- Research Article
- Open access
- Published:
A General Iterative Approach to Variational Inequality Problems and Optimization Problems
Fixed Point Theory and Applications volume 2011, Article number: 284363 (2011)
Abstract
We introduce a new general iterative scheme for finding a common element of the set of solutions of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem. Applications of the main result are also given.
1. Introduction
Let be a real Hilbert space with inner product
and induced norm
. Let
be a nonempty closed convex subset of
and
be self-mapping on
. We denote by
the set of fixed points of
and by
the metric projection of
onto
.
Let be a nonlinear mapping of
into
. The variational inequality problem is to find a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ1_HTML.gif)
We denote the set of solutions of the variational inequality problem (1.1) by . The variational inequality problem has been extensively studied in the literature; see [1–5] and the references therein.
Recently, in order to study the problem (1.1) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.1) and the set of fixed points of nonexpansive mappings; see [6–9] and the references therein. In particular, in 2005, Iiduka and Takahashi [8] introduced an iterative scheme for finding a common point of the set of fixed points of a nonexapansive mapping and the set of solutions of the problem (1.1) for an inverse-strong monotone mapping
:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ2_HTML.gif)
where and
. They proved that the sequence generated by (1.2) strongly converges strongly to
. In 2010, Jung [10] provided the following new composite iterative scheme for the fixed point problem and the problem (1.1):
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ3_HTML.gif)
where is a contraction with constant
,
,
, and
. He proved that the sequence
generated by (1.3) strongly converges strongly to a point in
, which is the unique solution of a certain variational inequality.
On the other hand, the following optimization problem has been studied extensively by many authors:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ4_HTML.gif)
where ,
are infinitely many closed convex subsets of
such that
,
,
is a real number,
is a strongly positive bounded linear operator on
(i.e., there is a constant
such that
, for all
), and
is a potential function for
(i.e.,
for all
). For this kind of optimization problems, see, for example, Deutsch and Yamada [11], Jung [10], and Xu [12, 13] when
and
for a given point
in
.
In 2007, related to a certain optimization problem, Marino and Xu [14] introduced the following general iterative scheme for the fixed point problem of a nonexpansive mapping:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ5_HTML.gif)
where and
. They proved that the sequence
generated by (1.5) converges strongly to the unique solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ6_HTML.gif)
which is the optimality condition for the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ7_HTML.gif)
where is a potential function for
. The result improved the corresponding results of Moudafi [15] and Xu [16].
In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem (1.1) for an inverse-strongly monotone mapping and the set of fixed points of a nonexapansive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem. Applications of the main result are also discussed. Our results improve and complement the corresponding results of Chen et al. [6], Iiduka and Takahashi [8], Jung [10], and others.
2. Preliminaries and Lemmas
Let be a real Hilbert space and let
be a nonempty closed convex subset of
. We write
to indicate that the sequence
converges weakly to
.
implies that
converges strongly to
.
First we recall that a mapping is a contraction on
if there exists a constant
such that  
,
. A mapping
is called nonexpansive if
. We denote by
the set of fixed points 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%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ8_HTML.gif)
for all .
is called the metric projection of
onto
. It is well known that
is nonexpansive and
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ9_HTML.gif)
for every . Moreover,
is characterized by the properties:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ10_HTML.gif)
In the context of the variational inequality problem for a nonlinear mapping , this implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ11_HTML.gif)
It is also well known that satisfies the Opial condition, that is, for any sequence
with
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ12_HTML.gif)
holds for every with
.
A mapping of
into
is called inverse-strongly monotone if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ13_HTML.gif)
for all ; see [4, 7, 17]. For such a case,
is called
-inverse-strongly monotone. We know that if
, where
is a nonexpansive mapping of
into itself and
is the identity mapping of
, then
is
-inverse-strongly monotone and
. A mapping
of
into
is called strongly monotone if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ14_HTML.gif)
for all . In such a case, we say
is
-strongly monotone. If
is
-strongly monotone and
-Lipschitz continuous, that is,
for all
, then
is
-inverse-strongly monotone. If
is an
-inverse-strongly monotone mapping of
into
, then it is obvious that
is
-Lipschitz continuous. We also have that for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ15_HTML.gif)
So, if , then
is a nonexpansive mapping of
into
. The following result for the existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda [9].
Proposition 2.1.
Let   be a bounded closed convex subset of a real Hilbert space and let
be an
-inverse-strongly monotone mapping of
into
. Then,
is nonempty.
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 mapping. It is known that a monotone mapping
is maximal if and only if for
,
for every
implies
. Let
be an inverse-strongly monotone mapping of
into
and let
be the normal cone to
at
, that is,
, and define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ16_HTML.gif)
Then is maximal monotone and
if and only if
; see [18, 19].
We need the following lemmas for the proof of our main results.
Lemma 2.2.
In a real Hilbert space , there holds the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ17_HTML.gif)
for all .
Lemma 2.3 (Xu [12]).
Let be a sequence of nonnegative real numbers satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ18_HTML.gif)
where and
satisfy the following conditions:
(i) and
or, equivalently,
;
(ii) or
;
(iii).
Then .
Lemma 2.4 (Marino and Xu [14]).
Assume that is a strongly positive linear bounded operator on a Hilbert space
with constant
and
. Then
.
The following lemma can be found in [20, 21] (see also Lemma 2.2 in [22]).
Lemma 2.5.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a proper lower semicontinunous differentiable convex function. If
is a solution to the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ19_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ20_HTML.gif)
In particular, if solves the optimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ21_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ22_HTML.gif)
where is a potential function for
.
3. Main Results
In this section, we present a new general composite iterative scheme for inverse-strongly monotone mappings and a nonexpansive mapping.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
such that
. Let
be an
-inverse-strongly monotone mapping of
into
and
a nonexpansive mapping of
into itself such that
. Let
and let
be a strongly positive bounded linear operator on
with constant
and
a contraction of
into itself with constant
. Assume that
and
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ23_HTML.gif)
where ,
, and
. Let
,
, and
satisfy the following conditions:
(i);
;
(ii) for all
and for some
;
(iii) for some
,
with
;
(iv),
,
.
Then converges strongly to
, which is a solution of the optimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ24_HTML.gif)
where is a potential function for
.
Proof.
We note that from the control condition (i), we may assume, without loss of generality, that . Recall that if
is bounded linear self-adjoint operator on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ25_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ26_HTML.gif)
which is to say that is positive. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ27_HTML.gif)
Now we divide the proof into several steps.
Step 1.
We show that is bounded. To this end, let
and
for every
. Let
. Since
is nonexpansive and
from (2.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ28_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ29_HTML.gif)
Now, set . Let
. Then, from (IS) and (3.4), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ30_HTML.gif)
From (3.5) and (3.6), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ31_HTML.gif)
By induction, it follows from (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ32_HTML.gif)
Therefore, is bounded. So
,
,
,
,
,
, and
are bounded. Moreover, since
and
,
and
are also bounded. And by the condition (i), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ33_HTML.gif)
Step 2.
We show that and
. Indeed, since
and
are nonexpansive and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ34_HTML.gif)
Similarly, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ35_HTML.gif)
Simple calculations show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ36_HTML.gif)
So, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ37_HTML.gif)
Also observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ38_HTML.gif)
By (3.11), (3.13), and (3.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ39_HTML.gif)
where ,
, and
. From the conditions (i) and (iv), it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ40_HTML.gif)
Applying Lemma 2.3 to (3.15), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ41_HTML.gif)
Moreover, by (3.10) and (3.13), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ42_HTML.gif)
Step 3.
We show that and
. Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ43_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ44_HTML.gif)
Obviously, by (3.9) and Step 2, we have as
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ45_HTML.gif)
By (3.9) and (3.21), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ46_HTML.gif)
Step 4.
We show that and
. To this end, let
. Since
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ47_HTML.gif)
So we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ48_HTML.gif)
Since from the condition (i) and
from Step 3, we have
. Moreover, from (2.4) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ49_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ50_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ51_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ52_HTML.gif)
Since ,
and
, we get
. Also by (3.21)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ53_HTML.gif)
Step 5.
We show that . In fact, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ54_HTML.gif)
from (3.9) and (3.29), we have .
Step 6.
We show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ55_HTML.gif)
where is a solution of the optimization problem (OP1). First we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ56_HTML.gif)
Since is bounded, we can choose a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ57_HTML.gif)
Without loss of generality, we may assume that converges weakly to
.
Now we will show that . First we show that
. Assume that
. Since
and
, by the Opial condition and Step 5, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ58_HTML.gif)
which is a contradiction. Thus we have .
Next, let us show that . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ59_HTML.gif)
Then is maximal monotone. Let
. Since
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ60_HTML.gif)
On the other hand, from , we have
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ61_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ62_HTML.gif)
Since in Step 4 and
is
-inverse-strongly monotone, we have
as
. Since
is maximal monotone, we have
and hence
.
Therefore, . Now from Lemma 2.5 and Step 5, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ63_HTML.gif)
By (3.9) and (3.39), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ64_HTML.gif)
Step 7.
We show that and
, where
is a solution of the optimization problem (OP1). Indeed from (IS) and Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ65_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ66_HTML.gif)
where ,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ67_HTML.gif)
From (i), in Steps 3, and 6, it is easily seen that
,
, and
. Hence, by Lemma 2.3, we conclude
as
. This completes the proof.
As a direct consequence of Theorem 3.1, we have the following results.
Corollary 3.2.
Let , and
be as in Theorem 3.1. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ68_HTML.gif)
where and
. Let
and
satisfy the conditions (i), (ii), and (iv) in Theorem 3.1. Then
converges strongly to
, which is a solution of the optimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ69_HTML.gif)
where is a potential function for
.
Corollary 3.3.
Let ,
,
,
,
,
,
,
,
, and
be as in Theorem 3.1. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ70_HTML.gif)
where ,
, and
. Let
,
and
satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then
converges strongly to
, which is a solution of the optimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ71_HTML.gif)
where is a potential function for
.
Remark 3.4.
4. Applications
In this section, as in [6, 8, 10], we prove two theorems by using Theorem 3.1. First of all, we recall the following definition.
A mapping is called strictly pseudocontractive if there exists
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ72_HTML.gif)
for every . If
, then
is nonexpansive. Put
, where
is a strictly pseudo-contractive mapping with constant
. Then
is
-inverse-strongly monotone; see [2]. Actually, we have, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ73_HTML.gif)
On the other hand, since is a real Hilbert space, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ74_HTML.gif)
Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ75_HTML.gif)
Using Theorem 3.1, we found a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping.
Theorem 4.1.
Let ,
,
,
,
,
,
,
,
, and
be as in Theorem 3.1. Let
be an
-strictly pseudo-contractive mapping of
into itself such that
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ76_HTML.gif)
where ,
, and
. Let
,
, and
satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then
converges strongly to
, which is a solution of the optimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ77_HTML.gif)
where is a potential function for
.
Proof.
Put . Then
is
-inverse-strongly monotone. We have
and
. Thus, the desired result follows from Theorem 3.1.
Using Theorem 3.1, we also obtain the following result.
Theorem 4.2.
Let be a real Hilbert space. Let
be an
-inverse-strongly monotone mapping of
into
and
a nonexpansive mapping of
into itself such that
. Let
,  and let
be a strongly positive bounded linear operator on
with constant
and
a contraction with constant
. Assume that
and
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ78_HTML.gif)
where ,
, and
. Let
,
, and
satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then
converges strongly to
, which is a solution of the optimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F284363/MediaObjects/13663_2010_Article_1395_Equ79_HTML.gif)
where is a potential function for
.
Proof.
We have . So, putting
, by Theorem 3.1, we obtain the desired result.
Remark 4.3.
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Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0017007).
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Jung, J. A General Iterative Approach to Variational Inequality Problems and Optimization Problems. Fixed Point Theory Appl 2011, 284363 (2011). https://doi.org/10.1155/2011/284363
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DOI: https://doi.org/10.1155/2011/284363