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.
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 :
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 ).
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
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 ).
Lemma 2.4 (Rockafellar ).
Lemma 2.5 (Xu ).
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 ).
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:
Lemma 2.11 (Kohsaka and Takahashi ).
Lemma 2.12 (Tan and Xu ).
3. Strong Convergence Theorem
Proof of Theorem 3.2.
We divide the proof into several steps.
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.
4. Weak Convergence Theorem
Before proving a weak convergence theorem, we need the following proposition.
Now, we are in a position to prove the following theorem.
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://firstname.lastname@example.org) 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://email@example.com) was partially supported by the Grant NSF 97-2115-M-110-001.
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