- Research Article
- Open Access
Iterative Schemes for Generalized Equilibrium Problem and Two Maximal Monotone Operators
© L. C. Zeng et al. 2009
- Received: 20 July 2009
- Accepted: 27 October 2009
- Published: 3 November 2009
The purpose of this paper is to introduce and study two new hybrid proximal-point algorithms for finding a common element of the set of solutions to a generalized equilibrium problem and the sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex Banach space. We established strong and weak convergence theorems for these two modified hybrid proximal-point algorithms, respectively.
- Hilbert Space
- Banach Space
- Variational Inequality
- Nonexpansive Mapping
- Lower Semicontinuous
Whenever a Hilbert space, problem (1.2) was very recently introduced and considered by Kamimura and Takahashi . Problem (1.2) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for example, [13, 14]. A mapping is called nonexpansive if for all . Denote by the set of fixed points of , that is, . Iterative schemes for finding common elements of EP and fixed points set of nonexpansive mappings have been studied recently; see, for example, [12, 15–17] and the references therein.
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  and generally studied by Rockafellar  in the framework of a Hilbert space . Later many authors studied (1.5) and its variants in a Hilbert space or in a Banach space ; see, for example, [13, 19–23] and the references therein.
Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the following conditions (A1)–(A4) which were imposed in :
(A1) for all ;
(A2) is monotone, that is, , for all ;
(A3)for all ;
(A4)for all is convex and lower semicontinuous.
Let be a maximal monotone operator such that
The purpose of this paper is to introduce and study two new iterative algorithms for finding a common element of the set of solutions for the generalized equilibrium problem (1.2) and the set for maximal monotone operators in a uniformly smooth and uniformly convex Banach space . First, motivated by Kamimura and Takahashi [12, Theorem ], Ceng et al. [16, Theorem ], and Zhang [17, Theorem ], we introduce a sequence that, under some appropriate conditions, is strongly convergent to in Section 3. Second, inspired by Kamimura and Takahashi [12, Theorem ], Ceng et al. [16, Theorem ], and Zhang [17, Theorem ], we define a sequence weakly convergent to an element , where in Section 4. Our results represent a generalization of known results in the literature, including Takahashi and Zembayashi , Kamimura and Takahashi , Li and Song , Ceng and Yao , and Ceng et al. . In particular, compared with Theorems and in , our results (i.e., Theorems 3.2 and 4.2 in this paper) extend the problem of finding an element of to the one of finding an element of . Meantime, the algorithms in this paper are very different from those in  (because of considering the complexity involving the problem of finding an element of ).
In the sequel, we denote the strong convergence, weak convergence and weak* convergence of a sequence to a point by , and , respectively.
The proof of the main results of Sections 3 and 4 will be based on the following assumption.
Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying the same conditions (A1)–(A4) as in Section 1. Let be two maximal monotone operators such that
It is clear that in a Hilbert space , (2.2) reduces to .
Let be a mapping from into itself. A point in is called an asymptotically fixed point of if contains a sequence which converges weakly to such that . The set of asymptotically fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive 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 (2.4), we have which implies that . From the definition of , we have and therefore, ; see  for more details.
We need the following lemmas for the proof of our main results.
Lemma 2.1 (Kamimura and Takahashi ).
Let be a smooth and uniformly convex Banach space and let and be two sequences of . If and either or is bounded, then .
Lemma 2.4 (Rockafellar ).
Let be a reflexive strictly convex and smooth Banach space and let be a multivalued operator. Then there hold the following hold:
(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 2.5 (Xu ).
for all and , where .
Lemma 2.6 (Kamimura and Takahashi ).
The following result is due to Blum and Oettli .
Lemma 2.7 (Blum and Oettli ).
Lemma 2.8 (Takahashi and Zembayashi ).
for all . Then, the following hold:
(i) is singlevalued;
(iv) is closed and convex.
Using Lemma 2.8, one has the following result.
Lemma 2.9 (Takahashi and Zembayashi ).
Utilizing Lemmas 2.7, 2.8 and 2.9 as previously mentioned, Zhang  derived the following result.
Proposition 2.10 (Zhang [21, Lemma ]).
Let be a smooth strictly convex and reflexive Banach space and let be a nonempty closed convex subset of . Let be an -inverse-strongly monotone mapping, let be a bifunction from to satisfying (A1)–(A4), and let . Then the following hold:
then the mapping has the following properties:
(i) is singlevalued,
(iv) is a closed convex subset of ,
(v) for all
Lemma 2.11 (Kohsaka and Takahashi ).
Lemma 2.12 (Tan and Xu ).
Let and be two sequences of nonnegative real numbers satisfying the inequality: for all . If , then exists.
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 .
where and is the duality mapping on . In particular, whenever a real Hilbert space, is a nonexpansive mapping on .
and satisfies . Then, the sequence converges strongly to provided for any sequence with , where is the generalized projection of onto .
In this case, we can remove the requirement that for any sequence with .
Proof of Theorem 3.2.
We divide the proof into several steps.
We claim that is closed and convex for each .
This implies that . Therefore, is closed and convex.
We claim that for each and that is well defined.
As by the induction assumption, the last inequality holds, in particular, for all . This, together with the definition of implies that . Hence (3.20) holds for all . So, for all . This implies that the sequence is well defined.
We claim that is bounded and that as .
We claim that , , and .
and therefore, . Since is uniformly norm-to-norm continuous on bounded subsets of and , then .
Thus, from (3.35) it follows that . Since is uniformly norm-to-norm continuous on bounded subsets of , it follows that .
Thus . Therefore, we obtain that by the arbitrariness of .
We claim that converges strongly to .
which implies that . Since has the Kadec-Klee property, then . Therefore, converges strongly to .
In this case, the previous Theorem 3.2 reduces to [20, Theorem ].
In this section, we present the following algorithm for finding a common element of the set of solutions for a generalized equilibrium problem and the set for two maximal monotone operators and .
where , and , is defined by (2.15).
Before proving a weak convergence theorem, we need the following proposition.
Then, converges strongly to , where is the generalized projection of onto .
for all . So, from , and Lemma 2.12, we deduce that exists. This implies that is bounded. Thus, is bounded and so are , and .
Hence, from (4.12), it follows that .
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.
and satisfies . If is weakly sequentially continuous, then converges weakly to , where .
From Lemma 2.5 and as in the proof of Theorem 3.2, there exists a continuous, strictly increasing, and convex function with such that
Thus, from , and , it follows that . In terms of Lemma 2.1, we derive .
Next, let us show that , where .
Then the maximality of the operators implies that .
From the strict convexity of , it follows that . Therefore, , where . This completes the proof.
Suppose that conditions (A1)–(A5) are fulfilled and let be a sequence defined by (4.65), where is defined in Lemma 2.8, satisfy the conditions and , and satisfies . If is weakly sequentially continuous, then converges weakly to , where .
The first author (http://email@example.com) was partially supported by the National Science Foundation of China (10771141), Ph. D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author (http://firstname.lastname@example.org) was partially supported by the Grant NSF 97-2115-M-110-001.
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