- Research Article
- Open access
- Published:
A Shrinking Projection Method for Generalized Mixed Equilibrium Problems, Variational Inclusion Problems and a Finite Family of Quasi-Nonexpansive Mappings
Journal of Inequalities and Applications volume 2010, Article number: 458247 (2010)
Abstract
The purpose of this paper is to consider a shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S.Takahashi and W. Takahashi (2008), and many others.
1. Introduction
Throughout this paper, we assume that is a real Hilbert space with inner product
and norm
, and let
be a nonempty closed convex subset of
. We denote weak convergence and strong convergence by notations
and
, respectively.
Recall that the following definitions.
(1)A mapping is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ1_HTML.gif)
(2)A mapping is said to be quasi-nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ2_HTML.gif)
We denote be the set of fixed points of
.
Let be a single-valued nonlinear mapping and
to a set-valued mapping. The variational inclusion problem is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ3_HTML.gif)
where is the zero vector in
. The set of solutions of problem (1.3) is denoted by
.
Definition 1.1.
A mapping is said to be a
-inverse-strongly monotone if there exists a constant
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ4_HTML.gif)
Remark 1.2.
It is obvious that any -inverse-strongly monotone mapping
is monotone and
-Lipschitz continuous. It is easy to see that if any
constant is in
, then the mapping
is nonexpansive, where
is the identity mapping on
A set-valued mapping is called monotone if for all
, and
implying
. 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 all
imply
.
Definition 1.3.
Let be a set-valued maximal monotone mapping, then the single-valued mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ5_HTML.gif)
is called the resolvent operator associated with , where
is any positive number and
is the identity mapping.
Remark 1.4.
(R1) The resolvent operator is single-valued and nonexpansive for all
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ6_HTML.gif)
(R2)The resolvent operator is 1-inverse strongly monotone; see [1], that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ7_HTML.gif)
(R3)The solution of problem (1.3) is a fixed point of the operator for all
; see also [2], that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ8_HTML.gif)
(R4)If , then the mapping
is nonexpansive.
(R5) is closed and convex.
Let be a nonlinear mapping, let
be a real-valued function and
a bifunction from
to
. We consider the following generalized mixed equilibrium problem.
Finding such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ9_HTML.gif)
The set of such is denoted by
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ10_HTML.gif)
It is easy to see that is solution of problem (1.9) implies that
.
(i)In the case of (:the zero mapping), then the generalized mixed equilibrium problem (1.9) is reduced to the mixed equilibrium problem. Finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ11_HTML.gif)
The set of solution of (1.11) isdenoted by
(ii)In the case of , then the generalized mixed equilibrium problem (1.9) is reduced to the generalized equilibrium problem. Finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ12_HTML.gif)
The set of solution of (1.12) is denoted by
(iii)In the case of (:the zero mapping) and
, then the generalized mixed equilibrium problem (1.9) is reduced to the equilibrium problem. Finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ13_HTML.gif)
The set of solution of (1.13) is denoted by
(iv)In the case of ,
and
then the generalized mixed equilibrium problem (1.9) is reduced to the variational inequality problem. Finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ14_HTML.gif)
The set of solution of (1.14) is denoted by
The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problems as special cases (see, e.g., [3–8]). Some methods have been proposed to solve the generalized mixed equilibrium problems, generalized equilibrium problems and equilibrium problems; see, for instance, [9–22].
In 2007, Takahashi et al. [23] proved the following strong convergence theorem for a nonexpansive mapping by using the shrinking projection method in mathematical programming. For a and
, they defined a sequence as follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ15_HTML.gif)
where . They proved that the sequence
generated by (1.15) converges weakly to
, where
In 2008, S. Takahashi and W. Takahashi [24] introduced the following iterative scheme for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space. Starting with arbitrary , define sequences
,
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ16_HTML.gif)
They proved that under certain appropriate conditions imposed on ,
and
, the sequence
generated by (1.16) converges strongly to
In 2008, Zhang et al. [25] introduced the following new iterative scheme for finding a common element of the set of solutions to the problem (1.3) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitrary , define sequences
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ17_HTML.gif)
where is the resolvent operator associated with
and a positive number
is a sequence in the interval
.
In 2008, Peng et al. [26] introduced the following iterative scheme by the viscosity approximation method for finding a common element of the set of solutions to the problem (1.3), the set of solutions of an equilibrium problems and the set of fixed points of nonexpansive mappings in a Hilbert space. Starting with an arbitrary , define sequences
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ18_HTML.gif)
They proved that under certain appropriate conditions imposed on and
, the sequence
generated by (1.18) converges strongly to
In 2010, Katchang and Kumam [27] introduced an iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings, and inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space.
In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed shrinking projection method under some suitable conditions. The results obtained in this paper extend and improve several recent results in this area.
2. Preliminaries
Let be a real Hilbert space and let
be a nonempty closed convex subset of
Recall that the (nearest point) projection
from
onto
assigns to each
the unique point in
satisfying the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ19_HTML.gif)
We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.1.
For a given and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ20_HTML.gif)
It is well known that is a firmly nonexpansive mapping of
onto
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ21_HTML.gif)
Lemma 2.2 (see [1]).
Let be a maximal monotone mapping and let
be a Lipshitz continuous mapping. Then the mapping
is a maximal monotone mapping.
Lemma 2.3 (see [28]).
Let be a closed convex subset of
and let
be a bounded sequence in
. Assume that
(1)the weak -limit set
,
(2)for each ,
exists.
Then is weakly convergent to a point in
.
Lemma 2.4 (see [29]).
Each Hilbert space satisfies Opial's condition, that is, for any sequence
with
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ22_HTML.gif)
holds for each with
.
Lemma 2.5 (see [30]).
Each Hilbert space satisfies the Kadec-Klee property, that is, for any sequence
with
and
together imply
.
For solving the generalized equilibrium problems, let us give the following assumptions for and the set
:
(A1) for all
(A2) is monotone, that is,
for all
(A3)for each is weakly upper semicontinuous;
(A4)for each is convex and lower semicontinuous;
(B1)for each and
there exists a bounded subset
and
such that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ23_HTML.gif)
(B2)C is bounded set.
Lemma 2.6 (see [31]).
Let be a nonempty closed convex subset of
and let
be a bifunction of
into
satisfying (A1)–(A4). Let
be a proper lower semicontinuous and convex function such that
. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ24_HTML.gif)
Assume that either or
holds. Then, the following conclusions hold:
(1)for each
(2) is single-valued;
(3) is firmly nonexpansive, that is, for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ25_HTML.gif)
(4)
(5) is closed and convex.
Remark 2.7.
Replacing with
in (2.5), then there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ26_HTML.gif)
3. Main Results
In this section, we will introduce an iterative scheme by using shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space.
Let be a finite family of nonexpansive mappings of
into itself, and let
be real numbers such that
for every
. We define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ27_HTML.gif)
Such a mapping is called the K-mapping generated by
and
; see [32].
We have the following crucial Lemma 3.1 and Lemma 3.2 concerning -mapping which can be found in [14]. Now we only need the following similar version in Hilbert spaces.
Lemma 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a finite family of quasi-nonexpansive mappings and
-Lipschitz mappings of
into itself with
and let
be real numbers such that
for every
,
and
. Let
be the K-mapping generated by
and
. Then, the followings hold:
(1) is quasi-nonexpansive and Lipschitz,
(2).
Lemma 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a finite family of quasi-nonexpansive mappings and
-Lipschitz mappings of
into itself and
sequences in
such that
Moreover, for every
, let
and
be the K-mappings generated by
and
, and
and
, respectively. Then, for every
, we have
Now we study the strong convergence theorem concerning the shrinking projection method.
Theorem 3.3.
Let be a nonempty closed convex subset of a real Hilbert space
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let
be a finite family of quasi-nonexpansive and
-Lipschitz mappings of
into itself, and let
be a
-inverse-strongly monotone mapping of
into
, let
a
-inverse-strongly monotone mapping of
into
and
be a maximal monotone mapping. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ28_HTML.gif)
Let be the
-mapping generated by
and
. Let
,
,
,
and
be sequences generated by
,
,
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ29_HTML.gif)
where satisfy the following conditions:
(i) for some
with
;
(ii),
for some
with
;
(iii) for some
with
.
Then, and
converge strongly to
Proof.
In the light of the definition of the resolvent, can be rewritten as
. Let
and using the fact
be a sequence of mappings defined as in Lemma 2.6,
is an
-inverse-strongly monotone and that
, where
for some
with
, we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ30_HTML.gif)
Next, we will divide the proof into six steps.
Step 1.
We first show that is well defined and
is closed and convex for any
.
From the assumption, we see that is closed and convex. Suppose that
is closed and convex for some
. Next, we show that
is closed and convex for some
. For any
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ31_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ32_HTML.gif)
Thus is closed and convex. Then,
is closed and convex for any
. This implies that
is well defined.
Step 2.
Next, we show by induction that for each
.
Taking and by condition (ii), we get that
is nonexpansive for all
. From the assumption, we see that
. Suppose
for some
. For any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ33_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ34_HTML.gif)
It follows that This implies that
for each
.
Step 3.
Next, we show that and
.
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ35_HTML.gif)
for each . Using
we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ36_HTML.gif)
So, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ37_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ38_HTML.gif)
From and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ39_HTML.gif)
From (3.13), we have, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ40_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ41_HTML.gif)
Thus the sequence is a bounded and nonincreasing sequence, so
exists, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ42_HTML.gif)
Indeed, from (3.13), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ43_HTML.gif)
From (3.16), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ44_HTML.gif)
Since we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ45_HTML.gif)
By (3.18), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ46_HTML.gif)
Step 4.
Next, we show that
For any given ,
. It is easy to see that
. As
is nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ47_HTML.gif)
Similarly, we can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ48_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ49_HTML.gif)
Substituting (3.21) into (3.23), and using conditions (i) and (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ50_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ51_HTML.gif)
Since , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ52_HTML.gif)
Since the resolvent operator is 1-inverse strongly monotone, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ53_HTML.gif)
which yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ54_HTML.gif)
Similarly, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ55_HTML.gif)
Substituting (3.28) into (3.23), and using condition (i), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ56_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ57_HTML.gif)
Applying and
as
to the last inequality, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ58_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ59_HTML.gif)
Substituting (3.22) into (3.33), and using conditions (i) and (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ60_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ61_HTML.gif)
Since , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ62_HTML.gif)
Substituting (3.29) into (3.33), and using conditions (i) and (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ63_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ64_HTML.gif)
Applying and
as
to the last inequality, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ65_HTML.gif)
From (3.32) and (3.39), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ66_HTML.gif)
From (3.33), (3.4), and condition (iii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ67_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ68_HTML.gif)
Since , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ69_HTML.gif)
On the other hand, in the light of Lemma 2.6(3), is firmly nonexpansvie, so we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ70_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ71_HTML.gif)
Using (3.41) again and (3.45), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ72_HTML.gif)
It follows from the condition (i) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ73_HTML.gif)
Since and
, it is implied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ74_HTML.gif)
From (3.39) and (3.48), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ75_HTML.gif)
By (3.3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ76_HTML.gif)
Since for some
with
, and
as
, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ77_HTML.gif)
From (3.40) and (3.48), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ78_HTML.gif)
Furthermore, by the triangular inequality, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ79_HTML.gif)
Applying (3.51) and (3.52), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ80_HTML.gif)
Let be the mapping defined by (3.1). Since
is bounded, applying Lemma 3.2 and (3.54), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ81_HTML.gif)
Step 5.
Next, we show that
Since is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we can assume that
. Since
and
is closed and convex,
is weakly closed and hence
. From
we obtain
-
(a)
First, we prove that
.
We observe that is a
-Lipschitz monotone mapping and
. From Lemma 2.2, we know that
is maximal monotone. Let
, that is,
. Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ82_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ83_HTML.gif)
By virtue of the maximal monotonicity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ84_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ85_HTML.gif)
It follows from ,
and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ86_HTML.gif)
It follows from the maximal monotonicity of that
, that is,
.
-
(b)
Next, we show that
. Since
dom
, we have
(3.61)
From (A2), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ88_HTML.gif)
And hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ89_HTML.gif)
For with
and
let
Since
and
we have
So, from (3.63), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ90_HTML.gif)
Since we have
. Further, from the inverse strongly monotonicity of
we have
So, from (A5), the weakly lower semicontinuity of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ91_HTML.gif)
as From (A1), (A4) and (3.65), we also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ92_HTML.gif)
Letting we have, for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ93_HTML.gif)
This implies that
-
(c)
Now, we prove that
.
Assume Since
and we know that
and
it follows by the Opial's condition (Lemma 2.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ94_HTML.gif)
which is a contradiction. Thus, we get .
The conclusion is .
Step 6.
Finally, we show that and
, where
Since is nonempty closed convex subset of
, there exists a unique
such that
Since
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ95_HTML.gif)
for all . From (3.69),
is bounded, so
. By the weak lower semicontinuity of the norm, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ96_HTML.gif)
However, Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ97_HTML.gif)
Using (3.69) and (3.70), we obtain . Thus
and
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ98_HTML.gif)
Thus, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ99_HTML.gif)
From , we obtain
. Using the Kadec-Klee property (Lemma 2.5) of
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ100_HTML.gif)
and hence in norm. Finally, noticing
we also conclude that
in norm. This completes the proof.
Corollary 3.4.
Let be a nonempty closed convex subset of a real Hilbert space
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let
be a finite family of nonexpansive mappings of
into itself, let
be a
-inverse-strongly monotone mapping of
into
, let
be a
-inverse-strongly monotone mapping of
into
and
a maximal monotone mapping. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ101_HTML.gif)
Let be the
-mapping generated by
and
. Let
,
,
,
and
be sequences generated by (3.3) satisfying the following conditions in Theorem 3.3. Then,
and
converge strongly to
From Theorem 3.3, we can obtain the following results.
Theorem 3.5.
Let be a nonempty closed convex subset of a real Hilbert space
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let
be a finite family of quasi-nonexpansive and
-Lipschitz mappings of
into itself, let
be a
-inverse-strongly monotone mapping of
into
and let
be a
-inverse-strongly monotone mapping of
into
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ102_HTML.gif)
Let be the
-mapping generated by
and
. Let
,
,
,
and
be sequences generated by
,
,
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ103_HTML.gif)
where satisfy the following conditions:
(i) for some
with
;
(ii),
for some
with
;
(iii) for some
with
.
Then, and
converge strongly to
Proof.
In Theorem 3.3 take , where
is the indicator function of
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ104_HTML.gif)
Then the variational inclusion problem (1.3) is equivalent to variational inequality problem (1.14), that is, to find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ105_HTML.gif)
Again, since , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ106_HTML.gif)
and so we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ107_HTML.gif)
We can obtain the desired conclusion from Theorem 3.3 immediately.
Next, we consider another class of important nonlinear mappings: strict pseudocontractions.
Definition 3.6.
A mapping is called strictly pseudocontraction if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ108_HTML.gif)
If , then
is nonexpansive.
In this case, let a
-strictly pseudocontraction. Putting
, then
is a
-inverse-strongly monotone mapping. In fact, from (3.82) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ109_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ110_HTML.gif)
Hence, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ111_HTML.gif)
This shows that is
-inverse-strongly monotone mapping.
Now, we get the following result.
Theorem 3.7.
Let be a nonempty closed convex subset of a real Hilbert space
, let
be a bifunction from
to
satisfying (A1)–(A4) and let
be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let
be a finite family of quasi-nonexpansive and
-Lipschitz mappings of
into itself, let
be a
-strictly pseudocontraction mapping of
into
and let
be a
-strictly pseudocontraction mapping of
into
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ112_HTML.gif)
Let be the
-mapping generated by
and
. Let
,
,
,
and
be sequences generated by
,
,
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ113_HTML.gif)
where satisfy the following conditions:
(i) for some
with
;
(ii),
for some
with
;
(iii) for some
with
.
Then, and
converge strongly to
Proof.
Taking and
, respectively. Then we see that
is
-inverse-strongly monotone and
is
-inverse-strongly monotone, respectively. We have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F458247/MediaObjects/13660_2010_Article_2155_Equ114_HTML.gif)
By using Theorem 3.5, it is easy to obtain the desired conclusion.
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Acknowledgments
The authors would like to express their thank to the referees for helpful suggestions. The first author was supported by the National Research Council of Thailand and the Faculty of Science and Technology RMUTT Research Fund. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute. The third author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044.
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Kumam, W., Jaiboon, C., Kumam, P. et al. A Shrinking Projection Method for Generalized Mixed Equilibrium Problems, Variational Inclusion Problems and a Finite Family of Quasi-Nonexpansive Mappings. J Inequal Appl 2010, 458247 (2010). https://doi.org/10.1155/2010/458247
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DOI: https://doi.org/10.1155/2010/458247