- Research Article
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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)
Define, 
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.
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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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Keywords
- Variational Inequality
- Positive Real Number
- Nonexpansive Mapping
- Contraction Mapping
- Maximal Monotone
