Research Article  Open  Published:
Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities
Journal of Inequalities and Applicationsvolume 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 inversestrongly monotone, [9, 10] if there exists a positive real number such that
If we say that is firmly nonexpansive. It is obvious that any inversestrongly monotone mapping is monotone and Lipschitz continuous.
(5) is calledrelaxedcocoercive 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 setvalued 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 inversestrongly 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 inversestrongly 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 inversestrongly monotone, Takahashi and Toyoda [30] introduced the following iterative scheme:
where is an inversestrongly 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 inversestrongly 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 singlevalued;
(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 inversestrongly 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.
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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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Keywords
 Variational Inequality
 Positive Real Number
 Nonexpansive Mapping
 Contraction Mapping
 Maximal Monotone