- Research Article
- Open access
- Published:
Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems
Fixed Point Theory and Applications volume 2010, Article number: 642584 (2010)
Abstract
We consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point such that
, for all
, where
,
are two monotone operators and
is the solution of the equilibrium problem of finding
such that
, for all
. We note that the problem (HEVP) includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving (HEVP), we propose a double-net algorithm which generates a net
. We prove that the net
hierarchically converges to the solution of (HEVP); that is, for each fixed
, the net
converges in norm, as
, to a solution
of the equilibrium problem, and as
, the net
converges in norm to the unique solution
of (HEVP).
1. Introduction
Let be a real Hilbert space with inner product
and norm
, respectively, and let
be a nonempty closed convex subset of
. Recall that a mapping
of
into
is called monotone if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ1_HTML.gif)
for all and
is called
-inverse strongly monotone mapping if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ2_HTML.gif)
for all . It is obvious that any
-inverse strongly monotone mapping
is monotone and
-Lipschitz continuous.
Recently, the following problem has attracted much attention: find hierarchically a fixed point of a nonexpansive mapping with respect to a nonexpansive mapping
, namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ3_HTML.gif)
Some algorithms for solving the hierarchical fixed point problem (1.3) have been introduced by many authors. For related works, please see, for instance, [1–9] and the references therein.
Remark 1.1.
It is not hard to check that solving (1.3) is equivalent to the fixed point problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ4_HTML.gif)
where stands for the metric projection on the closed convex set
. By using the definition of the normal cone to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ5_HTML.gif)
we easily prove that (1.3) is equivalent to the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ6_HTML.gif)
At this point, we wish to point out the link with some monotone variational inequalities and convex programming problems as follows.
Example 1.2.
Setting , where
is
-Lipschitzian and
-strongly monotone with
, then (1.3) reduces to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ7_HTML.gif)
a variational inequality studied by Yamada and Ogura [10].
Example 1.3.
Let be a maximal monotone operator. Taking
and
, where
is a convex function such that
is
-Lipschitzian (which is equivalent to the fact that
is
cocoercive) with
, and
. Then (1.3) reduces to the following mathematical program with generalized equation constraint:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ8_HTML.gif)
a problem considered by Luo et al. [11].
Example 1.4.
Taking , where
is the subdifferential of a lower semicontinuous convex function, then (1.8) reduces to the following hierarchical minimization problem considered in Cabot [12] and Solodov [13]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ9_HTML.gif)
Let be a nonlinear mapping, and let
be a bifunction of
into
. Consider the following equilibrium problem of finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ10_HTML.gif)
If , then (1.10) reduces to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ11_HTML.gif)
The solution set of equilibrium problems (1.10) and (1.11) are denoted by and
, respectively. The equilibrium problem (1.10) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, fixed point problems, minimax problems, Nash equilibrium problem in noncooperative games, and others. We remind the readers to refer to [14–30] and the references therein.
Motivated and inspired by the above works, in this paper, we consider the following hierarchical equilibrium problem and variational inequality problem: find a point such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ12_HTML.gif)
where are two monotone operators. The solution set of (1.12) is denoted by
.
Remark 1.5.
It is clear that the hierarchical variational inequality problem and equilibrium problem (1.12) includes the variational inequality problem studied by Yamada and Ogura [10], mathematical program studied by Luo et al. [11], hierarchical minimization problem considered by Cabot [12] and Solodov [13], as special cases.
For solving (1.12), we propose a double-net algorithm which generates a net . We prove that the net
hierarchically converges to the solution of (1.12); that is, for each fixed
, the net
converges in norm, as
, to a solution
of the equilibrium problem, and as
, the net
converges in norm to the unique solution
of (1.12).
2. Preliminaries
Let be a real Hilbert space. Throughout this paper, let us assume that a bifunction
  satisfies the following conditions:
(F1) for all
;
(F2) is monotone, that is,
for all
;
(F3) for each ,
;
(F4) for each ,
is convex and lower semicontinuous.
On the equilibrium problems, we have the following important lemma. You can find it in [31].
Lemma 2.1.
Let be a real Hilbert space, and let
be a bifunction of
into
satisfying conditions (F1)–(F4). Let
, and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ13_HTML.gif)
Further, if , 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%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ14_HTML.gif)
(3);
(4) is closed and convex.
Below we gather some basic facts that are needed in the argument of the subsequent sections.
Lemma 2.2 (see [32]).
Let be a real Hilbert space. Let the mapping
be α-inverse strongly monotone, and let
be a constant. Then, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ15_HTML.gif)
In particular, if , then
is nonexpansive.
Lemma 2.3 (demiclosedness principle for nonexpansive mappings, see [33]).
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a nonexpansive mapping with
. If
is a sequence in
weakly converging to
, and if
converges strongly to
, then
; in particular, if
, then
.
Lemma 2.4.
Let be a real Hilbert space. Let
be a
-contraction with coefficient
. Let the mapping
be α-inverse strongly monotone. Let
, and
. Then the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ16_HTML.gif)
is equivalent to the dual variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ17_HTML.gif)
Proof.
Assume that solves (2.4). For all
, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ18_HTML.gif)
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ19_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ20_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ21_HTML.gif)
Letting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ22_HTML.gif)
which is exactly (2.5).
Assume that solves (2.5). Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ23_HTML.gif)
Noting that and
are monotone, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ24_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ25_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ26_HTML.gif)
This implies that solves (2.4). The proof is completed.
3. Main Results
In this section, we first introduce our double-net algorithm.
Let be a real Hilbert space. Let
be a
-contraction with coefficient
. Let the mappings
be
-inverse strongly monotone and
-inverse strongly monotone, respectively. Let
be a bifunction from
  , and let
and
be two constants. For
, we define the following mapping:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ27_HTML.gif)
where is defined by Lemma 2.1. We note that the mapping
is a contraction. As a matter of fact, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ28_HTML.gif)
which implies that the mapping is contractive. Hence, by Banach's contraction principle,
has a unique fixed point which is denoted
; that is,
is the unique solution in
of the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ29_HTML.gif)
Below is our main result of this paper which displays the behavior of the net as
and
successively.
Theorem 3.1.
Let be a real Hilbert space. Let
be a
-contraction with coefficient
. Let the mappings
be α-inverse strongly monotone and β-inverse strongly monotone, respectively. Let
and
be two constants. Let
be a bifunction from
satisfying (F1)–(F4). Suppose the solution set
of (1.12) is nonempty. Let, for each
,
be defined implicitly by (3.3). Then, the net
hierarchically converges to the unique solution
of the hierarchical equilibrium problem and variational inequality problem (1.12). That is to say, for each fixed
, the net
converges in norm, as
, to a solution
of the equilibrium problem (1.10). Moreover, as
, the net
converges in norm to the unique solution
. Furthermore,
also solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ30_HTML.gif)
We divide our detailed proofs into several conclusions as follows. Throughout, we assume all assumptions of Theorem 3.1 are satisfied.
Conclusion.
For each fixed , the net
is bounded.
Proof.
Take any . It is clear that
. Set
for all
. Since
,
and
are nonexpansive (by Lemmas 2.1 and 2.2), we have from (3.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ31_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ32_HTML.gif)
It follows that for each fixed ,
is bounded, so are the nets
,
and
. Note that we use
as a positive constant which bounds all bounded terms appearing in the following.
Conclusion.
as
.
Proof.
From Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ33_HTML.gif)
By (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ34_HTML.gif)
This together with (3.7) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ35_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ36_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ37_HTML.gif)
Using Lemma 2.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ38_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ39_HTML.gif)
From (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ40_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ41_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ42_HTML.gif)
Next, we show that, for each fixed , the net
is relatively norm-compact as
. It follows from (3.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ43_HTML.gif)
It turns out that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ44_HTML.gif)
Assume that is such that
as
. By (3.18), we conclude immediately that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ45_HTML.gif)
Since is bounded, without loss of generality, we may assume that as
,
converges weakly to a point
. Note that
also converges weakly to a point
.
Now we show that . Since
, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ46_HTML.gif)
From the monotonicity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ47_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ48_HTML.gif)
Put for all
and
. From (3.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ49_HTML.gif)
Note that . Further, from monotonicity of
, we have
. Letting
in (3.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ50_HTML.gif)
From (F1), (F4), and (3.24), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ51_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ52_HTML.gif)
Letting in (3.26), we have, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ53_HTML.gif)
This implies that .
We can then substitute for
in (3.19) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ54_HTML.gif)
Consequently, the weak convergence of to
actually implies that
strongly. This has proved the relative norm-compactness of the net
as
.
Now we return to (3.19) and take the limit, as , to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ55_HTML.gif)
In particular, solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ56_HTML.gif)
or the equivalent dual variational inequality (see Lemma 2.4)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ57_HTML.gif)
Notice that (3.31) is equivalent to the fact that . That is,
is the unique element in
of the contraction
. Clearly, this is sufficient to conclude that the entire net
converges in norm to
as
.
Conclusion.
The net is bounded.
Proof.
In (3.31), we take any to deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ58_HTML.gif)
By virtue of the monotonicity of and the fact that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ59_HTML.gif)
It follows from (3.32) and (3.33) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ60_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ61_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ62_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ63_HTML.gif)
which implies that is bounded.
Conclusion.
The net which solves the variational inequality VI (3.4).
Proof.
First, we note that the solution of the variational inequality VI (3.4) is unique.
We next prove that ; namely, if
is a null sequence in
such that
weakly as
, then
. To see this, we use (3.31) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ64_HTML.gif)
However, since is monotone,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ65_HTML.gif)
Combining the last two relations yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ66_HTML.gif)
Letting as
in (3.40), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ67_HTML.gif)
which is equivalent to its dual variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ68_HTML.gif)
Namely, is a solution of VI (1.12); hence,
.
We further prove that , the unique solution of VI (3.4). As a matter of fact, we have by (3.36)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ69_HTML.gif)
Therefore, the weak convergence to of
implies that
in norm. Now we can let
in (3.36) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ70_HTML.gif)
It turns out that solves VI (3.4). By uniqueness, we have
. This is sufficient to guarantee that
in norm, as
. The proof is complete.
Proof.
By Conclusions 1–4, the proof of Theorem 3.1 is completed.
Take . Then (1.12) reduces to the following: find a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ71_HTML.gif)
The solution of (3.45) is denoted by .
Corollary 3.2.
Let be a real Hilbert space. Let
be a
-contraction with coefficient
. Let the mapping
be α-inverse strongly monotone. Let
be a constant. Let
be a bifunction from
satisfying (F1)–(F4). Suppose the solution set
is nonempty. Let, for each
,
be defined implicitly by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ72_HTML.gif)
Then, the net hierarchically converges to the unique solution
of the hierarchical equilibrium problem and variational inequality problem (3.45). That is to say, for each fixed
, the net
converges in norm, as
, to a solution
of the equilibrium problem (1.11). Moreover, as
, the net
converges in norm to the unique solution
. Furthermore,
solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ73_HTML.gif)
Taking in Theorem 3.1, we have the following corollary.
Corollary 3.3.
Let be a real Hilbert space. Let
be a
-contraction with coefficient
. Let the mapping
be β-inverse strongly monotone. Let
be a constant. Let
be a bifunction from
satisfying (F1)–(F4). Suppose that the solution set
of (1.10) is nonempty. Let, for each
,
be defined implicitly by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ74_HTML.gif)
Then, the net hierarchically converges to the unique solution
of the equilibrium problem (1.10). That is to say, for each fixed
, the net
converges in norm, as
, to a solution
of the equilibrium problem (1.10). Moreover, as
, the net
converges in norm to the unique solution
. Furthermore,
solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ75_HTML.gif)
Taking in Theorem 3.1, we have the following corollary.
Corollary 3.4.
Let be a real Hilbert space. Let
be a
-contraction with coefficient
. Let
be a bifunction from
satisfying (F1)–(F4). Suppose the solution set
of (1.11) is nonempty. Let, for each
,
be defined implicitly by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ76_HTML.gif)
Then, the net hierarchically converges to the unique solution
of the equilibrium problem (1.11). That is to say, for each fixed
, the net
converges in norm, as
, to a solution
of the equilibrium problem (1.11). Moreover, as
, the net
converges in norm to the unique solution
. Furthermore,
solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F642584/MediaObjects/13663_2010_Article_1320_Equ77_HTML.gif)
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Acknowledgment
The work of the second author was partially supported by the Grant NSC 98-2923-E-110-003-MY3 and the work of the third author was partially supported by the Grant NSC 98-2221-E-110-064.
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Yao, Y., Liou, YC. & Chen, CP. Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems. Fixed Point Theory Appl 2010, 642584 (2010). https://doi.org/10.1155/2010/642584
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DOI: https://doi.org/10.1155/2010/642584