- Research Article
- Open Access
- Published:
Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities
Journal of Inequalities and Applications volume 2010, Article number: 728028 (2010)
Abstract
We introduce a new iterative scheme for finding the common element of the set of solutions of the generalized equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inequality problems for a relaxed -cocoercive and
-Lipschitz continuous mapping in a real Hilbert space. Then, we prove the strong convergence of a common element of the above three sets under some suitable conditions. Our result can be considered as an improvement and refinement of the previously known results.
1. Introduction
Variational inequalities introduced by Stampacchia [1] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences. It is well known that the variational inequalities are equivalent to the fixed point problems. This alternative equivalent formulation has been used to suggest and analyze in variational inequalities. In particular, the solution of the variational inequalities can be computed using the iterative projection methods. It is well known that the convergence of a projection method requires the operator to be strongly monotone and Lipschitz continuous. Gabay [2] has shown that the convergence of a projection method can be proved for cocoercive operators. Note that cocoercivity is a weaker condition than strong monotonicity.
Equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization which has been extended and generalized in many directions using novel and innovative technique; see [3, 4]. Related to the equilibrium problems, we also have the problem of finding the fixed points of the nonexpansive mappings. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of a set of the solutions of the equilibrium problems and a set of the fixed points of infinitely (finitely) many nonexpansive mappings; see [5–7] and the references therein. In this paper, we suggest and analyze a new iterative method for finding a common element of a set of the solutions of generalized equilibrium problems and a set of fixed points of an infinite family of nonexpansive mappings and the set solution of the variational inequality problems for a relaxed -cocoercive mapping in a real Hilbert space.
Let be a real Hilbert space and let
be a nonempty closed convex subset of
and
is the metric projection of
onto
Recall that a mapping
is contraction on
if there exists a constant
such that
for all
A mapping
of
into itself is called nonexpansive if
for all
We denote by
the set of fixed points of
, that is,
. If
is nonempty, bounded, closed, and convex and
is a nonexpansive mapping of
into itself, then
is nonempty; see, for example, [8]. We recalled some definitions as follows.
Definition 1.1.
Let be a mapping. Then one has the following.
(1) is calledmonotone if
for all
(2) is called
-strongly monotone if there exists a positive real number
such that

(3) is called
-Lipschitz continuous if there exists a positive real number
such that

(4) is called
-inverse-strongly monotone, [9, 10] if there exists a positive real number
such that

If we say that
is firmly nonexpansive. It is obvious that any
-inverse-strongly monotone mapping
is monotone and
-Lipschitz continuous.
(5) is calledrelaxed
-cocoercive if there exists a positive real number
such that

For ,
is
-strongly monotone. This class of maps is more general than the class of strongly monotone maps. It is easy to see that we have the following implication:
-strongly monotonicity
relaxed
-cocoercivity.
(6)A set-valued mapping is called monotone if for all
,
and
imply
. A monotone mapping
is maximal if the graph of
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 mapping of
into
and let
be the normal cone to
at
, that is,

Define

Then is the maximal monotone and
if and only if
; see [11, 12]
In addition, let be a inverse-strongly monotone mapping. Let
be a bifunction of
into
, where
is the set of real numbers. The generalized equilibrium problem for
is to find
such that

The set of such is denoted by
that is,

Special Cases
() If
(:the zero mapping), then the problem (1.7) is reduced to the equilibrium problem:

The set of solutions of (1.9) is denoted by that is,

() If
, the problem (1.7) is reduced to the variational inequality problem:

The set of solutions of (1.11) is denoted by , that is,

The generalized equilibrium problem (1.7) is very general in the sense that it includes, as special case, some optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, economics, and others (see, e.g., [4, 13]). Some methods have been proposed to solve the equilibrium problem and the generalized equilibrium problem; see, for instance, [5, 14–28]. Recently, Combettes and Hirstoaga [29] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Very recently, Moudafi [24] introduced an itertive method for finding an element of
, where
is an inverse-strongly monotone mapping and then proved a weak convergence theorem.
For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem for an -inverse-strongly monotone, Takahashi and Toyoda [30] introduced the following iterative scheme:

where is an
-inverse-strongly monotone mapping,
is a sequence in (0, 1), and
is a sequence in
. They showed that if
is nonempty, then the sequence
generated by (1.13) converges weakly to some
. On the other hand, Shang et al. [31] introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a relaxed
-cocoercive mapping in a real Hilbert space. Let
be a nonexpansive mapping. Starting with arbitrary initial
defined sequences
recursively by

They proved that under certain appropriate conditions imposed on ,
,
and
, the sequence
converges strongly to
, where
In 2008, S. Takahashi and W. Takahashi [27] introduced the following iterative scheme for finding an element of under some mild conditions. Let
be a nonempty closed convex subset of a real Hilbert space
. Let
be an
-inverse-strongly monotone mapping of
into
and let
be a nonexpansive mapping of
into itself such that
Suppose
and let
,
, and
by sequences generated by

where ,
and
satisfy some parameters controlling conditions. They proved that the sequence
defined by (1.15) converges strongly to a common element of
.
On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [32–35] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences.
A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping in a real Hilbert space :

where is the fixed point set of a nonexpansive mapping
on
and
is a given point in
. Assume that
is a strongly positive bounded linear operator on
; that is, there exists a constant
such that

In 2006, Marino and Xu [36] considered the following iterative method:

They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence
generated by (1.18) converges strongly to the unique of the variational inequality

which is the optimality condition for the minimization problem

where is a potential function for
(i.e.,
for
).
In 2008, Qin et al. [26] proposed the following iterative algorithm:

where is a strongly positive linear bounded operator and
is a relaxed cocoercive mapping of
into
. They prove that if the sequences
,
and
of parameters satisfy appropriate condition, then the sequences
and
both converge to the unique solution
of the variational inequality

which is the optimality condition for the minimization problem

where is a potential function for
(i.e.,
for
).
Furthermore, for finding approximate common fixed points of an infinite family of nonexpansive mappings under very mild conditions on the parameters, we need the following definition.
Definition 1.2 (see [37]).
Let be a sequence of nonexpansive mappings of
into itself and let
be a sequence of nonnegative numbers in
. For each
, define a mapping
of
into itself as follows:

Such a mappings is called the
-mapping generated by
and
. It is obvious that
is nonexpansive, and if
then
.
On the other hand, Yao et al. [38] introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite family of nonexpansive mappings on . Starting with an arbitrary initial
, define sequences
and
recursively by

where is a sequence in
. It is proved [38] that under certain appropriate conditions imposed on
and
, the sequence
generated by (1.25) converges strongly to
. Very recently, Qin et al. [6] introduced an iterative scheme for finding a common fixed points of a finite family of nonexpansive mappings, the set of solutions of the variational inequality problem for a relaxed cocoercive mapping, and the set of solutions of the equilibrium problems in a real Hilbert space. Starting with an arbitrary initial
, define sequences
and
recursively by

where is a relaxed
-cocoercive mapping and
is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on
and
, the sequences
and
generated by (1.26) converge strongly to some point
, which is a unique solution of the variation inequality:

and is also the optimality for some minimization problems.
In this paper, motivated by iterative schemes considered in (1.15), (1.25), and (1.26) we will introduce a new iterative process (3.4) below for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the generalized equilibrium problem, and the set of solutions of variational inequality problem for a relaxed -cocoercive mapping in a real Hilbert space. The results obtained in this paper improve and extend the recent ones announced by Yao et al. [38], S. Takahashi and W. Takahashi [27], and Qin et al. [6] and many others.
2. Preliminaries
Let be a real Hilbert space with inner product
and norm
. Let
be a nonempty closed convex subset of
We denote weak convergence and strong convergence by notations
and
, respectively. Recall that the (nearest point) projection
from
to
assigns each
the unique point in
satisfying the property

The following characterizes the projection .
We need some facts tools in a real Hilbert space which are listed as follows.
Lemma 2.1.
For any ,

It is well known that is a firmly nonexpansive mapping of
onto
and satisfies

Moreover, is characterized by the following properties:
and for all

Lemma 2.2 (see [39]).
Let be a Hilbert space, let
be a nonempty closed convex subset of
and let
be a mapping of
into
. Let
. Then for
,

where is the metric projection of
onto
.
It is clear from Lemma 2.2 that variational inequality and fixed point problem are equivalent. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.
Lemma 2.3 (see [40]).
Each Hilbert space satisfies Opials condition; that is, for any sequence
with
, the inequality

holds for each with
.
Lemma 2.4 (see [36]).
Assume that is a strongly positive linear bounded operator on
with coefficient
and
. Then
.
For solving the equilibrium problem for a bifunction , let us assume that
satisfies the following conditions:
, for all
is monotone, that is,
, for all
, for all
for each is convex and lower semicontinuous.
The following lemma appears implicitly in [4].
Lemma 2.5 (see [4]).
Let be a nonempty closed convex subset of
and let
be a bifunction of
into
satisfying (A1)–(A4). Let
and
. Then, there exists
such that

The following lemma was also given in [5].
Lemma 2.6 (see [5]).
Assume that satisfies (A1)–(A4). For
and
, define a mapping
as follows:

for all . Then, the following holds:
(1) is single-valued;
(2) is firmly nonexpansive, that is, for any

(3)
(4) is closed and convex.
Remark 2.7.
Replacing with
in (2.7), then there exists
, such that

Lemma 2.8 (see [41]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be nonexpansive mappings of
into itself such that
is nonempty, and let
be real numbers such that
for every
. Then, for every
and
, the limit
exists.
Using Lemma 2.8, one can define a mapping of
into itself as follows:

for every . Such a
is called the
-mapping generated by
and
. Throughout this paper, we will assume that
for every
. Then, we have the following results.
Lemma 2.9 (see [41]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be nonexpansive mappings of
into itself such that
is nonempty, let
be real numbers such that
for every
. Then,
.
Lemma 2.10 (see [7]).
If is a bounded sequence in
, then
.
Lemma 2.11 (see [42]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in
with
Suppose
for all integers
and
Then,
Lemma 2.12.
Let be a real Hilbert space. Then the following inequality holds:
(1),
(2)
for all .
Lemma 2.13 (see [43]).
Assume that is a sequence of nonnegative real numbers such that

where is a sequence in
and
is a sequence in
such that
(1),
(2) or
Then
3. Main Results
In this section, we prove a strong convergence theorem of a new iterative method (3.4) for an infinite family of nonexpansive mappings and relaxed -cocoercive mappings in a real Hilbert space.
We first prove the following lemmas.
Lemma 3.1.
Let be a real 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

So, is a nonexpansive mapping of
into
.
Lemma 3.2.
Let be a real Hilbert space, let
be a nonempty closed convex subset of
and let
be a relaxed
-cocoercive and
-Lipschitz continuous. If
,
, then
is a nonexpansive mapping in
.
Proof.
For any and
,
.
Putting , we obtain

that is, . It follows that

for all . Thus
.
So, is a nonexpansive mapping of
into
.
Now, we prove the following main theorem.
Theorem 3.3.
Let be a nonempty closed convex subset of a real Hilbert space
, and let
be a bifunction satisfying (A1)–(A4). Let
(1) be an infinite family of nonexpansive mappings of
into
;
(2) be an
-inverse strongly monotone mappings of
into
;
(3) be relaxed
-cocoercive and
-Lipschitz continuous mappings of
into
.
Assume that . Let
be a contraction mapping with
and let
be a strongly positive linear bounded operator on
with coefficient
and
. Let
,
,
and
be sequences generated by

where is the sequence generated by (1.24) and
,
,
and
are sequences in
satisfy the following conditions:



and
,
,
,
for some
with
,
,
for some
with
.
Then, and
converge strongly to a point
, where
, which solves the variational inequality

which is the optimality condition fot the minimization problem

where is a potential function for
(i.e.,
for
).
Proof.
Since by the condition (C1) and
, we may assume, without loss of generality, that
. Since
is a strongly positive bounded linear operator on
, then

Observe that

that is to say is positive. It follows that

We will divide the proof of Theorem 3.3 into six steps.
Step 1.
We prove that there exists such that
.
Let . Note that
is a contraction mapping of
into itself with coefficient
. Then, we have

Therefore, is a contraction mapping of
into itself. Therefore by the Banach Contraction Mapping Principle guarantee that
has a unique fixed point, say
. That is,
.
Step 2.
We prove that is bounded.
Since

we obtain

From Lemma 2.6, we have for all
.
For any , it follows from
that

So, we have

By Lemma 2.6 again, we have for all
. If follows that

If we applied Lemma 3.2, we get and
are nonexpansive. Since
and
is a nonexpansive, we have
, and we have

It follows that

which yields that

This in turn implies that

Therefore, is bounded. We also obtain that
,
,  
,
,
,  
,
,
,
and
are all bounded.
Step 3.
We claim that and
.
From Lemma 2.6, we have and
. Let
, we get
,
, and so


Putting in (3.20) and
in (3.21), we have

So, from the monotonicity of , we get

and hence

Without loss of generality, let us assume that there exists a real number such that
for all
Then, we have

and hence

where .
Put ,
and
. Since
,
and
are nonexpansive, then we have the following some estimates:

Similarly, we can prove that


Since and
are nonexpansive, we deduce that, for each
,

where is a constant such that
for all
Similarly, we can obtain that there exist nonnegative numbers ,
such that

and so are

Observing that

we obtain

which yields that

Substitution of (3.27) and (3.30) into (3.35) yields that

where is an appropriate constant such that
.
Observing that

we obtain

which yields that

Substitution of (3.28) and (3.32) into (3.39) yields that

where is an appropriate constant such that
.
Substituting (3.26) and (3.36) into (3.40), we obtain

where is an appropriate constant such that
.
Substituting (3.41) into (3.29), we obtain

where is an appropriate constant such that
.
Define

Observe that from the definition , we obtain

It follows from (3.32), (3.42), and (3.44) that

where is an appropriate constant such that
.
It follows from conditions (C1), (C2), (C3), (C4), (C5), and for all

Hence, by Lemma 2.11, we obtain

It follows that

Applying (3.48) and conditions in Theorem 3.3 to (3.26), (3.41), and (3.42), we obtain that

From (3.49), (C2), (C5), and for all
, we also have

Since , we have

that is,

By (C1), (C3), and (3.48) it follows that

Step 4.
We claim that the following statements hold:
(i);
(ii);
(iii).
Since is relaxed
-cocoercive and
-Lipschitz continuous mappings, by the assumptions imposed on
for any
, we have

Similarly, we have

Observe that

where

It follows from condition (C1) that

Substituting (3.54) into (3.56), and using condition (C6), we have

It follows that

Since as
and (3.48), we obtain

Note that


Using (3.56) again, we have

Substituting (3.62) into (3.64) and using condition (C2) and (C6), we have

It follows that

Since as
and (3.48), we obtain

In a similar way, we can prove

By (2.3), we also have

which yields that

Substituting (3.70) into (3.56), we have

It follows that

Applying ,
and
as
to the last inequality, we have

On the other hand, we have

which yields that

Similarly, we can prove


Substituting (3.75) into (3.56), we have

which yields that

Applying (3.48) and (3.61) to the last inequality, we have

Using (3.64) again, we have

which implies that

From (3.48) and (3.67), we obtain

By using the same argument, we can prove that

Note that

Since and
as
, respectively, we also have

On the other hand, we observe

Applying (3.73), (3.83), (3.84), and (3.86), we have

On the other hand, we have

Substituting (3.89) into (3.64) and using conditions (C2) and (C7), we have

This implies that

In view of the restrictions (C2) and (C7), we obtain that

Let . Since
and
is firmly nonexpansive (Lemma 2.6), then we obtain

So, we obtain

Therefore, we have

It follows that

Using ,
as
, (3.48), (3.88), and (3.92), we obtain

Since , we obtain

Note that

and thus from (3.88) and (3.97), we have

Observe that

Applying (3.53) and (3.100), we obtain

Let be the mapping defined by (2.11). Since
is bounded, applying Lemma 2.10 and (3.102), we have

Step 5.
We claim that where
is the unique solution of the variational inequality
for all
Since is a unique solution of the variational inequality (3.5), to show this inequality, we choose a subsequence
of
such that

Since is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we can assume that
From
we obtain
. Next, We show that
, where
.
-
(a)
First, we prove
.
Since , we know that

From (A2), we also have

Replacing by
, we have

For any with
and
let
Since
and
we have
So, from (3.107) we have

Since is Lipschitz continuous, from (3.97), we have
as
.
Further, from the monotonicity of , we get that

It follows from (A4) and (3.108) that

From (A1), (A4), and (3.110), we also have

and hence

Letting in the above inequality, we have, for each
,

Thus
-
(b)
Next, we show that
By Lemma 2.9, we have . Assume
Since
we know that
and
and it follows by the Opial's condition (Lemma 2.3) that

that is a contradiction. Thus, we have .
-
(c)
Finally, Now we prove that
Define,
(3.115)
Since is relaxed
-cocoercive and condition (C6), we have

which yields that is monotone. Then,
is maximal monotone. Let
Since
and
we have
On the other hand, from
we have

and hence

Therefore, we have

which implies that

Since is maximal monotone, we have
and hence
That is,
, where
. Since
, it follows that

On the other hand, we have

From (3.53) and (3.121), we obtain that

Step 6.
Finally, we show that and
converge strongly to
.
Indeed, from (3.4) and Lemma 2.4, we obtain

Since ,
and
are bounded, we can take a constant
such that

for all . It then follows that

where

Using (C1), (3.121), and (3.123), we get ,
and
. Applying Lemma 2.13 to (3.126), we conclude that
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 satisfying (A1)–(A4), let
be relaxed
-cocoercive and
-Lipschitz continuous mappings, and let
be an infinite family of nonexpansive mappings of
into itself such that
. Let
be a contraction mapping of
into itself with
. Let
,
,
and
be sequences generated by

where is the sequence generated by (1.24) and
,
,
, and
are sequences in
and
is a real sequence in
satisfying the following conditions:
,




,
,
for some
with
,
.
Then, and
converge strongly to a point
, where
.
Proof.
Put ,
,
(:the zero mapping) and
in Theorem 3.3. Then
and for any
, we see that

Let be a sequence satisfying the restriction:
, where
. Then we can obtain the desired conclusion easily from Theorem 3.3.
Corollary 3.5.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be an infinite family of nonexpansive mappings of
into itself and let
be relaxed
-cocoercive and
-Lipschitz continuous mappings such that
. Let
be a contraction mapping with
and let
be a strongly positive linear bounded operator on
with coefficient
and
. Let
,
and
be sequences generated by

where is the sequence generated by (1.24) and
,
,
and
are sequences in
satisfying the following conditions:



,
,
,
for some
with
,
.
Then, converges strongly to a point
, where
, which solves the variational inequality

which is the optimality condition fot the minimization problem

where is a potential function for
(i.e.,
for
).
Proof.
Put ,
for all
and
for all
in Theorem 3.3. Then, we have
. So, by Theorem 3.3, we can conclude the desired conclusion easily.
References
Stampacchia G: Formes bilinéaires coercitives sur les ensembles convexes. Comptes Rendus Academy of Sciences 1964, 258: 4413–4416.
Gabay D: Applications of the method of multipliers to variational inequalities. In Augmented Lagrangian Methods. Edited by: Fortin M, Glowinski R. North-Holland, Amsterdam, The Netherlands; 1983:299–331.
Noor MA, Oettli W: On general nonlinear complementarity problems and quasi-equilibria. Le Matematiche 1994, 49(2):313–331.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994, 63(1–4):123–145.
Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997, 78(1):29–41.
Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Mathematical and Computer Modelling 2008, 48(7–8):1033–1046. 10.1016/j.mcm.2007.12.008
Yao Y, Liou Y-C, Yao J-C: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory and Applications 2007, 2007:-12.
Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Applications. Yokohama, Yokohama, Japan; 2000:iv+276.
Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods. Journal of Optimization Theory and Applications 2004, 121(1):203–210.
Verma RU: General convergence analysis for two-step projection methods and applications to variational problems. Applied Mathematics Letters 2005, 18(11):1286–1292. 10.1016/j.aml.2005.02.026
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
Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2005, 61(3):341–350. 10.1016/j.na.2003.07.023
Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and Mathematical Systems. Volume 477. Springer, Berlin, Germany; 1999:187–201.
Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(1):99–112. 10.1016/j.na.2009.06.042
Cho YJ, Qin X, Kang SM: Some results for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Computational Analysis and Applications 2009, 11(2):294–316.
Cho YJ, Qin X, Kang JI: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):4203–4214. 10.1016/j.na.2009.02.106
Hu CS, Cai G: Viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(3–4):1792–1808. 10.1016/j.na.2009.09.021
Huang N-J, Lan H-Y, Teo KL: On the existence and convergence of approximate solutions for equilibrium problems in Banach spaces. Journal of Inequalities and Applications 2007, 2007:-14.
Jaiboon C, Kumam P: Strong convergence theorems for solving equilibrium problems and fixed point problems of -strict pseudo-contraction mappings by two hybrid projection methods. Journal of Computational and Applied Mathematics. In press
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, Chantarangsi W, Kumam P: A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings. Nonlinear Analysis: Hybrid Systems 2010, 4(1):199–215. 10.1016/j.nahs.2009.09.009
Kangtunyakarn A, Suantai S: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(10):4448–4460. 10.1016/j.na.2009.03.003
Liu Q-Y, Zeng W-Y, Huang N-J: An iterative method for generalized equilibrium problems, fixed point problems and variational inequality problems. Fixed Point Theory and Applications 2009, 2009:-20.
Moudafi A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. Journal of Nonlinear and Convex Analysis 2008, 9(1):37–43.
Peng J-W, Wang Y, Shyu DS, Yao J-C: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems. Journal of Inequalities and Applications 2008, 2008:-15.
Qin X, Shang M, Su Y: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):3897–3909. 10.1016/j.na.2007.10.025
Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(3):1025–1033. 10.1016/j.na.2008.02.042
Zeng W-Y, Huang N-J, Zhao C-W: Viscosity approximation methods for generalized mixed equilibrium problems and fixed points of a sequence of nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-15.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005, 6: 117–136.
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
Shang M, Su Y, Qin X: Strong convergence theorem for nonexpansive mappings and relaxed cocoercive mappings. International Journal of Applied Mathematics and Mechanics 2007, 3(4):24–34.
Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numerical Functional Analysis and Optimization 1998, 19(1–2):33–56.
Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Studies in Computational Mathematics. Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.
Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002, 66(1):240–256. 10.1112/S0024610702003332
Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003, 116(3):659–678. 10.1023/A:1023073621589
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
Chang S-S, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(9):3307–3319. 10.1016/j.na.2008.04.035
Yao Y, Noor MA, Liou Y-C: On iterative methods for equilibrium problems. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):497–509. 10.1016/j.na.2007.12.021
Cho YJ, Qin X: Generalized systems for relaxed cocoercive variational inequalities and projection methods in Hilbert spaces. Mathematical Inequalities & Applications 2009, 12(2):365–375.
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001, 5(2):387–404.
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
Acknowledgments
The authors would like to express their thanks to the Faculty of Science KMUTT Research Fund for their financial support. The first author was supported by the Faculty of Applied Liberal Arts RMUTR Research Fund and King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT. The second author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5180034. Moreover, the authors are extremely grateful to the referees for their helpful suggestions that improved the content of the paper.
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Jaiboon, C., Kumam, P. Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities. J Inequal Appl 2010, 728028 (2010). https://doi.org/10.1155/2010/728028
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DOI: https://doi.org/10.1155/2010/728028
Keywords
- Variational Inequality
- Positive Real Number
- Nonexpansive Mapping
- Contraction Mapping
- Maximal Monotone