Strong and Weak Convergence Theorems for Common Solutions of Generalized Equilibrium Problems and Zeros of Maximal Monotone Operators
© L.-C. Zeng et al. 2010
Received: 27 October 2009
Accepted: 12 January 2010
Published: 21 January 2010
The purpose of this paper is to introduce and study two modified hybrid proximal-point 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 proximal-point algorithms are established.
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 Kadec-Klee property, that is,
It is well known that if is strictly convex, then is single-valued. In the sequel, we shall still denote the single-valued normalized duality mapping by . Let be a nonempty closed convex subset of , a bifunction, and a nonlinear mapping. Very recently, Zhang  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
Whenever a Hilbert space, problem (1.3) was very recently introduced and considered by S. Takahashi and W. Takahashi . 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  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:
Zhang  proved the strong convergence of the sequence to under appropriate conditions.
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 further studied by Rockafellar  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  introduced and studied the following iterative scheme:
Very recently, utilizing the ideas of the above algorithms in [15, 16, 18, 21, 22, 24], we  introduced two iterative methods for finding an element of and established the following strong and weak convergence theorems.
Theorem 1.1 (see ).
Theorem 1.2 (see ).
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 , we introduce a sequence that converges strongly to under some appropriate conditions.
Secondly, inspired by Theorem 1.2 and a result of Zhang , 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:
(ii)the algorithms in this paper are very different from those in  because of considering the complexity involving the problem of finding an element of .
Let be a uniformly smooth and uniformly convex Banach space and let be a nonempty closed convex subset of . Let be an -inverse-strongly monotone mapping and let be a bifunction satisfying the conditions (A1)–(A4). Let be two maximal monotone operators such that:
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  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  by
The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping ; See, for example, . In a Hilbert space, . From , 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  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 .
We need the following lemmas for the proof of our main results.
Lemma 2.2 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.7 (see ).
The following result is due to Blum and Oettli .
Lemma 2.8 (see ).
Lemma 2.9 (see ).
Using Lemma 2.9, we have the following result.
Lemma 2.10 (see ).
Utilizing Lemmas 2.8, 2.9, and 2.10, Zhang  derived the following result.
Proposition 2.11 (see ).
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 single-valued 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 ).
Lemma 2.13 (see ).
3. Strong Convergence Theorem
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 (3.14) holds for all . So, for all . This implies that the sequence is well defined.
Observe first that
Since is uniformly norm-to-norm 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 norm-to-norm continuous on bounded subsets of , it follows that .
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
In this case, Theorem 3.1 reduces to [17, Theorem ].
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.
Now let us show that
Indeed, from (4.6e) we get
From (4.6d) it follows that
From (4.6c) it follows that
From (4.6c) it follows that
From (4.6a) it follows that
On the other hand, let us show that
Also, observe that
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.
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 98-2115-M-110-001.
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