- Research Article
- Open access
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A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions, Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces
Fixed Point Theory and Applications volume 2009, Article number: 567147 (2009)
Abstract
We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of J. W. Peng et al. (2008) and many others.
1. Introduction
Let be a real Hilbert space whose inner product and norm are denoted by
and
, respectively. Let
be a nonempty closed convex subset of
, and 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%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by Recently, Combettes and Hirstoaga [1] introduced an iterative scheme of finding the best approximation to the initial data when
is nonempty and proved a strong convergence theorem. Let
be a nonlinear map. The classical variational inequality which is denoted by
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ2_HTML.gif)
The variational inequality has been extensively studied in literature. See, for example, [2, 3] and the references therein. Recall that a mapping of
into itself is called nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ3_HTML.gif)
A mapping is called contractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ4_HTML.gif)
We denote by the set of fixed points of
.
Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping; see, for instance, [3–6] and the references therein. Recently, Plubtieng and Punpaeng [6] introduced the following iterative scheme. Let and let
, and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ5_HTML.gif)
They proved that if the sequences and
of parameters satisfy appropriate conditions, then the sequences
and
both converge strongly to the unique solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ6_HTML.gif)
which is the optimality condition for the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ7_HTML.gif)
where is a potential function for
Let be a single-valued nonlinear mapping, and let
be a set-valued mapping. We consider the following variational inclusion, which is to find a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ8_HTML.gif)
where is the zero vector in
. The set of solutions of problem (1.8) is denoted by
. If
, then problem (1.8) becomes the inclusion problem introduced by Rockafellar [7]. If
, where
is a nonempty closed convex subset of
and
is the indicator function of C, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ9_HTML.gif)
then the variational inclusion problem (1.8) is equivalent to variational inequality problem (1.2). It is known that (1.8) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, game theory. Also various types of variational inclusions problems have been extended and generalized (see [8] and the references therein.)
Very recently, Peng et al. [9] introduced the following iterative scheme for finding a common element of the set of solutions to the problem (1.8), the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with , define sequence,
,
, and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ10_HTML.gif)
for all , where
,
and
. They proved that under certain appropriate conditions imposed on
and
, the sequences
,
, and
generated by (1.10) converge strongly to
, where
.
Motivated and inspired by Plubtieng and Punpaeng [6], Peng et al. [9] and Aoyama et al. [10], we introduce an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem (1.8) with multi-valued maximal monotone mapping and inverse-strongly monotone mappings, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with an arbitrary define sequences
,
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ11_HTML.gif)
for all , where
,
, and let
;
be a strongly bounded linear operator on
, and
is a sequence of nonexpansive mappings on
. Under suitable conditions, some strong convergence theorems for approximating to this common elements are proved. Our results extend and improve some corresponding results in [3, 9] and the references therein.
2. Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the next section.
Let be a real Hilbert space with inner product
and norm
, respectively.
It is wellknown that for all and
, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ12_HTML.gif)
Let be a nonempty closed convex subset of
. Then, for any
, there exists a unique nearest point of
, denoted by
, such that
for all
. Such a
is called the metric projection from
into
. We know that
is nonexpansive. It is also known that,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ13_HTML.gif)
It is easy to see that (2.2) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ14_HTML.gif)
For solving the equilibrium problem for a bifunction , let us assume that
satisfies the following conditions:
(A1)  for all
(A2) is monotone, that is,
(A3)for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ15_HTML.gif)
(A4)for each is convex and lower semicontinuous.
The following lemma appears implicitly in [11] and [1].
Let be a nonempty closed convex subset of
and let
be a bifunction of
in to
satisfying (A1)–(A4). Let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ16_HTML.gif)
Define a mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ17_HTML.gif)
for all . Then, the following hold:
(1) is single-valued;
(2) is firmly nonexpansive, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ18_HTML.gif)
(3)
(4) is closed and convex.
We also need the following lemmas for proving our main result.
Lemma 2.2 (See [12]).
Let be a Hilbert space,
a nonempty closed convex subset of
,
a contraction with coefficient
, and
a strongly positive linear bounded operator with coefficient
. Then :
(1)if   , then
(2)if   , then
.
Lemma (See [13]).
Assume is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ19_HTML.gif)
where is a sequence in
and
is a sequence in
such that
(1)
(2) or
Then
Recall that a mapping is called
-inverse-strongly monotone, if there exists a positive number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ20_HTML.gif)
Let be the identity mapping on
. It is well known that if
is
-inverse-strongly monotone, then
is
-Lipschitz continuous and monotone mapping. In addition, if
, then
is a nonexpansive mapping.
A set-valued is called monotone if for all
and
imply
. A monotone mapping
is maximal if its graph
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 the set-valued mapping be maximal monotone. We define the resolvent operator
associated with
and
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ21_HTML.gif)
where is a positive number. It is worth mentioning that the resolvent operator
is single-valued, nonexpansive and
-inverse-strongly monotone, see for example [14] and that a solution of problem (1.8) is a fixed point of the operator
for all
, see for instance [15].
Lemma 2.4 (See [14]).
Let be a maximal monotone mapping and
be a Lipschitz-continuous mapping. Then the mapping
is a maximal monotone mapping.
Remark (See [9]).
Lemma 2.4 implies that is closed and convex if
is a maximal monotone mapping and
be an inverse strongly monotone mapping.
Lemma 2.6 (See [10]).
Let be a nonempty closed subset of a Banach space and let
a sequence of mappings of
into itself. Suppose that
. Then, for each
,
converges strongly to some point of
. Moreover, let
be a mapping from
into itself defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ22_HTML.gif)
Then .
3. Main Results
We begin this section by proving a strong convergence theorem of an implicit iterative sequence obtained by the viscosity approximation method for finding a common element of the set of solutions of the variational inclusion, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.
Throughout the rest of this paper, we always assume that is a contraction of
into itself with coefficient
, and
is a strongly positive bounded linear operator with coefficient
and
. Let
be a nonexpansive mapping of
into
. Let
be an
-inverse-strongly monotone mapping,
be a maximal monotone mapping and let
be defined as in (2.10). Let
be a sequence of mappings defined as Lemma 2.1. Consider a sequence of mappings
on
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ23_HTML.gif)
where By Lemma 2.2, we note that
is a contraction. Therefore, by the Banach contraction principle,
has a unique fixed point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ24_HTML.gif)
Theorem 3.1.
Let be a real Hilbert space, let
be a bifunction from
satisfying (A1)–(A4) and let
be a nonexpansive mapping on
. Let
be an
-inverse-strongly monotone mapping,
be a maximal monotone mapping such that
Let
be a contraction of
into itself with a constant
and let
be a strongly bounded linear operator on
with coefficient
and
. Let
,
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ25_HTML.gif)
where ,
and
satisfy
and
. Then,
,
and
converges strongly to a point
in
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ26_HTML.gif)
Equivalently, we have
Proof.
First, we assume that . By Lemma 2.2, we obtain
. Let
Since
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ27_HTML.gif)
We note from that
. As
is nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ28_HTML.gif)
for all Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ29_HTML.gif)
It follows that Hence
is bounded and we also obtain that
,
,
,
and
are bounded. Next, we show that
. Since
, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ30_HTML.gif)
Moreover, it follows from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ31_HTML.gif)
and hence Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ32_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ33_HTML.gif)
Since is bounded and
it follows that
as
Put . From (3.10), it follows by the nonexpansive of
and the inverse strongly monotonicity of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ34_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ35_HTML.gif)
Since , we have
as
. Since
is
–inverse-strongly monotone and
is nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ36_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ37_HTML.gif)
From (3.5), (3.10), and (3.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ38_HTML.gif)
Thus, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ39_HTML.gif)
Since ,
as
, we have
as
. It follows from the inequality
that
as
. Moreover, we have
as
.
Put . Since both
and
are nonexpansive, we have
is a nonexpansive mapping on
and then we have
for all
. It follows by Theorem 3.1 of Plubtieng and Punpaeng [6] that
converges strongly to
, where
and
, for all
. We will show that
. Since
converges strongly to
, we also have
. Let us show
Assume
Since
and
, we have
Since
it follows by the Opial's condition that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ40_HTML.gif)
This is a contradiction. Hence We now show that
. In fact, since
is
–inverse-strongly monotone,
is an
-Lipschitz continuous monotone mapping and
. It follows from Lemma 2.4 that
is maximal monotone. Let
, that is,
. Again since
, we have
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ41_HTML.gif)
By the maximal monotonicity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ42_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ43_HTML.gif)
It follows from ,
and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ44_HTML.gif)
Since is maximal monotone, this implies that
that is,
. Hence,
. Since
, we have
. It implies that
is the unique solution of the variational inequality (3.4).
Now we prove the following theorem which is the main result of this paper.
Theorem 3.2.
Let be a real Hilbert space, let
be a bifunction from
satisfying (A1)–(A4) and let
be a sequence of nonexpansive mappings on
. Let
be an
-inverse-strongly monotone mapping,
be a maximal monotone mapping such that
Let
be a contraction of
into itself with a constant
and let
be a strongly bounded linear operator on
with coefficient
and
. Let
,
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ45_HTML.gif)
for all , where
,
and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ46_HTML.gif)
Suppose that for any bounded subset
of
. Let
be a mapping of
into itself defined by
and suppose that
. Then,
,
and
converges strongly to
, where
is a unique solution of the variational inequalities (3.4).
Proof.
Since , we may assume that
for all
. First we will prove that
is bonded. Let
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ47_HTML.gif)
It follows from (3.25) and induction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ48_HTML.gif)
Hence is bounded and therefore
,
and
are also bounded. Next, we show that
. Since
is nonexpansive, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ49_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ50_HTML.gif)
where . On the other hand, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ51_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ52_HTML.gif)
Putting in (3.29) and
in (3.30),
and
By (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ53_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ54_HTML.gif)
Since we assume that there exists a real number
such that
for all
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ55_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ56_HTML.gif)
where . From (3.28), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ57_HTML.gif)
Since is bounded, it follows that
. Hence, by Lemma 2.3, we have
as
. From (3.34) and
, we have
. Moreover, we have from (3.27) that
.
We note from (3.23) that . Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ58_HTML.gif)
Since ,
and
, we get
. From the proof of Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ59_HTML.gif)
for all . Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ60_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ61_HTML.gif)
Since is bounded,
and
, it follows that
as
Put . It follows from (3.38), the nonexpansive of
and the inverse strongly monotonicity of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ62_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ63_HTML.gif)
Since and
, we have
as
.From (3.5), (3.15) and (3.38), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ64_HTML.gif)
Thus, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ65_HTML.gif)
Since ,
and
, we have
as
. It follows from the inequality
that
as
. Moreover, we note that
as
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ66_HTML.gif)
for all , it follows that
. Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ67_HTML.gif)
where is a unique solution of the variational inequality (3.4). To show this inequality, we choose a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ68_HTML.gif)
Since is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we can assume that
From
we obtain
. Let us show
. It follows by (3.23) and (A2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ69_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ70_HTML.gif)
Since and
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%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ71_HTML.gif)
and hence . From (A3), we have
for all
and hence
By the same argument as in proof of Theorem 3.1, we have
and hence
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ72_HTML.gif)
Finally we prove that . From (3.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ73_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ74_HTML.gif)
where and
. It easily verified that
,
and
. Hence, by Lemma 2.1, the sequence
converges strongly to
.
As in [10, Theorem  4.1], we can generate a sequence of nonexpansive mappings satisfying condition
. for any bounded
of
by using convex combination of a general sequence
of nonexpansive mappings with a common fixed point.
Corollary 3.3.
Let be a real Hilbert space, let
be a bifunction from
satisfying (A1)–(A4) and let
be an
-inverse-strongly monotone mapping,
be a maximal monotone mapping. Let
be a contraction of
into itself with a constant
and let
be a strongly bounded linear operator on
with coefficient
and
. Let
be a family of nonnegative numbers with indices
with
such that
(i)
(ii)
(iii)
Let be a sequence of nonexpansive mappings on
with
and let
,
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ75_HTML.gif)
for all , where
,
and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ76_HTML.gif)
Then, ,
and
converges strongly to
in
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ77_HTML.gif)
If and
in Theorem 3.2, we obtain the following corollary.
Corollary 3.4 (see Peng et al. [9]).
Let be a real Hilbert space, let
be a bifunction from
satisfying (A1)–(A4) and let
be a nonexpansive mapping on
. Let
be an
-inverse-strongly monotone mapping,
be a maximal monotone mapping such that
Let
be a contraction of
into itself with a constant
. Let
,
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ78_HTML.gif)
for all , where
,
and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ79_HTML.gif)
Then, ,
and
converges strongly to
, where
.
If and
in Theorem 3.2, we obtain the following corollary.
Corollary 3.5 (see S. Plubtieng and R. Punpaeng [6]).
Let be a real Hilbert space, let
be a bifunction from
satisfying (A1)–(A4) and let
be a nonexpansive mapping on
such that
Let
be a contraction of
into itself with a constant
and let
be a strongly bounded linear operator on
with coefficient
and
. Let
,
and be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ80_HTML.gif)
for all , where
and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ81_HTML.gif)
Then, and
converges strongly to a point
in
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F567147/MediaObjects/13663_2009_Article_1155_Equ82_HTML.gif)
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Acknowledgments
The first author thank the National Research Council of Thailand to Naresuan University, 2009 for the financial support. Moreover, the second author would like to thank the "National Centre of Excellence in Mathematics", PERDO, under the Commission on Higher Education, Ministry of Education, Thailand.
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Plubtieng, S., Sriprad, W. A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions, Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces. Fixed Point Theory Appl 2009, 567147 (2009). https://doi.org/10.1155/2009/567147
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DOI: https://doi.org/10.1155/2009/567147