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Strong and Weak Convergence Theorems for Common Solutions of Generalized Equilibrium Problems and Zeros of Maximal Monotone Operators
Fixed Point Theory and Applications volume 2010, Article number: 590278 (2010)
Abstract
The purpose of this paper is to introduce and study two modified hybrid proximalpoint algorithms for finding a common element of the solution set EP of a generalized equilibrium problem and the set for two maximal monotone operators and defined on a Banach space . Strong and weak convergence theorems for these two modified hybrid proximalpoint algorithms are established.
1. Introduction
Let be a real Banach space with its dual . The mapping defined by
is called the normalized duality mapping. From the HahnBanach theorem, it follows that for each .
A Banach space is said to be strictly convex, if for all with . is said to be uniformly convex if for each , there exists such that for all with . Recall that each uniformly convex Banach space has the KadecKlee property, that is,
It is well known that if is strictly convex, then is singlevalued. In the sequel, we shall still denote the singlevalued normalized duality mapping by . Let be a nonempty closed convex subset of , a bifunction, and a nonlinear mapping. Very recently, Zhang [1] considered and studied the generalized equilibrium problem of finding such that
The set of solutions of (1.3) is denoted by . Problem (1.3) and related problems have been studied and investigated extensively in the literature; See, for example, [2–12] and references therein. Whenever , problem (1.3) reduces to the equilibrium problem of finding such that
The set of solutions of (1.4) is denoted by . Whenever , problem (1.3) reduces to the variational inequality problem of finding such that
The set of solutions of (1.5) is denoted by .
Whenever a Hilbert space, problem (1.3) was very recently introduced and considered by S. Takahashi and W. Takahashi [13]. Problem (1.3) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; See, for example, [1, 2, 4, 6–9, 14–17] which are references therein.
A mapping is called nonexpansive if for all . Denote by the set of fixed points of , that is, . Very recently, W. Takahashi and K. Zembayashi [18] proposed an iterative algorithm for finding a common element of the solution set of the equilibrium problem (1.4) and the set of fixed points of a relatively nonexpansive mapping in a Banach space . They also studied the strong and weak convergence of the sequences generated by their algorithm. In particular, they proposed the following iterative algorithm:
where for all , and for some . They proved that the sequence generated by the above algorithm converges strongly to , where is the generalized projection of onto . They have also studied the weak convergence of the sequence generated by the following algorithm:
to , where .
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be an inversestrongly monotone mapping and a bifunction satisfying the following conditions:
(A1) for all ;
(A2) is monotone, that is, , for all ;
(A3) for all , ;
(A4) for all , is convex and lower semicontinuous.
Let be two relatively nonexpansive mappings such that . Let be the sequence generated by
Zhang [1] proved the strong convergence of the sequence to under appropriate conditions.
On the other hand, a classic method of solving in a Hilbert space is the proximal point algorithm which generates, for any starting point , a sequence in by the iterative scheme
where is a sequence in , for each is the resolvent operator for , and is the identity operator on . This algorithm was first introduced by Martinet [19] and further studied by Rockafellar [20] in the framework of a Hilbert space . Later several authors studied (1.9) and its variants in the setting of a Hilbert space or in a Banach space ; See, for example, [15, 21–25] and references therein. Very recently, Li and Song [24] introduced and studied the following iterative scheme:
where and is the duality mapping on .
Algorithm (1.10) covers, as special cases, the algorithms introduced by Kohsaka and Takahashi [23] and Kamimura et al. [22] in a smooth and uniformly convex Banach space .
Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty closed convex subset of . Let be a maximal monotone operator such that:
(A5).
In addition, for each , define a mapping as follows:
for all .
Very recently, utilizing the ideas of the above algorithms in [15, 16, 18, 21, 22, 24], we [17] introduced two iterative methods for finding an element of and established the following strong and weak convergence theorems.
Theorem 1.1 (see [17]).
Suppose that conditions (A1)–(A5) are satisfied and let be chosen arbitrarily. Consider the sequence
where
is defined by (1.11), satisfy , , and satisfies . Then, the sequence converges strongly to , where is the generalized projection of onto .
Theorem 1.2 (see [17]).
Suppose that conditions (A1)–(A5) are satisfied and let be chosen arbitrarily. Consider the sequence
where is defined by (1.11), satisfy the conditions and , and satisfies . If is weakly sequentially continuous, then converges weakly to an element , where .
The purpose of this paper is to introduce and study two new iterative methods for finding a common element of the solution set of generalized equilibrium problem (1.3) and the set for maximal monotone operators and in a uniformly smooth and uniformly convex Banach space . Firstly, motivated by Theorem 1.1 and a result of Zhang [1], we introduce a sequence that converges strongly to under some appropriate conditions.
Secondly, inspired by Theorem 1.2 and a result of Zhang [1], we define a sequence that converges weakly to an element , where (Section 4).
Our results represent a generalization of known results in the literature, including those in [16–18, 24]. Our Theorems 3.1 and 4.2 are the extension and improvements of Theorems 1.1 and 1.2 in the following way:
(i)the problem of finding an element of includes the one of finding an element of as a special case;
(ii)the algorithms in this paper are very different from those in [17] because of considering the complexity involving the problem of finding an element of .
2. Preliminaries
Throughout the paper, we denote the strong convergence, weak convergence, and weak convergence of a sequence to a point by , and , respectively.
Assumption 2.1.
Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be an inversestrongly monotone mapping and let be a bifunction satisfying the conditions (A1)–(A4). Let be two maximal monotone operators such that:
(A5).
Recall that if is a nonempty closed convex subset of a Hilbert space , then the metric projection of onto is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. In this connection, Alber [26] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.
Consider the functional defined as in [26] by
It is clear that in a Hilbert space , (2.1) reduces to .
The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem
The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping ; See, for example, [27]. In a Hilbert space, . From [26], in a smooth, strictly convex and reflexive Banach space , we have
Moreover, by the property of subdifferential of convex functions, we easily get the following inequality:
Let be a mapping from into itself. A point in is called an asymptotic fixed point of [28] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of is denoted by . A mapping from into itself is called relatively nonexpansive [18, 29, 30] if and , for all and .
Observe that, if is a reflexive, strictly convex and smooth Banach space, then for any if and only if . To this end, it is sufficient to show that if , then . Actually, from (2.3), we have , which implies that . From the definition of , we have and therefore, . For further details, we refer to [31].
We need the following lemmas for the proof of our main results.
Lemma 2.2 (see [32]).
Let be a smooth and uniformly convex Banach space and let and be two sequences of . If and either or is bounded, then .
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , and . Then
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space . Then
Lemma 2.5 (see [33]).
Let be a reflexive, strictly convex and smooth Banach space and let be a multivalued operator. Then
(i) is closed and convex if is maximal monotone such that ;
(ii) is maximal monotone if and only if is monotone with for all .
Lemma 2.6 (see [34]).
Let be a uniformly convex Banach space and let . Then there exists a strictly increasing, continuous and convex function such that and
for all and , where .
Lemma 2.7 (see [32]).
Let be a smooth and uniformly convex Banach space and let . Then there exists a strictly increasing, continuous, and convex function such that and
The following result is due to Blum and Oettli [14].
Lemma 2.8 (see [14]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , a bifunction satisfying conditions (A1)–(A4), and and . Then, there exists such that
Motivated by a result in [35] in a Hilbert space setting, Takahashi and Zembayashi [18] established the following lemma.
Lemma 2.9 (see [18]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space , and a bifunction satisfying conditions (A1)–(A4). For and , define a mapping as follows:
for all . Then
(i) is singlevalued;
(ii) is a firmly nonexpansivetype mapping, that is, for all ,
(iii);
(iv) is closed and convex.
Using Lemma 2.9, we have the following result.
Lemma 2.10 (see [18]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , a bifunction satisfying conditions (A1)–(A4), and . Then, for and ,
Utilizing Lemmas 2.8, 2.9, and 2.10, Zhang [1] derived the following result.
Proposition 2.11 (see [1]).
Let be a smooth, strictly convex and reflexive Banach space and let be a nonempty closed convex subset of . Let be an inversestrongly monotone mapping, a bifunction satisfying conditions (A1)–(A4), and . Then

(I)
for , there exists such that
(2.13)

(II)
if is additionally uniformly smooth and is defined as
(2.14)
then the mapping has the following properties:
(i) is singlevalued,
(ii) is a firmly nonexpansivetype mapping, that is,
(iii),
(iv) is a closed convex subset of ,
(v) for all .
Proof.
Define a bifunction by
It is easy to verify that satisfies the conditions (A1)–(A4). Therefore, the conclusions (I) and (II) follow immediately from Lemmas 2.8, 2.9, and 2.10.
Let be two maximal monotone operators in a smooth Banach space . We denote the resolvent operators of and by and for each , respectively. Then and are two singlevalued mappings. Also, and for each , where and are the sets of fixed points of and , respectively. For each , the Yosida approximations of and are defined by and , respectively. It is known that
Lemma 2.12 (see [23]).
Let be a reflexive, strictly convex and smooth Banach space, and let be a maximal monotone operator with . Then,
Lemma 2.13 (see [36]).
Let and be two sequences of nonnegative real numbers such that for all . If , then exists.
3. Strong Convergence Theorem
In this section, we prove a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators and .
Theorem 3.1.
Suppose that Assumption 2.1 is satisfied. Let be chosen arbitrarily. Consider the sequence
where
is defined by (2.14), satisfy
and satisfies . Then, the sequence converges strongly to , where is the generalized projection of onto .
Proof.
For the sake of simplicity, we define
so that
We divide the proof into several steps.
Step 1.
We claim that is closed and convex for each .
Indeed, it is obvious that is closed and is closed and convex for each . Let us show that is convex. For and , put . It is sufficient to show that . We first write for each . Next, we prove that
is equivalent to
Indeed, from (2.1) we deduce that there hold the following:
which combined with (3.6) yield that (3.6) is equivalent to (3.7). Thus we have
This implies that . Therefore, is closed and convex.
Step 2.
We claim that for each and that is well defined.
Indeed, take arbitrarily. Note that is equivalent to
Then from Lemma 2.12, we obtain
Moreover, we have
and hence by Proposition 2.11,
So for all . Now, let us show that
We prove this by induction. For , we have . Assume that . Since is the projection of onto , by Lemma 2.3 we have
As by the induction assumption, the last inequality holds, in particular, for all . This, together with the definition of implies that . Hence (3.14) holds for all . So, for all . This implies that the sequence is well defined.
Step 3.
We claim that is bounded and that as .
Indeed, it follows from the definition of that . Since and , so for all , that is, is nondecreasing. It follows from and Lemma 2.4 that
for each for each . Therefore, is bounded, which implies that the limit of exists. Since
so is bounded. From Lemma 2.4, we have
for each . This implies that
Step 4.
We claim that , , and .
Indeed, from , we have
Therefore, from and , it follows that .
Since and is uniformly convex and smooth, we have from Lemma 2.2 that
and, therefore, . Since is uniformly normtonorm continuous on bounded subsets of and , then .
Let us set . Then, according to Lemma 2.5 and Proposition 2.11, we know that is a nonempty closed convex subset of such that . Fix arbitrarily. As in the proof of Step 2, we can show that ,
Hence it follows from the boundedness of that , and are also bounded. Let . Since is a uniformly smooth Banach space, we know that is a uniformly convex Banach space. Therefore, by Lemma 2.6 there exists a continuous, strictly increasing, and convex function , with , such that
for and . So, we have that
and hence
for all . Consequently, we have
Since and is uniformly normtonorm continuous on bounded subsets of , we obtain . From and , we have
Therefore, from the properties of , we get
recalling that is uniformly normtonorm continuous on bounded subsets of . Next let us show that
Observe first that
Since , and is bounded, so it follows that . Also, observe that
Since , and the sequences are bounded, so it follows that . Meantime, observe that
and hence
Since and , it follows from the boundedness of that . Thus, in terms of Lemma 2.2, we have that and so . Furthermore, it follows from (3.25) that
and hence
Since is uniformly normtonorm continuous on bounded subsets of , it follows from that . Thus from , , and the boundedness of both and , we deduce that . Utilizing the properties of , we have that . Since is uniformly normtonorm continuous on bounded subsets of , it follows that .
Step 5.
We claim that , where
Indeed, since is bounded and is reflexive, we know that . Take arbitrarily. Then there exists a subsequence of such that . Hence it follows from , , and that and converge weakly to the same point . On the other hand, from (3.28a), (3.28b) and , we obtain that
If and , then it follows from (2.17) and the monotonicity of the operators that for all
Letting , we have that and . Then the maximality of the operators implies that and .
Next, let us show that . Since
from and Proposition 2.11 it follows that
Also, since
so we get
So, from (3.39), , and , we have .
Since is uniformly convex and smooth, we conclude from Lemma 2.2 that
From , , and (3.42), we have and .
Since is uniformly normtonorm continuous on bounded subsets of , from (3.42) we derive
From , it follows that
By the definition of , we have
where
Replacing by , we have from (A2) that
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (3.44) and (A4), we have
For , with , and , let . Since and , then and hence . So, from (A1) we have
Dividing by , we have
Letting , from (A3) it follows that
So, . Therefore, we obtain that by the arbitrariness of .
Step 6.
We claim that converges strongly to .
Indeed, from and , it follows that
Since the norm is weakly lower semicontinuous, then
From the definition of , we have . Hence , and
which implies that . Since has the KadecKlee property, then . Therefore, converges strongly to .
Remark 3.2.
In Theorem 3.1, let , , and . Then, for all and , we have that
Moreover, there hold the following
and hence
In this case, Theorem 3.1 reduces to [17, Theorem ].
4. Weak Convergence Theorem
In this section, we present the following algorithm for finding a common element of the solution set of a generalized equilibrium problem and the set for two maximal monotone operators and .
Let be chosen arbitrarily and consider the sequence generated by
where , and is defined by (2.14).
Before proving a weak convergence theorem, we need the following proposition.
Proposition 4.1.
Suppose that Assumption 2.1 is fulfilled and let be a sequence defined by (4.1), where satisfy the following conditions:
Then, converges strongly to , where is the generalized projection of onto .
Proof.
We set and
so that
Then, in terms of Lemma 2.5 and Proposition 2.11, is a nonempty closed convex subset of such that . We first prove that is bounded. Fix . Note that by the first and third of (4.3), and
Here, each is relatively nonexpansive. Then from Proposition 2.11, we obtain
and hence by Proposition 2.11
Consequently, the last two inequalities yield that
for all . So, from , , and Lemma 2.13, we deduce that exists. This implies that is bounded. Thus, is bounded and so are , , , and .
Define for all . Let us show that is bounded. Indeed, observe that
for each . This, together with the boundedness of , implies that is bounded and so is . Furthermore, from and (4.6e), we have
Since is the generalized projection, then, from Lemma 2.4 we obtain
Hence, from (4.8), it follows that .
Note that , , and is bounded, so that . Therefore, is a convergent sequence. On the other hand, from (4.6e) we derive, for all ,
In particular, we have
Consequently, from and Lemma 2.4, we have
and hence
Let . From Lemma 2.7, there exists a continuous, strictly increasing, and convex function with such that
So, we have
Since is a convergent sequence, is bounded and is convergent; from the property of , we have that is a Cauchy sequence. Since is closed, converges strongly to . This completes the proof.
Now, we are in a position to prove the following theorem.
Theorem 4.2.
Suppose that Assumption 2.1 is fulfilled and let be a sequence defined by (4.1), where satisfy the following conditions:
and satisfies . If is weakly sequentially continuous, then converges weakly to , where .
Proof.
We consider the notations (4.3). As in the proof of Proposition 4.1, we have that , and are bounded sequences. Let
From Lemma 2.6 and as in the proof of Theorem 3.1, there exists a continuous, strictly increasing, and convex function with such that
for and . Observe that for ,
Hence,
Consequently, the last two inequalities yield that
Thus, we have
By the proof of Proposition 4.1, it is known that is convergent; since , , , and , then we have
Taking into account the properties of , as in the proof of Theorem 3.1, we have
since is uniformly normtonorm continuous on bounded subsets of .
Now let us show that
Indeed, from (4.6e) we get
which, together with , yields that
From (4.6d) it follows that
which, together with , yields that
From (4.6c) it follows that
which, together with , yields that
From (4.6c) it follows that
which together with
yields that
From (4.6a) it follows that
which, together with , yields that
On the other hand, let us show that
Indeed, let . From Lemma 2.7, there exists a continuous, strictly increasing, and convex function with such that
Since and , we deduce from Proposition 2.11 that for ,
This implies that
Since is uniformly normtonorm continuous on bounded subsets of , from the properties of , we obtain
Note that
Since , it follows from (4.24) and (4.41) that and .
Also, observe that
and hence
Thus, from , and , it follows that . In terms of Lemma 2.2, we derive .
Next, let us show that , where .
Indeed, since is bounded, there exists a subsequence of such that . Hence it follows from (4.24), (4.41), and that and converge weakly to the same point . Furthermore, from and (4.24), we have that
If and , then it follows from (2.17) and the monotonicity of the operators that for all
Letting , we obtain that
Then the maximality of the operators implies that .
Now, by the definition of , we have
where . Replacing by , we have from (A2) that
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting in the last inequality, from (4.41) and (A4), we have
For , with , and , let . Since and , then and hence . So, from (A1), we have
Dividing by , we get . Letting , from (A3) it follows that . So, . Therefore, . Let . From Lemma 2.3 and , we get
From Proposition 4.1, we also know that . Note that . Since is weakly sequentially continuous, then as . In addition, taking into account the monotonicity of , we conclude that . Hence
From the strict convexity of , it follows that . Therefore, , where . This completes the proof.
Remark 4.3.
Compared with the algorithm of Theorem 1.2, the above algorithm (4.1) can be applied to find an element of . But, the algorithm of Theorem 1.2 cannot be applied. Therefore, algorithm (4.1) develops and improves the algorithm of Theorem 1.2.
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Acknowledgments
In this research, the first author was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). The Fourth author was partially supported by a grant NSC 982115M110001.
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Zeng, L., Ansari, Q., Shyu, D. et al. Strong and Weak Convergence Theorems for Common Solutions of Generalized Equilibrium Problems and Zeros of Maximal Monotone Operators. Fixed Point Theory Appl 2010, 590278 (2010). https://doi.org/10.1155/2010/590278
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Keywords
 Hilbert Space
 Banach Space
 Convex Function
 Equilibrium Problem
 Lower Semicontinuous