- Research Article
- Open access
- Published:
A System of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Pseudocontractive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2010, Article number: 361512 (2010)
Abstract
We introduce and analyze a new iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of a system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Furthermore, we prove new strong convergence theorems for a new iterative algorithm under some mild conditions. Finally, we also apply our results for solving convex feasibility problems in Hilbert spaces. The results obtained in this paper improve and extend the corresponding results announced by Qin and Kang (2010) and the previously known results in this area.
1. Introduction
Let be 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. Let
be a mapping. In the sequel, we will use
to denote the set of fixed points of
, that is,
.
Definition 1.1.
Let be a mapping. Then
is called
contraction if there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ1_HTML.gif)
nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ2_HTML.gif)
Remark 1.2.
It is well known that if is nonempty, bounded, closed, and convex and
is a nonexpansive mapping on
then
is nonempty; see, for example, [1].
strongly pseudocontractive with the coefficient if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ3_HTML.gif)
strictly pseudocontractive with the coefficient if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ4_HTML.gif)
for such a case, is also said to be a
-strict pseudocontraction, and if
, then
is a nonexpansive mapping,
pseudocontractive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ5_HTML.gif)
The class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of fixed points for strict pseudocontractions.
In 1967, Browder and Petryshyn [2] introduced a convex combination method to study strict pseudocontractions in Hilbert spaces. On the other hand, Marino and Xu [3] and Zhou [4] introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. More precisely, take and define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ6_HTML.gif)
where is a strict pseudocontraction. Under appropriate restrictions on
, it is proved the mapping
is nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudocontractions.
The domain of the function is the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ7_HTML.gif)
Let be a proper extended real-valued function and let
be a bifunction of
into
such that
, where
is the set of real numbers.
There exists the generalized mixed equilibrium problem for finding such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ8_HTML.gif)
The set of solutions of (1.8) is denoted by that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ9_HTML.gif)
We see that is a solution of problem (1.8) implies that
Special Examples
(1)If , problem (1.8) is reduced into the mixed equilibrium problem for finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ10_HTML.gif)
Problem (1.10) was studied by Ceng and Yao [5]. The set of solutions of (1.10) is denoted by .
-
(2)
If
, problem (1.8) is reduced into the generalized equilibrium problem for finding
such that
(1.11)
Problem (1.11) was studied by Takahashi and Toyoda [6]. The set of solutions of (1.11) is denoted by .
-
(3)
If
and
, problem (1.8) is reduced into the equilibrium problem for finding
such that
(1.12)
Problem (1.12) was studied by Blum and Oettli [7]. The set of solutions of (1.12) is denoted by .
-
(4)
If
, problem (1.8) is reduced into the mixed variational inequality of Browder type for finding
such that
(1.13)
Problem (1.13) was studied by Browder [8]. The set of solutions of (1.13) is denoted by .
-
(5)
If
and
, problem (1.8) is reduced into the variational inequality problem for finding
such that
(1.14)
Problem (1.14) was studied by Hartman and Stampacchia [9]. The set of solutions of (1.14) is denoted by . The variational inequality has been extensively studied in the literature. See, for example, [7, 10, 11] and the references therein.
-
(6)
If
and
, problem (1.8) is reduced into the minimize problem for finding
such that
(1.15)
The set of solutions of (1.15) is denoted by .
The generalized mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.8). In 1997, Combettes and Hirstoaga [12] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. Many authors have proposed some useful methods for solving the
,
and
; see, for instance, [5, 12–23].
Definition 1.3.
Let be a nonlinear mapping. Then
is called
(1)monotone if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ16_HTML.gif)
(2)-strongly monotone if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ17_HTML.gif)
-
(3)
 
-Lipschitz continuous if there exists a positive real number
such that
(1.18)
(4)-inverse-strongly monotone if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ19_HTML.gif)
Remark 1.4.
It is obvious that any -inverse-strongly monotone mappings
are monotone and
-Lipschitz continuous.
For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solution of variational inequalities for a -inverse-strongly monotone mapping, Takahashi and Toyoda [6] introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ20_HTML.gif)
where is the metric projection of
onto
,
is a
-inverse-strongly monotone mapping,
is a sequence in
, and
is a sequence in
. They showed that if
is nonempty, then the sequence
generated by (1.20) converges weakly to some
.
On the other hand, Y. Yao and J.-C Yao [24] introduced the following iterative process defined recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ21_HTML.gif)
where is a
-inverse-strongly monotone mapping,
and
are sequences in the interval
, and
is a sequence in
. They showed that if
is nonempty, then the sequence
generated by (1.21) converges strongly to some
.
Let be a strongly positive linear bounded operator on
if there is a constant
with property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ22_HTML.gif)
A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ23_HTML.gif)
where is a linear bounded operator,
is the fixed point set of a nonexpansive mapping
on
and
is a given point in
Moreover, it is shown in [25] that the sequence
defined by the scheme
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ24_HTML.gif)
converges strongly to Recently, Plubtieng and Punpaeng [26] proposed the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ25_HTML.gif)
They proved that if the sequences and
of parameters satisfy appropriate condition, then the sequences
and
both converge to the unique solution
of the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ26_HTML.gif)
which is the optimality condition for the minimization problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ27_HTML.gif)
where is a potential function for
(i.e.,
for
).
Very recently, Ceng et al. [27] introduced iterative scheme for finding a common element of the set of solutions of equilibrium problems and the of fixed points of a -strict pseudocontraction mapping defined in the setting of real Hilbert space
:
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ28_HTML.gif)
where for some
and
satisfies
. Further, they proved that
and
converge weakly to
, where
.
On the other hand, for finding a common element of the set of fixed points of a -strict pseudocontraction mapping and the set of solutions of an equilibrium problems in a real Hilbert space, Liu [28] introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ29_HTML.gif)
where is a
-strict pseudocontraction mapping and
and
are sequences in
They proved that under certain appropriate conditions over
,
, and
, the sequences
and
both converge strongly to some
, which solves some variational inequality problems (1.26).
In 2008, Ceng and Yao [5] introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem (1.8) in Hilbert spaces and obtained the strong convergence theorem which used the following condition.
 : 
is
-strongly convex with constant
and its derivative
is sequentially continuous from the weak topology to the strong topology. We note that the condition
for the function
:
is a very strong condition. We also note that the condition
does not cover the case
and
for each
. Very recently, Wangkeeree and Wangkeeree [29] introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a
-strict pseudocontraction mapping, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the condition
for the sequences generated by these processes.
In 2009, Qin and Kang [30] introduced an explicit viscosity approximation method for finding a common element of the set of fixed points of strict pseudocontraction and the set of solutions of variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Let be a sequence generated by the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ30_HTML.gif)
Then, they proved that under certain appropriate conditions imposed on ,
,
,
,
, and
, the sequence
generated by (1.30) converges strongly to
, where
.
In the present paper, motivated and inspired by Qin and Kang [30], Peng and Yao [21], Plubtieng and Punpaeng [26], and Liu [28], we introduce a new general iterative scheme for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. The results in this paper extend and improve the corresponding recent results of Qin and Kang [30], Peng and Yao [21], Plubtieng and Punpaeng [26], and Liu [28] and many others.
2. Preliminaries
Let be a real Hilbert space and let
be a nonempty closed convex subset of
. In a real Hilbert space
, it is well known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ31_HTML.gif)
For any , there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ32_HTML.gif)
The mapping is called the metric projection of
onto
It is well known that is a firmly nonexpansive mapping of
onto
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ33_HTML.gif)
Moreover, is characterized by the following properties:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ34_HTML.gif)
for all .
Lemma 2.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Given
and
then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ35_HTML.gif)
Lemma 2.2.
Let be a Hilbert space, let
be a nonempty closed convex subset of
and let
be a mapping of
into
Let
. Then for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ36_HTML.gif)
where is the metric projection of
onto
.
A set-valued mapping is called amonotone if for all
,
and
imply
. A monotone mapping
is called 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 a monotone map of
into
,
-Lipschitz continuous mappings and let
be the normal cone to
when
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ37_HTML.gif)
and define a mapping on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ38_HTML.gif)
Then is the maximal monotone and
if and only if
see [31].
Lemma 2.3.
Let be a Hilbert space, let
be a nonempty closed convex subset of
and let
be
-inverse-strongly monotone. It
, then
is a nonexpansive mapping in
Proof.
For all and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ39_HTML.gif)
So, is a nonexpansive mapping of
into
.
Lemma 2.4 (see [32]).
Let be an inner product space. Then, for all
and
with
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ40_HTML.gif)
Lemma 2.5 (see [25]).
Let be a nonempty closed convex subset of
let
be a contraction of
into itself with
, and let
be a strongly positive linear bounded operator on
with coefficient
. Then, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ41_HTML.gif)
That is, is strongly monotone with coefficient
.
Lemma 2.6 (see [25]).
Assume that is a strongly positive linear bounded operator on
with coefficient
and
. Then
.
Lemma 2.7 (see [4]).
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a
-strict pseudocontraction mapping with a fixed point. Then
is closed and convex. Define
by
for each
. Then
is nonexpansive such that
.
Lemma 2.8 (see [33]).
Let be a closed convex subset of a Hilbert space
and let
 : 
be a nonexpansive mapping. Then
is demiclosed at zero, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ42_HTML.gif)
Lemma 2.9 (see [34]).
Let be a nonempty closed convex subset of a strictly convex Banach space
. Let
be a sequence of nonexpansive mappings on
. Suppose that
is nonempty. Let
be a sequence of positive number with
. Then a mapping
on
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ43_HTML.gif)
for is well defined and nonexpansive and
holds.
For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction , the function
and the set
:
for all
is monotone, that is,
for all
for each   
for each is convex and lower semicontinuous;
for each is weakly upper semicontinuous;
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%2F361512/MediaObjects/13663_2010_Article_1265_Equ44_HTML.gif)
is a bounded set.
By similar argument as in the proof of Lemma in [35], we have the following lemma appearing.
Lemma 2.10.
Let be a nonempty closed convex subset of
. Let
be a bifunction satisfies (A1)–(A5) and let
be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ45_HTML.gif)
for all . Then, the following holds:
(i)for each ;
(ii) is single-valued;
(iii) is firmly nonexpansive, that is, for any
 
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ46_HTML.gif)
(iv) 
(v) is closed and convex.
Remark 2.11.
We remark that Lemma 2.10 is not a consequence of Lemma in [5], because the condition of the sequential continuity from the weak topology to the strong topology for the derivative
of the function
does not cover the case
.
Lemma 2.12 (see [36]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in
with
Suppose
for all integers
and
Then,
Lemma 2.13 (see [37]).
Assume that is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ47_HTML.gif)
where is a sequence in
and
is a sequence in
such that
(1)
(2) or
Then
Lemma 2.14.
Let be a real Hilbert space. Then for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ48_HTML.gif)
3. Main Results
In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of a common solutions of the variational inequalities for inverse-strongly monotone mappings in a real Hilbert space.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
and
be two bifunctions from
to
satisfying
and let
be a proper lower semicontinuous and convex function with either (B1) or (B2). Let
 : 
be a
-inverse-strongly monotone mapping, let
 : 
be a
-inverse-strongly monotone mapping, let
 : 
be an
-inverse-strongly monotone mapping, and let
 : 
be a
-inverse-strongly monotone mapping. Let
 : 
be an
-contraction with coefficient
and let
be a strongly positive linear bounded operator on
with coefficient
and
. Let
 : 
be a
-strict pseudocontraction with a fixed point. Define a mapping
 : 
by
for all
. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ49_HTML.gif)
Let be a sequence generated by the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ50_HTML.gif)
where ,
,
, and
are sequences in
, where
,
,
, and
and
are positive sequences. Assume that the control sequences satisfy the following restrictions:
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_IEq407_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_IEq409_HTML.gif)
and
, where
are two positive constants,
, where
.
Then, converges strongly to a point
which is the unique solution of the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ51_HTML.gif)
Equivalently, one has
Proof.
Since , as
, we may assume, without loss of generality, that
for all
. By Lemma 2.6, we know that if
, then
. We will assume that
. Since
is a strongly positive bounded linear operator on
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ52_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ53_HTML.gif)
and so this shows that is positive. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ54_HTML.gif)
Since is a contraction of
into itself with
, then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ55_HTML.gif)
Since , it follows that
is a contraction of
into itself. Therefore the Banach Contraction Mapping Principle implies that there exists a unique element
such that
Next, we will divide the proof into five steps.
Step 1.
We claim that is bounded.
Indeed, let and by Lemma 2.10, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ56_HTML.gif)
Note that dom
and
dom
; we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ57_HTML.gif)
Put and
. For each
and
by Lemma 2.3, we get that
and
are nonexpansive. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ58_HTML.gif)
From Lemma 2.7, we have that is nonexpansive with
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ59_HTML.gif)
which yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ60_HTML.gif)
Hence, is bounded, and so are
,
,
,
,
,
,
and
.
Step 2.
We claim that and
Observing that and
dom
, by the nonexpansiveness of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ61_HTML.gif)
Similarly, let dom
and
dom
; we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ62_HTML.gif)
From and
; thus, we compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ63_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ64_HTML.gif)
Also noticing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ65_HTML.gif)
we compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ66_HTML.gif)
Substitution of (3.13), (3.14), (3.15), and (3.16) into (3.18) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ67_HTML.gif)
where is an appropriate constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ68_HTML.gif)
Putting for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ69_HTML.gif)
Then, we compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ70_HTML.gif)
It follows from (3.19) and (3.22) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ71_HTML.gif)
This together with (C2), (C3), (C4), and (C6) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ72_HTML.gif)
Hence, by Lemma 2.12, we obtain as
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ73_HTML.gif)
Moreover, we also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ74_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ75_HTML.gif)
By conditions (C2), (C3), and (3.25), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ76_HTML.gif)
Step 3.
We claim that the following statements hold:
;
;
;
.
For , we compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ77_HTML.gif)
By the same way, we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ78_HTML.gif)
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ79_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ80_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ81_HTML.gif)
Substituting (3.29), (3.30), (3.31), and (3.32) into (3.33), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ82_HTML.gif)
It follows from (3.2) and (3.34) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ83_HTML.gif)
It follows from (C5) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ84_HTML.gif)
From (C2), (C6), and (3.25), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ85_HTML.gif)
Since , we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ86_HTML.gif)
From (C2), (C6), and (3.25), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ87_HTML.gif)
Similarly, from (3.37) and (3.39), we can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ88_HTML.gif)
On the other hand, let for each
we get
. By Lemma 2.10(iii), that is,
is firmly nonexpansive, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ89_HTML.gif)
So, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ90_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ91_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ92_HTML.gif)
By using the same argument in (3.42) and (3.44), we can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ93_HTML.gif)
Substituting (3.42), (3.44), and (3.45) into (3.33), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ94_HTML.gif)
From Lemma 2.4, (3.2), and (3.46), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ95_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ96_HTML.gif)
From (C2), (C6), (3.37), (3.39), (3.40), and as
, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ97_HTML.gif)
From (3.47) and by using the same argument above, we can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ98_HTML.gif)
Applying (3.28), (3.49), and (3.50), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ99_HTML.gif)
Step 4.
We claim that where
is the unique solution of the variational inequality
for all
To show the above inequality, we choose a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ100_HTML.gif)
Since is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we can assume that
We claim that
.
That is, we will prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ101_HTML.gif)
Assume also that and
.
Define a mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ102_HTML.gif)
where , where
. Since
and by Lemma 2.9, we have that
is nonexpansive and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ103_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ104_HTML.gif)
where is an appropriate constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ105_HTML.gif)
From (C4), (C6), and (3.28), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ106_HTML.gif)
Since is a contraction with the coefficient
, we have that there exists a unique fixed point. We use
to denote the unique fixed point to the mapping
. That is,
. Since
is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we may assume that
. It follows from (3.58), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ107_HTML.gif)
It follows from Lemma 2.8 that . By (3.55), we have
.
Hence from (3.52) and (2.4), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ108_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ109_HTML.gif)
From (3.25) and (3.60), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ110_HTML.gif)
Step 5.
We claim that .
Indeed, by (3.2) and using Lemmas 2.6 and 2.14, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ111_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ112_HTML.gif)
Taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ113_HTML.gif)
then, we can rewrite (3.64) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ114_HTML.gif)
and we can see that and
. Applying Lemma 2.13 to (3.66), we conclude that
converges strongly to
in norm. This completes the proof.
If the mapping is nonexpansive, then
. We can obtain the following result from Theorem 3.1 immediately.
Corollary 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
and
be two bifunctions from
to
satisfying (A1)–(A5) and let
 : 
be a proper lower semicontinuous and convex function with either (B1) or (B2). Let
 : 
be a
-inverse-strongly monotone mapping, let
 : 
be a
-inverse-strongly monotone mapping, let
 : 
be an
-inverse-strongly monotone mapping and let
 : 
be a
-inverse-strongly monotone mapping. Let
be an
-contraction with coefficient
and let
be a strongly positive linear bounded operator on
with coefficient
and
. Let
be a nonexpansive mapping with a fixed point. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ115_HTML.gif)
Let be a sequence generated by the following iterative algorithm (3.2), where
,
,
, and
are sequences in
, where
,
,
, and
and
are positive sequences. Assume that the control sequences satisfy (C1)–(C6) in Theorem 3.1. Then,
converges strongly to a point
which is the unique solution of the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ116_HTML.gif)
Equivalently, one has
If and
in Theorem 3.1, then we can obtain the following result immediately.
Corollary 3.3.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
and
be two bifunctions from
to
satisfying (A1)–(A4). Let
be a
-inverse-strongly monotone mapping and let
be a
-inverse-strongly monotone mapping. Let
be an
-contraction with coefficient
and let
be a strongly positive linear bounded operator on
with coefficient
and
. Let
be a
-strict pseudocontraction with a fixed point. Define a mapping
by
for all
. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ117_HTML.gif)
Let be a sequence generated by the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ118_HTML.gif)
where ,
,
, and
are sequences in
, where
,
,
, and
and
are positive sequences. Assume that the control sequences satisfy the condition (C1)–(C6) in Theorem 3.1 and
. Then,
converges strongly to a point
, where
If and
in Corollary 3.3, then
and we get
and
; hence we can obtain the following result immediately.
Corollary 3.4.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a
-strict pseudocontraction with a fixed point. Define a mapping
by
for all
. Suppose that
Let
be a sequence generated by the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ119_HTML.gif)
where ,
,
, and
are sequences in
. Assume that the control sequences satisfy the conditions (C2) and (C3),
in Theorem 3.1, and
. Then,
converges strongly to a point
, where
.
Finally, we consider the following Convex Feasibility Problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ120_HTML.gif)
where is an integer and each
is assumed to be the of solutions of equilibrium problem with the bifunction
and the solution set of the variational inequality problem. There is a considerable investigation on
in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [38, 39], computer tomography [40], and radiation therapy treatment planning [41].
The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.
Theorem 3.5.
Let be a nonempty closed convex subset of a real Hilbert space
. Let be a
bifunction from
to
satisfying (A1)–(A5) and let
 : 
be a proper lower semicontinuous and convex function with either (B1) or (B2). Let
 : 
be an
-inverse-strongly monotone mapping for each
. Let
 : 
be a contraction mapping with coefficient
and let
be a strongly positive linear bounded operator on
with coefficient
and
. Let
 : 
be a
-strict pseudocontraction with a fixed point. Define a mapping
 : 
by
for all
. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ121_HTML.gif)
Let be a sequence generated by the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ122_HTML.gif)
where such that
,
are positive sequences, and
and
are sequences in
. Assume that the control sequences satisfy the following restrictions:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_IEq682_HTML.gif)
for each
,
, where
is some positive constant for each
,
, for each
.
Then, converges strongly to a point
which is the unique solution of the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F361512/MediaObjects/13663_2010_Article_1265_Equ123_HTML.gif)
Equivalently, one has
References
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Marino G, Xu H-K: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,329(1):336–346. 10.1016/j.jmaa.2006.06.055
Zhou H: Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis 2008,69(2):456–462. 10.1016/j.na.2007.05.032
Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022
Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
Browder FE: Existence and approximation of solutions of nonlinear variational inequalities. Proceedings of the National Academy of Sciences of the United States of America 1966, 56: 1080–1086. 10.1073/pnas.56.4.1080
Hartman P, Stampacchia G: On some nonlinear elliptic differential-functional equations. Acta Mathematica 1966, 115: 271–310. 10.1007/BF02392210
Yao J-C, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity. Volume 76. Edited by: Haddjissas N, Schaible S. Springer, New York, NY, USA; 2005:501–558. 10.1007/0-387-23393-8_12
Zeng LC, Schaible S, Yao JC: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s10957-004-1182-z
Combettes PL, Hirstoaga SA: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997,78(1):29–41.
Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Analysis 2010,72(1):99–112. 10.1016/j.na.2009.06.042
Gao X, Guo Y: Strong convergence of a modified iterative algorithm for mixed-equilibrium problems in Hilbert spaces. Journal of Inequalities and Applications 2008, 2008:-23.
Jaiboon C, Kumam P: A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Fixed Point Theory and Applications 2009, 2009:-32.
Jaiboon C, Kumam P: Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities. Journal of Inequalities and Applications 2010, 2010:-43.
Jaiboon C, Kumam P: A general iterative method for addressing mixed equilibrium problems and optimization problems. Nonlinear Analysis 2010,73(5):1180–1202. 10.1016/j.na.2010.04.041
Jaiboon C, Kumam P, Humphries UW: Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems. Bulletin of the Malaysian Mathematical Sciences Society 2009,32(2):173–185.
Jung JS: Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. Applied Mathematics and Computation 2009,213(2):498–505. 10.1016/j.amc.2009.03.048
Kumam P, Jaiboon C: A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems. Nonlinear Anal: Hybrid Systems 2009,3(4):510–530. 10.1016/j.nahs.2009.04.001
Peng J-W, Yao J-C: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems. Mathematical and Computer Modelling 2009,49(9–10):1816–1828. 10.1016/j.mcm.2008.11.014
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036
Yao Y, Liou Y-C, Yao J-C: A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems. Fixed Point Theory and Applications 2008, 2008:-15.
Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2007,186(2):1551–1558. 10.1016/j.amc.2006.08.062
Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028
Plubtieng S, Punpaeng R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,336(1):455–469. 10.1016/j.jmaa.2007.02.044
Ceng L-C, Al-Homidan SA, Ansari QH, Yao J-C: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. Journal of Computational and Applied Mathematics 2009,223(2):967–974. 10.1016/j.cam.2008.03.032
Liu Y: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis 2009,71(10):4852–4861. 10.1016/j.na.2009.03.060
Wangkeeree R, Wangkeeree R: A general iterative method for variational inequality problems, mixed equilibrium problems, and fixed point problems of strictly pseudocontractive mappings in Hilbert spaces. Fixed Point Theory and Applications 2009, 2009:-32.
Qin X, Kang SM: Convergence theorems on an iterative method for variational inequality problems and fixed point problems. Bulletin of the Malaysian Mathematical Sciences Society 2010,33(1):155–167.
Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5
Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Computers & Mathematics with Applications 2000,40(4–5):559–567. 10.1016/S0898-1221(00)00179-6
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill, 1968). American Mathematical Society, Providence, RI, USA; 1976:1–308.
Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.
Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
Combettes PL: The convex feasibility problem: in image recovery. In Advances Imaging and Electron Physics. Volume 95. Edited by: Hawkes P. Academic Press, Orlando, Fla, USA; 1996:155–270.
Kotzer T, Cohen N, Shamir J: Images to ration by a novel method of parallel projection onto constraint sets. Optics Letters 1995, 20: 1172–1174. 10.1364/OL.20.001172
Sezan MI, Stark H: Application of convex projection theory to image recovery in tomograph and related areas. In Image Recovery: Theory and Application. Edited by: Stark H. Academic Press, Orlando, Fla, USA; 1987:155–270.
Censor Y, Zenios SA: Parallel Optimization, Numerical Mathematics and Scientific Computation. Oxford University Press, New York, NY, USA; 1997:xxviii+539.
Acknowledgments
The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper. The first author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kumam, P., Jaiboon, C. A System of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Pseudocontractive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2010, 361512 (2010). https://doi.org/10.1155/2010/361512
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/361512