the results in [10, 20] are dropped.
(3)A single mapping of the results in [3] is replaced by a finite family of mappings.
2. Preliminaries and Lemmas
A Banach space is said to have G
teaux differentiable norm if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ9_HTML.gif)
exists for each , where
. The norm of
is uniformly G
teaux differentiable   if for each
, the limit is attained uniformly for
. The norm of
is uniformly Fréchet differentiable (
is also called uniformly smooth) if the limit is attained uniformly for each
. It is well known that if
is uniformly G
teaux differentiable norm, then the duality mapping
is single-valued and
uniformly continuous on each bounded subset of
.
A Banach space is called strictly convex if for
,
, and
, we have
for
,
and
for
. In a strictly convex Banach space
, we have that if
, for
,
,
and
, then
.
Lemma 2.1 (The Resolvent Identity).
For and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ10_HTML.gif)
We denote by the set of all natural numbers, and let
be a mean on
, that is, a continuous linear functional
on
satisfying
. We know that
is a mean on
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ11_HTML.gif)
for each . In general, we use
instead of
. Let
with
, and let
be a Banach limit on
. Then
. Further, we know the following result.
Let be a nonempty closed convex subset of a Banach space
with uniformly G
teaux differentiable norm. Assume that
is a bounded sequence in
. Let
, and let
a Banach limit. Then
if and only if
,
.
Let be a closed convex and, let
a mapping of
onto
. Then
is said to be sunny [12, 13] if
for all
and
. A mapping
of
onto
is said to be retraction if
; If a mapping
is a retraction then
for any
, the range of
. A subset
of
is said to be a sunny nonexpansive retraction of
if there exists a sunny nonexpansive retraction of
onto
, and it is said to be a nonexpansive retraction of
if there exists a nonexpansive retraction of
onto
. In a smooth Banach space
, it is known ([5, Page 48]) that
is a sunny nonexpansive retraction if and only if the following condition holds:
,
and
.
Lemma 2.3 (see [14]).
Let and
be bounded sequences in a Banach space
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ12_HTML.gif)
where is a sequence in
such that
. Assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ13_HTML.gif)
Then .
Lemma 2.4.
Let be a real Banach space. Then for all
in
and
, the following inequality holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ14_HTML.gif)
Lemma 2.5 (see [18]).
Let is a sequence of nonnegative real number such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ15_HTML.gif)
where is a sequence in
and
is a sequence in
satisfying the following conditions:
(i);
(ii) or
.
Then  .
Lemma 2.6 ([8]).
Let be a nonempty closed convex subset of a strictly convex Banach space
. Suppose that
is a finite family of accretive operators such that
and satisfies the range conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ16_HTML.gif)
Let be real numbers in
with
and
, where
and
. Then
is nonexpansive and
.
3. Main Results
For the sake of convenience, we list the assumptions to be used in this paper as follows.
(i) is a strictly convex Banach space which has uniformly G
teaux differentiable norm, and
is a nonempty closed convex subset of
which has the fixed point property for nonexpansive mappings.
(ii)The real sequence satisfies the conditions: (C1).
and (C2).
.
We will employ the viscosity approximation methods [11, 19] to obtain a strong convergence theorem. The method of proof is closely related to [2, 3, 19].
Theorem 3.1.
Let be a finite family of accretive operators satisfying the following range conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ17_HTML.gif)
Assume that . Let
be a
-contraction with
. For
, the net
is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ18_HTML.gif)
where with
, for
,
and
. If
, then the net
converges strongly to
, as
, where
is the unique solution of a variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ19_HTML.gif)
Proof.
Put , for all
and
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ20_HTML.gif)
and so is a contraction of
into itself. Hence, for each
, there exists a unique element
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ21_HTML.gif)
Thus the net is well defined.
Lemma 2.6 implies that . Taking
, we have for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ22_HTML.gif)
Consequently, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ23_HTML.gif)
that is, the net is bounded, and so are
and
. Rewriting (I) to find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ24_HTML.gif)
and hence for any , it yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ25_HTML.gif)
Obviously, estimate (I) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ26_HTML.gif)
In view of the Resolvent Identity, we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ27_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ28_HTML.gif)
Combining (3.8) and the above inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ29_HTML.gif)
Assume , as
. Set
and define
(
is the set of all real numbers) by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ30_HTML.gif)
where LIM is a Banach limit on . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ31_HTML.gif)
It is easy to see that is a nonempty closed convex and bounded subset of
and
is invariant under
. Indeed, as
, we have for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ32_HTML.gif)
and so is an element of
. Since
has the fixed point property for nonexpansive mappings,
has a fixed point
in
. Using Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ33_HTML.gif)
Clearly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ34_HTML.gif)
Consequently, by (3.15), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ35_HTML.gif)
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ36_HTML.gif)
and there exists a subsequence which is still denoted by such that
.
On the other hand, let of
be such that
. Now (3.7) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ37_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ38_HTML.gif)
Interchange and
to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ39_HTML.gif)
Addition of (3.20) and (3.21) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ40_HTML.gif)
and so we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ41_HTML.gif)
Since , it follows that
. Consequently
as
. Likewise, using (3.7), it implies for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ42_HTML.gif)
Letting yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ43_HTML.gif)
for all .
Remark 3.2.
In addition,  if is a uniformly smooth Banach space in Theorem 3.1 and we define  
,   then we obtain from Theorem 3.1 and [19, Theorem 4.1] that the net  
converges strongly to  
,   as
,   where  
  and  
is a sunny nonexpansive retraction of  
  onto  
.
Theorem 3.3.
Let be a finite family of accretive operators satisfying the following range conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ44_HTML.gif)
Assume that . Let
be a
-contraction with
. For any
, the sequence
is generated by (1.7). Suppose further that sequences in the iterative sequence (1.7) satisfy the conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ45_HTML.gif)
Then the sequence converges strongly to
, where
is the unique solution of a variational inequality
.
Proof.
Lemma 2.6 implies that . Rewrite (1.7) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ46_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ47_HTML.gif)
Taking , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ48_HTML.gif)
Therefore, the sequence is bounded, and so are the sequences
,
,
,
and,
. We estimate from (3.29)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ49_HTML.gif)
In view of the Resolvent Identity, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ50_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ51_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ52_HTML.gif)
 and  
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ53_HTML.gif)
Consequently, by Lemma 2.3, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ54_HTML.gif)
From (3.29), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ55_HTML.gif)
and so it follows from (3.36) and (3.37) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ56_HTML.gif)
Using the Resolvent Identity and , we discover
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ57_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ58_HTML.gif)
It follows from Theorem 3.1 that generated by
converges strongly to
, as
, where
is the unique solution of a variational inequality
. Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ59_HTML.gif)
In view of Lemma 2.4, we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ60_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ61_HTML.gif)
Since the sequences ,
, and
are bounded and  
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ62_HTML.gif)
where . We also know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ63_HTML.gif)
From the facts that , as
,
is bounded and the duality mapping
is
uniformly continuous on bounded subset of
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ64_HTML.gif)
Combining (3.44), (3.45), and the two results mentioned above, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ65_HTML.gif)
Similarly, from (3.29) and the duality mapping is
uniformly continuous on bounded subset of
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ66_HTML.gif)
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ67_HTML.gif)
and apply Lemma 2.4 to find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ68_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ69_HTML.gif)
From (3.47), (3.48), (C1), (C2), and , it follows that
and
. Consequently applying Lemma 2.5 to (3.50), we conclude that
.
If we take , for all
, in the iteration (1.7), then, from Theorem 3.3, we have what follows
Corollary 3.4.
Let ,
,
, and
be as in Theorem 3.3. For any
, the sequence
is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ70_HTML.gif)
where with
,  for
,
and
. Then the sequence
converges strongly to
.
Remark 3.5.
Theorem 3.3 and Corollary 3.4 prove strong convergence results of the new iterative sequences which are different from the iterative sequences (1.4) and (1.5). In contrast to [20], the restriction: (C3).  or  
is removed.
If we consider the case of an accretive operator , then as a direct consequence of Theorem 3.1 and Theorem 3.3, we have the following corollaries.
Corollary 3.6 ([3, Theorem 3.1]).
Let (not strictly convex) be an accretive operator satisfying the following range condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ71_HTML.gif)
Assume that . Let
be a
-contraction with
. For
, the net
is given by:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ72_HTML.gif)
where . If
,  for some
, then
converges strongly to
, as
, where
is the unique solution of a variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_IEq386_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_IEq387_HTML.gif)
Corollary 3.7.
Let (not strictly convex) be an accretive operator satisfying the following range condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ73_HTML.gif)
Assume that . Let
be a
-contraction with
. Suppose that
and
are real sequences in
and
is a sequence in
, satisfying the conditions:
and
, for some
. For any
, the sequence
is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F491583/MediaObjects/13663_2009_Article_1147_Equ74_HTML.gif)
where . Then the sequence
converges strongly to
, where
is the unique solution of a variational inequality
.
Remark 3.8.
(i)Corollary 3.7 describes strong convergence result in Banach spaces for a modification of Mann iteration scheme in contrast to the weak convergence result on Hilbert spaces given in [9, Theorem 3].
(ii)In contrast to the result [10, Theorem 4.2], the iterative sequence in Corollary 3.7 is different from the iteration (1.3), and the conditions and
are not required.
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Acknowledgments
The work was supported partly by NNSF of China (no. 60872095), the NSF of Zhejiang Province (no. Y606093), K. C. Wong Magna Fund of NIngbo University, NIngbo Natural Science Foundation (no. 2008A610018), and Subject Foundation of Ningbo University (no. XK109050).
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Hu, LG., Wang, JP. Strong Convergence of a New Iteration for a Finite Family of Accretive Operators. Fixed Point Theory Appl 2009, 491583 (2009). https://doi.org/10.1155/2009/491583
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DOI: https://doi.org/10.1155/2009/491583