- Research Article
- Open access
- Published:
An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings
Fixed Point Theory and Applications volume 2008, Article number: 134148 (2008)
Abstract
We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. (2007) and many others.
1. Introduction
Let be a real Hilbert space with inner product
and norm
, and let
be a closed convex subset of
. Let
be a bifunction of
into
, where
is the set of real numbers. The equilibrium problem for
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by Given a mapping
, let
for all
. Then
if and only if
for all
that is,
is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). In 1997, Flåm and Antipin [1] introduced an iterative scheme of finding the best approximation to initial data when
is nonempty and proved a strong convergence theorem.
Let be a mapping. The classical variational inequality, denoted by
, is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ2_HTML.gif)
for all The variational inequality has been extensively studied in the literature. See, for example, [2, 3] and the references therein. A mapping
of
into
is called
-inverse-strongly monotone [4, 5] if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ3_HTML.gif)
for all . It is obvious that any
-inverse-strongly monotone mapping
is monotone and Lipschitz continuous. A mapping
of
into itself is called nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ4_HTML.gif)
for all . We denote by
the set of fixed points of
. For finding an element of
, under the assumption that a set
is nonempty, closed, and convex, a mapping
is nonexpansive and a mapping
is
-inverse-strongly monotone, Takahashi and Toyoda [6] introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ5_HTML.gif)
for every where
is a sequence in (0, 1), and
is a sequence in
. They proved that if
, then the sequence
generated by (1.5) converges weakly to some
. Recently, motivated by the idea of Korpelevič's extragradient method [7], Nadezhkina and Takahashi [8] introduced an iterative scheme for finding an element of
and the weak convergence theorem is presented. Moreover, Zeng and Yao [9] proposed some new iterative schemes for finding elements in
and obtained the weak convergence theorem for such schemes. Very recently, Yao et al. [10] introduced the following iterative scheme for finding an element of
under some mild conditions. Let
be a closed convex subset of a real Hilbert space
a monotone,
-Lipschitz continuous mapping, and
a nonexpansive mapping of
into itself such that
Suppose that
and
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ6_HTML.gif)
where and
satisfy some parameters controlling conditions. They proved that the sequence
defined by (1.6) converges strongly to a common element of
.
On the other hand, S. Takahashi and W. Takahashi [11] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let be a nonexpansive mapping. Starting with arbitrary initial
define sequences
and
recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ7_HTML.gif)
They proved that under certain appropriate conditions imposed on and
, the sequences
and
converge strongly to
, where
Moreover, Aoyama et al. [12] introduced an iterative scheme for finding a common fixed point of a countable family of nonexpansive mappings in Banach spaces and obtained the strong convergence theorem for such scheme.
In this paper, motivated by Yao et al. [10], S. Takahashi and W. Takahashi [11] and Aoyama et al. [12], we introduce a new extragradient method (4.2) which is mixed the iterative schemes considered in [10–12] for finding a common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the solution set of the classical variational inequality problem for a monotone -Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Further, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. The results obtained in this paper improve and extend the recent ones announced by Yao et al. results [10] and many others.
2. Preliminaries
Let be a real Hilbert space with norm
and inner product
and let
be a closed convex subset of
. For every point
, there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ8_HTML.gif)
is called the metric projection of
onto
It is well known that
is a nonexpansive mapping of
onto
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ9_HTML.gif)
for every Moreover,
is characterized by the following properties:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ11_HTML.gif)
for all . For more details, see [13]. It is easy to see that the following is true:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ12_HTML.gif)
A set-valued mapping is called monotone if for all
and
imply
. A monotone mapping
is maximal if the graph of
of
is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping
is maximal if and only if for
for every
implies
. Let
be a monotone map of
into
,
-Lipschitz continuous mapping and let
be the normal cone to
at
, that is,
for all
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ13_HTML.gif)
Then is the maximal monotone and
if and only if
; see [14].
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 (see [15]).
Let be an inner product space. Then for all
and
with
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ14_HTML.gif)
Lemma 2.2 (see [16]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in
with
Suppose
for all integers
and
Then,
Lemma 2.3 (see [17]).
Assume is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ15_HTML.gif)
where is a sequence in
and
is a sequence in
such that
(i) and
(ii) or
Then
Lemma 2.4 (see [12, Lemma 3.2]).
Let be a nonempty closed subset of a Banach space and let
be a sequence of mappings of
into itself. Suppose that
. Then, for each
,
converges strongly to some point of
. Moreover, let
be a mapping of
into itself defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ16_HTML.gif)
Then .
For solving the equilibrium problem for a bifunction , let us assume that
satisfies the following conditions:
(A1) for all
(A2) is monotone, that is,
for all
(A3)for each
(A4)for each is convex and lower semicontinuous.
The following lemma appears implicitly in [18].
Lemma 2.5 (see [18]).
Let be a nonempty closed convex subset of
and let
be a bifunction of
into
satisfying (A1)–(A4). Let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ17_HTML.gif)
The following lemma was also given in [1].
Lemma 2.6 (see [1]).
Assume that satisfies (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ18_HTML.gif)
for all . Then, the following hold:
(i) is single-valued;
(ii) is firmly nonexpansive, that is, for any
(iii)
(iv) is closed and convex.
3. Main Results
In this section, we prove a strong convergence theorem.
Theorem 3.1.
Let be a closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A4),
a monotone
-Lipschitz continuous mapping and let
be a sequence of nonexpansive mappings of
into itself such that
Let the sequences
, and
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ19_HTML.gif)
where ,
, and
satisfy the following conditions:
(C1),
(C2)
(C3)
(C4)
(C5).
Suppose that for any bounded subset
of
. Let
be a mapping of
into itself defined by
and suppose that
. Then the sequences
, and
converge strongly to the same point
, where
.
Proof.
Let . Since
is a contraction with
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ20_HTML.gif)
Therefore, is a contraction of
into itself, which implies that there exists a unique element
such that
. Then we divide the proof into several steps.
Step 1 ( is bounded).
Indeed, put for all
. Let
. From (2.5) we have
. Also it follows from (2.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ21_HTML.gif)
Since is monotone and
is a solution of the variational inequality problem
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ22_HTML.gif)
This together with (3.3) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ23_HTML.gif)
From (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ24_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ25_HTML.gif)
Hence it follows from (3.5) and (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ26_HTML.gif)
Since as
, there exists a positive integer
such that
, when
. Hence it follows from (3.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ27_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ28_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ29_HTML.gif)
Thus, we can calculate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ30_HTML.gif)
It follows from induction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ31_HTML.gif)
Therefore, is bounded. Hence, so are
, and
.
Step 2 ().
Indeed, we observe that for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ32_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ33_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ34_HTML.gif)
On the other hand, from and
we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ35_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ36_HTML.gif)
Putting in (3.17) and
in (3.18), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ37_HTML.gif)
So, from (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ38_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ39_HTML.gif)
Without loss of generality, let us assume that there exists a real number such that
for all
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ40_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ41_HTML.gif)
where . It follows from (3.16) and the last inequality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ42_HTML.gif)
Setting , we obtain
for all
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ43_HTML.gif)
It follows from (3.24) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ44_HTML.gif)
Combining (3.25) and (3.26), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ45_HTML.gif)
This together with (C1)–(C5) and implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ46_HTML.gif)
Hence, by Lemma 2.2, we obtain as
. It then follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ47_HTML.gif)
By (3.23) and (3.24), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ48_HTML.gif)
Step 3 ().
Indeed, pick any , to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ49_HTML.gif)
Therefore, . From Lemma 2.1 and (3.9), we obtain, when
, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ50_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ51_HTML.gif)
It now follows from the last inequality, (C1), (C2), (C3) and (3.29), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ52_HTML.gif)
Noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ53_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ54_HTML.gif)
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ55_HTML.gif)
Using (3.37), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ56_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ57_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ58_HTML.gif)
It now follows from (3.36) and (3.40) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ59_HTML.gif)
Applying Lemma 2.4 and (3.41), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ60_HTML.gif)
It follows from the last inequality and (3.36) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ61_HTML.gif)
Step 4 ().
Indeed, we choose a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ62_HTML.gif)
Without loss of generality, we may assume that converges weakly to
. From
we obtain
Now, we will show that
. Firstly, we will show
. Indeed, we observe that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ63_HTML.gif)
From (A2), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ64_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ65_HTML.gif)
From and
we get
. Since
it follows by (A4) that
for all
For
with
and
let
Since
and
we have
and hence
So, from (A1) and (A4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ66_HTML.gif)
and hence . From (A3), we have
for all
, and hence
By the Opial's condition, we can obtain that
Next we will show that
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ67_HTML.gif)
Then is maximal monotone (see [14]). Let
. Since
and
we have
. On the other hand, from
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ68_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ69_HTML.gif)
Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ70_HTML.gif)
Noting that as
,
is Lipschitz continuous and (3.52), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ71_HTML.gif)
Since is maximal monotone, we have
, and hence
. Hence
The property of the metric projection implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ72_HTML.gif)
Step 5 ().
Indeed, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ73_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ74_HTML.gif)
Setting , we have
. Applying Lemma 2.3 to (3.56), we conclude that
converges strongly to
. Consequently,
and
converge strongly to
. This completes the proof.
As in [12, Theorem 4.1], we can generate a sequence of nonexpansive mappings satisfying condition
for any bounded subset
of
by using convex combination of a general sequence
of nonexpansive mappings with a common fixed point.
Corollary 3.2.
Let be a closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A4),
a monotone,
-Lipschitz continuous mapping and let
be a family of nonnegative numbers with indices
with
such that
(i) for all
;
(ii) for every
;
(iii).
Let be a sequence of nonexpansive mappings of
into itself with
Let
and
and
be the sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ75_HTML.gif)
where , and
satisfy the following conditions:
(C1),
(C2)
(C3)
(C4)
(C5).
Then the sequences , and
converge strongly to the same point
, where
.
Setting and
in Theorem 3.1, we have the following result.
Corollary 3.3 (see [10, Theorem 3.1]).
Let be a closed convex subset of a real Hilbert space
. Let
be a monotone,
-Lipschitz continuous mapping, and let
be a nonexpansive mapping of
into itself such that
Suppose
and
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ76_HTML.gif)
where are sequences in
satisfying the following conditions:
(i),
(ii)
(iii)
(iv).
Then converges strongly to
Proof.
Put for all
and
in Theorem 3.1. Thus, we have
. Then the sequence
generated in Corallary 3.3 converges strongly to
.
4. Applications
In this section, we consider the problem of finding a zero of a monotone operator. A multivalued operator with domain
and range
is said to be monotone if for each
and
we have
. A monotone operator
is said to be maximal if its graph
is not properly contained in the graph of any other monotone operator. Let
denote the identity operator on
and let
be a maximal monotone operator. Then we can define, for each
, a nonexpansive single-valued mapping
by
. It is called the resolvent (or the proximal mapping) of
. We also define the Yosida approximation
by
. We know that
and
for all
. We also know that
for all
; see, for instance, Rockafellar [19] or Takahashi [20].
Lemma 4.1 (the resolvent identity).
For , there holds the identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ77_HTML.gif)
By using Theorem 3.1 and Lemma 4.1, we may obtain the following improvement.
Theorem 4.2.
Let be a maximal monotone operator. Let
be a bifunction from
to
satisfying (A1)–(A4),
a monotone
-Lipschitz continuous mapping of
into
such that
and
a contraction of
into itself with coefficient
. Let the sequences
and
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ78_HTML.gif)
where , and
satisfy the following conditions:
(C1),
(C2)
(C3)
(C4)
(C5).
Then converges strongly to
.
Proof.
We first verify that for any bounded subset
of
. Let
be a bounded subset of
. Since
for each
is bounded. It follows from Lemma 4.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ79_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ80_HTML.gif)
for each and
where
. Hence we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F134148/MediaObjects/13663_2008_Article_1063_Equ81_HTML.gif)
By the assumption that , we obtain
for some
. Since
, we obtain that
for all
. Since
for all
, we have
. Therefore, by Theorem 3.1,
converges strongly to
.
References
Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997, 78(1):29–41.
Yao J-C, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications. Volume 76. Edited by: Hadjisavvas N, Komlósi S, Schaible S. Springer, New York, NY, USA; 2005:501–558.
Zeng LC, Schaible S, Yao JC: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005, 124(3):725–738. 10.1007/s10957-004-1182-z
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20(2):197–228. 10.1016/0022-247X(67)90085-6
Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis 1998, 6(4):313–344. 10.1023/A:1008643727926
Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003, 118(2):417–428. 10.1023/A:1025407607560
Korpelevič GM: An extragradient method for finding saddle points and for other problems. Èkonomika i Matematicheskie Metody 1976, 12(4):747–756.
Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2006, 128(1):191–201. 10.1007/s10957-005-7564-z
Zeng L-C, Yao J-C: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2006, 10(5):1293–1303.
Yao Y, Liou Y-C, Yao J-C: An extragradient method for fixed point problems and variational inequality problems. Journal of Inequalities and Applications 2007, 2007:-12.
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007, 331(1):506–515. 10.1016/j.jmaa.2006.08.036
Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(8):2350–2360. 10.1016/j.na.2006.08.032
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970, 149(1):75–88. 10.1090/S0002-9947-1970-0282272-5
Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Computers & Mathematics with Applications 2000, 40(4–5):559–567. 10.1016/S0898-1221(00)00179-6
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005, 305(1):227–239. 10.1016/j.jmaa.2004.11.017
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004, 298(1):279–291. 10.1016/j.jmaa.2004.04.059
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994, 63(1–4):123–145.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976, 14(5):877–898. 10.1137/0314056
Takahashi W: Nonlinear Functional Analysis. Kindai Kagaku sha, Tokyo, Japan; 1988.
Acknowledgments
The author would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper. This research was partially supported by the Commission on Higher Education.
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Wangkeeree, R. An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings. Fixed Point Theory Appl 2008, 134148 (2008). https://doi.org/10.1155/2008/134148
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DOI: https://doi.org/10.1155/2008/134148