- Research Article
- Open access
- Published:
An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems
Fixed Point Theory and Applications volume 2009, Article number: 531308 (2009)
Abstract
We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for -inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. (2007).
1. Introduction
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a bifunction, where
is the set of real numbers. Let
be a nonlinear mapping. The generalized equilibrium problem (GEP) for
and
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ1_HTML.gif)
The set of solutions for the problem (1.1) is denoted by , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ2_HTML.gif)
If in (1.1), then GEP(1.1) reduces to the classical equilibrium problem (EP) and
is denoted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ3_HTML.gif)
If in (1.1), then GEP(1.1) reduces to the classical variational inequality and
is denoted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ4_HTML.gif)
It is well known that GEP(1.1) contains as special cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalities (see, e.g., [1–6] and the reference therein).
A mapping is called
-inverse-strongly monotone [7], if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ5_HTML.gif)
for all . It is obvious that any
-inverse-strongly monotone mapping
is monotone and Lipschitz continuous. A mapping
is called nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ6_HTML.gif)
for all . We denote by
the set of fixed points of
, that is,
. If
is bounded, closed and convex and
is a nonexpansive mappings of
into itself, then
is nonempty (see [8]).
In 1997, Flåm and Antipin [9] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. In 2003, Iusem and Sosa [10] presented some iterative algorithms for solving equilibrium problems in finite-dimensional spaces. They have also established the convergence of the algorithms. Recently, Huang et al. [11] studied the approximate method for solving the equilibrium problem and proved the strong convergence theorem for approximating the solutions of the equilibrium problem.
On the other hand, for finding an element of , Takahashi and Toyoda [12] introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ7_HTML.gif)
where ,
is metric projection of
onto
,
is a sequence in
and
is a sequence in
. Further, Iiduka and Takahashi [13] introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ8_HTML.gif)
where , and proved the strong convergence theorems for iterative scheme (1.8) under some conditions on parameters. In 2007, S. Takahashi and W. Takahashi [14] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also proved a strong convergence theorem which is connected with Combettes and Hirstoaga's result [3] and Wittmann's result [15]. Tada and W. Takahashi [16] introduced the Mann type iterative algorithm for finding a common element of the set of solutions of the
and the set of common fixed points of nonexpansive mapping and obtained the weak convergence of the Mann type iterative algorithm. Yao et al. [17] introduced an iteration process for finding a common element of the set of solutions of the
and the set of common fixed points of infinitely many nonexpansive mappings in Hilbert spaces. They proved a strong-convergence theorem under mild assumptions on parameters. Very recently, Moudafi [18] proposed an iterative algorithm for finding a common element of
, where
is an
-inverse-strongly monotone mapping, and obtained a weak convergence theorem. There are some related works, we refer to [19–22] and the references therein.
Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative process for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP(1.1), and the solution set of the variational inequality problem for an -inverse-strongly monotone mapping in real Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. [17].
2. Preliminaries
Let be a real Hilbert space with inner product
and norm
, and let
be a closed convex subset of
. Then, for any
, there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ9_HTML.gif)
is called the metric projection of
onto
. It is well known that
is a nonexpansive mapping and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ10_HTML.gif)
for all . Furthermore,
is characterized by the following properties:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ11_HTML.gif)
for all and
. It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ12_HTML.gif)
where is a parameter in
.
A set-valued mapping is called monotone if for all
,
and
imply
. A monotone mapping
is maximal if the graph
of
is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping
is maximal if and only if for
,
for all
implies
. Let
be a monotone,
-Lipschitz continuous mappings and let
be the normal cone to
at
, that is,
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ13_HTML.gif)
Then is the maximal monotone and
if and only if
; see [23].
Let be a sequence of nonexpansive mappings of
into itself and let
be a sequence of nonnegative numbers in
. For any
, define a mapping
of
into itself as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ14_HTML.gif)
Such a mapping is called the
-mapping generated by
and
see [24]. It is obvious that
is nonexpansive and if
then
.
Lemma 2.1 (see [24]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive mappings of
into itself such that
and let
be a sequence in
for some
. Then, for every
and
, the limit
exists.
Remark 2.2 (see [17]).
It can be known from Lemma 2.1 that if is a nonempty bounded subset of
, then for
, there exists
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ15_HTML.gif)
Using Lemma 2.1, one can define a mapping of
into itself as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ16_HTML.gif)
for every . Such a mapping
is called the
-mapping generated by
and
Since
is nonexpansive,
is also nonexpansive. If
is a bounded sequence in
, then we put
. Hence, it is clear from Remark 2.2 that for an arbitrary
there exists
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ17_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ18_HTML.gif)
Since and
are nonexpansive, we deduce that, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ19_HTML.gif)
for some constant .
Lemma 2.3 (see [24]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive mappings of
into itself such that
and let
be a sequence in
for some
. Then,
.
For solving the generalized equilibrium problem, we assume that the bifunction 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 [1].
Lemma 2.4 (see [1]).
Let be a nonempty closed convex subset of
and let
be a bifunction from
into
satisfying (a1)–(a4). Let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ20_HTML.gif)
The following lemma was also given in [3].
Lemma 2.5 (see [3]).
Assume that satisfies (a1)–(a4). For
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ21_HTML.gif)
for all . Then, the following hold:
(b1) is single-valued;
(b2) is firmly nonexpansive, that is, for any
,
(b3);
(b4) is closed and convex.
Remark 2.6.
Replacing with
in (2.12), then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ22_HTML.gif)
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.7 (see [25]).
Let and
be bounded sequences in Banach space
and let
be a sequence in
. Suppose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ23_HTML.gif)
for all integers . If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ24_HTML.gif)
then .
Lemma 2.8 (see [26]).
Assume is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ25_HTML.gif)
where is a sequence in (0,1) and
is a sequence in
such that
(1);
(2) or
.
Then .
3. Main Results
In this section, we deal with an iterative scheme by the approximation method for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP(1.1), and the solution set of the variational inequality problem for an -inverse-strongly monotone mapping in real Hilbert spaces.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
into
satisfying (a1)–(a4),
an inverse-strongly monotone mapping with constant
,
an inverse-strongly monotone mapping with constant
,
a contraction mapping with constant
. Let
be a
-mapping generated by
and
and
, where sequence
is nonexpansive and
is a sequence in
for some
. For
, suppose that
,
and
are generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ26_HTML.gif)
for all , where
,
, and
are three sequences in
,
is a sequence in
for some
and
for some
satisfying
(i);
(ii) and
;
(iii);
(iv) and
;
(v) and
.
Then ,
, and
converge strongly to the point
, where
.
Proof.
For any and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ27_HTML.gif)
which implies that is nonexpansive. Remark 2.6 implies that the sequences
and
are well defined. In view of the iterative sequence (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ28_HTML.gif)
It follows from Lemma 2.5 that for all
. Let
. For each
, we have
. By Lemma 2.5,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ29_HTML.gif)
and so (3.2) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ30_HTML.gif)
For , we have
from (2.4). Since
is a nonexpansive mapping and
is an inverse-strongly monotone mapping with constant
, by (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ31_HTML.gif)
Thus, (3.5) and (3.6) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ32_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ33_HTML.gif)
This implies that is bounded. Therefore,
,
,
,
and
are also bounded.
From and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ35_HTML.gif)
Putting in (3.9) and
in (3.10), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ36_HTML.gif)
Adding the above two inequalities, the monotonicity of implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ37_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ38_HTML.gif)
It follows from (3.2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ39_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ40_HTML.gif)
From (3.1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ41_HTML.gif)
Putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ42_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ43_HTML.gif)
Obviously, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ44_HTML.gif)
From (2.11) and (3.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ45_HTML.gif)
for some constant . Combining (3.15), (3.19), and (3.20), we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ46_HTML.gif)
It is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ47_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ48_HTML.gif)
Thus, by Lemma 2.7, we obtain . It then follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ49_HTML.gif)
By (3.15) and (3.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ50_HTML.gif)
Since , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ51_HTML.gif)
On the other hand, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ52_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ53_HTML.gif)
It is easy to see that and hence
.
From (3.5) and (3.6), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ54_HTML.gif)
Since , without loss of generality, we may assume that there exists a real number
such that
for all
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ55_HTML.gif)
Since ,
and
is bounded, (3.30) implies that
as
. From (2.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ56_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ57_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ58_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ59_HTML.gif)
Since ,
,
, and the sequences
,
and
are bounded, it follows from (3.34) that
. On the other hand, from (3.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ60_HTML.gif)
The same as in (3.30), we have as
. Likewise, using (3.5), we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ61_HTML.gif)
The same as in (3.34), we have . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ62_HTML.gif)
we get as
. From (2.10) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ63_HTML.gif)
we get .
Next, we show , where
. To show this inequality, we choose a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ64_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
. We now show that
. Indeed, we observe that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ65_HTML.gif)
From (a2), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ66_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ67_HTML.gif)
Form , we get
. Put
for all
and
. Consequently, we get
. From (3.42), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ68_HTML.gif)
From the Lipschitz continuous of and
, we obtain
. Since
is monotone, we know that
. Further,
. It follows from (a4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ69_HTML.gif)
Owing to (a1) and (a4), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ70_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ71_HTML.gif)
Letting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ72_HTML.gif)
This implies that .
Furthermore, we prove that . Assume
, since
, we have
. From Opial's theorem [27], we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ73_HTML.gif)
This is a contradiction. Hence, .
Now, we will show that . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ74_HTML.gif)
Then is a maximal monotone [23]. Let
, since
and
, we have
. From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ75_HTML.gif)
This is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ76_HTML.gif)
Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ77_HTML.gif)
Noting that and
is Lipschitz continuous, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ78_HTML.gif)
Since is maximal monotone, we have
and so
. Thus,
. The property of the metric projection implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ79_HTML.gif)
From (3.1) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ80_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ81_HTML.gif)
Setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ82_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ83_HTML.gif)
Applying Lemma 2.8 to (3.56), we conclude that converges strongly to
. Consequently,
and
converge strongly to
. This completes the proof.
As direct consequences of Theorem 3.1, we have the following two corollaries.
Corollary 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a contraction mapping with Lipschitz constant
and let
be an inverse-strongly monotone mapping with constant
. Suppose
and
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ84_HTML.gif)
for all , where
,
,
are three sequences in
,
is a sequence in
for some
satisfying conditions (i)–(iv). Then
converges strongly to the point
, where
.
Proof.
Let and
for all
and
in Theorem 3.1. Then
for
Letting
(the identity mapping) for all
, then
for
It is easy to see that all conditions of Theorem 3.1 hold. Therefore, we know that the sequence
generated by (3.59) converges strongly to
. This completes the proof.
Remark 3.3.
From Corollary 3.2, we can get an iterative scheme for finding the solution of the variational inequality involving the -inverse-strongly monotone mapping
.
Corollary 3.4 (see [17, Theorem  3.5]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
into
satisfying (a1)–(a4),
a contraction mapping with constant
. Let
be an
-mapping generated by
and
and
, where sequence
is nonexpansive and
is a sequence in
for some
. Suppose
and
,
are generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F531308/MediaObjects/13663_2009_Article_1153_Equ85_HTML.gif)
for all , where
,
,
are three sequences in
, and
is a sequence in
satisfying conditions (i)–(iii) and (v). Then, the sequences
and
converge strongly to the point
, where
.
Proof.
Let for
and
and
for all
in Theorem 3.1. Since
, we get that
. It follows from Theorem 3.1 that the sequences
and
converge strongly to the point
. This completes the proof.
Remark 3.5.
The main result of Yao et al. [17, Corollary  3.2] improved and extended the corresponding theorems in Combettes and Hirstoaga [3] and S. Takahashi and W. Takahashi [14]. Our Theorem 3.1 improves and extends Theorem  3.5 of Yao et al. [17] in the following aspects:
(1)the equilibrium problem is extended to the generalized equilibrium problem;
(2)our iterative process (3.1) is different from Yao et al. iterative process (3.60) because there are a project operator and an -inverse-strongly monotone mapping;
(3)our iterative process (3.1) is more general than Yao et al. iterative process (3.60) because it can be applied to solving the problem of finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for -inverse-strongly monotone mapping.
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Acknowledgments
This work was supported by the National Natural Science Foundation of China (50674078, 50874096, 10671135, 70831005) and the Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).
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Liu, Qy., Zeng, Wy. & Huang, Nj. An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems. Fixed Point Theory Appl 2009, 531308 (2009). https://doi.org/10.1155/2009/531308
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DOI: https://doi.org/10.1155/2009/531308