Abstract
Using -strongly accretive and
-strictly pseudocontractive mapping, we introduce a general iterative method for finding a common fixed point of a semigroup of non-expansive mappings in a Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded real-valued functions of the semigroup. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality.
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1. Introduction
Let be a real Hilbert space. A mapping
of
into itself is called non-expansive if
, for all
. By
, we denote the set of fixed points of
(i.e.,
).
Mann [1] introduced an iteration procedure for approximation of fixed points of a non-expansive mapping on a Hilbert space as follows. Let
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ1_HTML.gif)
where is a sequence in
. See also [2].
On the other hand, Moudafi [3] introduced the viscosity approximation method for fixed point of non-expansive mappings (see [4] for further developments in both Hilbert and Banach spaces). Let be a contraction on a Hilbert space
(i.e.,
for all
and
). Starting with an arbitrary initial
, define a sequence
recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ2_HTML.gif)
where is sequence in
. It is proved in [3, 4] that, under appropriate condition imposed on
, the sequence
generated by (1.2) converges strongly to the unique solution
in
of the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ3_HTML.gif)
Assume that is strongly positive, that is, there is a constant
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ4_HTML.gif)
In [4] (see also [5]), it is proved that the sequence defined by the iterative method below, with the initial guess
chosen arbitrarily,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ5_HTML.gif)
converges strongly to the unique solution of the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ6_HTML.gif)
provided that the sequence satisfies certain conditions. Marino and Xu [6] combined the iterative (1.5) with the viscosity approximation method (1.2) and considered the following general iterative methods:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ7_HTML.gif)
where . They proved that if
is a sequence in
satisfying the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_IEq37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_IEq39_HTML.gif)
either or
,
then, the sequence generated by (1.7) converges strongly, as
, to the unique solution of the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ8_HTML.gif)
which is the optimality condition for minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ9_HTML.gif)
where is a potential function for
(i.e.,
, for all
).
Let be the topological dual of a Banach space
. The value of
at
will be denoted by
or
. With each
, we associate the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ10_HTML.gif)
Using the Hahn-Banach theorem, it is immediately clear that for each
. The multivalued mapping
from
into
is said to be the (normalized) duality mapping. A Banach space
is said to be smooth if the duality mapping
is single valued. As it is well known, the duality mapping is the identity when
is a Hilbert space; see [7].
Let and
be two positive real numbers such that
. Recall that a mapping
with domain
and range
in
is called
-strongly accretive if, for each
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ11_HTML.gif)
Recall also that a mapping is called
-strictly pseudo-contractive if, for each
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ12_HTML.gif)
It is easy to see that (1.12) can be rewritten as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ13_HTML.gif)
see [8].
In this paper, motivated and inspired by Atsushiba and Takahashi [9], Lau et al. [10], Marino and Xu [6] and Xu [4, 11], we introduce the iterative below, with the initial guess chosen arbitrarily,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ14_HTML.gif)
where is
-strongly accretive and
-strictly pseudo-contractive with
,
is a contraction on a Hilbert space
with coefficient
,
is a positive real number such that
, and
is a non-expansive semigroup on
such that the set
of common fixed point of
is nonempty,
is a subspace of
such that
and the mapping
is an element of
for each
, and
is a sequence of means on
. Our purpose in this paper is to introduce this general iterative algorithm for approximating a common fixed points of semigroups of non-expansive mappings which solves some variational inequality. We will prove that if
is left regular and
is a sequence in
satisfying the conditions
and
, then
converges strongly to
, which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ15_HTML.gif)
Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of non-expansive mappings are also presented. It is worth mentioning that we obtain our result without assuming condition .
2. Preliminaries
Let be a semigroup and let
be the space of all bounded real-valued functions defined on
with supremum norm. For
and
, we define elements
and
in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ16_HTML.gif)
Let be a subspace of
containing
, and let
be its dual. An element
in
is said to be a mean on
if
. We often write
instead of
for
and
. Let
be left invariant (resp., right invariant), that is,
(resp.,
) for each
. A mean
on
is said to be left invariant (right invariant) if
(resp.
) for each
and
.
is said to be left (resp., right) amenable if
has a left (resp., right) invariant mean.
is amenable if
is both left and right amenable. As it is well known,
is amenable when
is a commutative semigroup; see [12]. A net
of means on
is said to be left regular if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ17_HTML.gif)
for each , where
is the adjoint operator of
.
Let be a nonempty closed and convex subset of a reflexive Banach space
. A family
of mapping from
into itself is said to be a non-expansive semigroup on
if
is non-expansive and
for each
. We denote by
the set of common fixed points of
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ18_HTML.gif)
The open ball of radius centered at
is denoted by
. For subset
of
, by
, we denote the closed convex hull of
. Weak convergence is denoted by
, and strong convergence is denoted by
.
Let f be a function of semigroup into a reflexive Banach space
such that the weak closure of
is weakly compact, and let
be a subspace of
containing all functions
with
. Then, for any
, there exists a unique element
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ19_HTML.gif)
for all . Moreover, if
is a mean on
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ20_HTML.gif)
One can write by
Lemma 2.2 (see [13]).
Let be a closed convex subset of a Hilbert space
,
a semigroup from
into
such that
, the mapping
an element of
for each
and
, and
a mean on
. If one writes
instead of
, then the following holds.
(i) is non-expansive mapping from
into
.
(ii) for each
.
(iii) for each
.
(iv)If is left invariant, then
is a non-expansive retraction from
onto
.
Let be a nonempty subset of a normed space
, and let
. An element
is said to be the best approximation to
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ21_HTML.gif)
where . The number
is called the distance from
to
or the error in approximating
by
. The (possibly empty) set of all best approximation from
to
is denoted by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ22_HTML.gif)
This defines a mapping from
into
and is called metric (the nearest point) projection onto
.
Lemma 2.3 (see [7]).
Let be a nonempty convex subset of a smooth Banach space
and let
and
. Then, the following is equivalent.
(i) is the best approximation to
.
(ii) is a solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ23_HTML.gif)
Let be a nonempty subset of a Banach space
and
a mapping. Then
is said to be demiclosed at
if, for any sequence
in
, the following implication holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ24_HTML.gif)
Lemma 2.4 (see [14]).
Let be a nonempty closed convex subset of a Hilbert space
and suppose that
is non-expansive. Then, the mapping
is demiclosed at zero.
The following lemma is well known.
Lemma 2.5.
Let be a real Hilbert space. Then, for all
(i)
(ii)
Lemma 2.6 (see [11]).
Let be a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ25_HTML.gif)
where and
are sequences of real numbers satisfying the following conditions:
(i)
(ii)either or
Then,
The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.
Lemma 2.7.
Let be a real smooth Banach space and
a mapping.
(i)If is
-strongly accretive and
-strictly pseudo-contractive with
, then,
is contractive with constant
.
(ii)If is
-strongly accretive and
-strictly pseudo-contractive with
, then, for any fixed number
,
is contractive with constant
.
Proof.
-
(i)
From (1.11) and (1.13), we obtain
(2.11)Because
, we have
(2.12)and, therefore,
is contractive with constant
.
-
(ii)
Because
is contractive with constant
, for each fixed number
, we have
(2.13)
This shows that is contractive with constant
.
Throughout this paper, will denote a
-strongly accretive and
-strictly pseudo-contractive mapping with
, and
is a contraction with coefficient
on a Hilbert space
. We will also always use
to mean a number in
.
3. Strong Convergence Theorem
The following is our main result.
Theorem 3.1.
Let be a non-expansive semigroup on a real Hilbert space
such that
. Let
be a left invariant subspace of
such that
, and the function
is an element of
for each
. Let
be a left regular sequence of means on
, and let
be a sequence in
such that
and
. Let
and
be generated by the iteration algorithm (1.14). Then,
converges strongly, as
, to
, which is a unique solution of the variational inequality (1.15). Equivalently, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ29_HTML.gif)
Proof.
First, we claim that is bounded. Let
; by Lemmas 2.2 and 2.7 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ30_HTML.gif)
By induction,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ31_HTML.gif)
Therefore, is bounded and so is
.
Set . We remark that
is
-invariant bounded closed convex set and
. Now we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ32_HTML.gif)
Let . By [15, Theorem
], there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ33_HTML.gif)
Also by [15, Corollary ], there exists a natural number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ34_HTML.gif)
for all and
. Let
. Since
is strongly left regular, there exists
such that
for
and
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ35_HTML.gif)
By Lemma 2.2 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ36_HTML.gif)
It follows from (3.5), (3.6), (3.7), and (3.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ37_HTML.gif)
for all and
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ38_HTML.gif)
Since is arbitrary, we get (3.4). In this stage, we will show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ39_HTML.gif)
Let and
. Then, there exists
, which satisfies (3.5). Take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ40_HTML.gif)
From and (3.4) there exists
such that
and
, for all
. By Lemma 2.7, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ41_HTML.gif)
for all . Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ42_HTML.gif)
for all . This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ43_HTML.gif)
Since is arbitrary, we get (3.11).
Let . Then
is a contraction of
into itself. In fact, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ44_HTML.gif)
and hence is a contraction due to
Therefore, by Banach contraction principal, has a unique fixed point
. Then using Lemma 2.3,
is the unique solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ45_HTML.gif)
We show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ46_HTML.gif)
Indeed, we can choose a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ47_HTML.gif)
Because is bounded, we may assume that
. In terms of Lemma 2.4 and (3.11), we conclude that
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ48_HTML.gif)
Finally, we prove that as
. By Lemmas 2.5 and 2.7 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ49_HTML.gif)
On the other hand
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ50_HTML.gif)
Since and
are bounded, we can take a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ51_HTML.gif)
So from the above, we reach the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ52_HTML.gif)
Substituting (3.24) in (3.21), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ53_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ54_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ55_HTML.gif)
Since is bounded and
, by (3.18), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ56_HTML.gif)
Consequently, applying Lemma 2.6, to (3.26), we conclude that .
Corollary 3.2.
Let ,
,
, and
be as in Theorem 3.1. Suppose that
a strongly positive bounded linear operator on
with coefficient
and
. Let
be defined by the iterative algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ57_HTML.gif)
Then, converges strongly, as
, to
, which is a unique solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ58_HTML.gif)
Proof.
Because is strongly positive bounded linear operator on
with coefficient
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ59_HTML.gif)
Therefore, is
-strongly accretive. On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ60_HTML.gif)
Since is strongly positive if and only if
is strongly positive, we may assume, with no loss of generality, that
, so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ61_HTML.gif)
This shows that is
-strictly pseudo-contractive. Now apply Theorem 3.1 to conclude the result.
Corollary 3.3.
Let ,
,
and
be as in Theorem 3.1. Suppose
and define a sequence
by the iterative algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ62_HTML.gif)
Then, converges strongly, as
, to a
, which is a unique solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ63_HTML.gif)
Proof.
It is sufficient to take and
in Theorem 3.1.
4. Some Application
Corollary 4.1.
Let and
be non-expansive mappings on a Hilbert space
with
such that
. Let
be a sequence in
satisfying conditions
and
. Let
,
and define a sequence
by the iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ64_HTML.gif)
Then, converges strongly, as
, to
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ65_HTML.gif)
Proof.
Let for each
. Then
is a semigroup of non-expansive mappings on
. Now, for each
and
, we define
Then,
is regular sequence of means [16]. Next, for each
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ66_HTML.gif)
Therefore, applying Theorem 3.1, the result follows.
Corollary 4.2.
Let be a strongly continuous semigroup of non-expansive mappings on a Hilbert space
such that
. Let
be a sequence in
satisfying conditions
and
. Let
and
. Let
be a sequence defined by the iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ67_HTML.gif)
where is an increasing sequence in
such that
and
. Then,
converges strongly, as
, to
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ68_HTML.gif)
Proof.
For , we define
for each
, where
denotes the space of all real-valued bounded continuous functions on
with supremum norm. Then,
is regular sequence of means [16]. Furthermore, for each
, we have
. Now, apply Theorem 3.1 to conclude the result.
Corollary 4.3.
Let be a strongly continuous semigroup of non-expansive mappings on a Hilbert space
such that
. Let
be a sequence in
satisfying conditions
and
. Let
and
. Let
be a sequence defined by the iterative algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ69_HTML.gif)
where is an decreasing sequence in
such that
. Then
converges strongly, as
, to
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ70_HTML.gif)
Proof.
For , we define
for each
. Then
is regular sequence of means [16]. Furthermore, for each
, we have
. Now, apply Theorem 3.1 to conclude the result.
Corollary 4.4.
Let be a non-expansive mapping on a Hilbert space
such that
. Let
be a sequence in
satisfying conditions
and
and let
be a strongly regular matrix. Let
and
. Let
be a sequence defined by the iterative algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ71_HTML.gif)
Then, converges strongly, as
, to
which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ72_HTML.gif)
Proof.
For each , we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F907275/MediaObjects/13663_2010_Article_1362_Equ73_HTML.gif)
for each . Since
is a strongly regular matrix, for each
, we have
, as
; see [17]. Then, it is easy to see that
is regular sequence of means. Furthermore, for each
, we have
Now, apply Theorem 3.1 to conclude the result.
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Acknowledgments
The authors thank the referee(s) for the helpful comments, which improved the presentation of this paper. This paper is dedicated to Professor Anthony To Ming Lau. This paper is based on final report of the research project of the Ph.D. thesis which is done with financial support of research office of the University of Tabriz.
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Piri, H., Vaezi, H. Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2010, 907275 (2010). https://doi.org/10.1155/2010/907275
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DOI: https://doi.org/10.1155/2010/907275