An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings
© Rabian Wangkeeree. 2008
Received: 28 February 2008
Accepted: 13 July 2008
Published: 14 July 2008
We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. (2007) and many others.
The set of solutions of (1.1) is denoted by Given a mapping , let for all . Then if and only if for all that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). In 1997, Flåm and Antipin  introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem.
where and satisfy some parameters controlling conditions. They proved that the sequence defined by (1.6) converges strongly to a common element of .
They proved that under certain appropriate conditions imposed on and , the sequences and converge strongly to , where
Moreover, Aoyama et al.  introduced an iterative scheme for finding a common fixed point of a countable family of nonexpansive mappings in Banach spaces and obtained the strong convergence theorem for such scheme.
In this paper, motivated by Yao et al. , S. Takahashi and W. Takahashi  and Aoyama et al. , we introduce a new extragradient method (4.2) which is mixed the iterative schemes considered in [10–12] for finding a common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the solution set of the classical variational inequality problem for a monotone -Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Further, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. The results obtained in this paper improve and extend the recent ones announced by Yao et al. results  and many others.
Then is the maximal monotone and if and only if ; see .
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,
Lemma 2.3 (see ).
where is a sequence in and is a sequence in such that
Lemma 2.4 (see [12, Lemma 3.2]).
For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:
(A1) for all
(A2) is monotone, that is, for all
(A4)for each is convex and lower semicontinuous.
The following lemma appears implicitly in .
Lemma 2.5 (see ).
The following lemma was also given in .
Lemma 2.6 (see ).
for all . Then, the following hold:
(i) is single-valued;
(ii) is firmly nonexpansive, that is, for any
(iv) is closed and convex.
3. Main Results
In this section, we prove a strong convergence theorem.
where , , and satisfy the following conditions:
Suppose that for any bounded subset of . Let be a mapping of into itself defined by and suppose that . Then the sequences , and converge strongly to the same point , where .
Therefore, is a contraction of into itself, which implies that there exists a unique element such that . Then we divide the proof into several steps.
Step 1 ( is bounded).
Therefore, is bounded. Hence, so are , and .
Step 2 ( ).
Step 3 ( ).
Step 4 ( ).
Step 5 ( ).
Setting , we have . Applying Lemma 2.3 to (3.56), we conclude that converges strongly to . Consequently, and converge strongly to . This completes the proof.
As in [12, Theorem 4.1], we can generate a sequence of nonexpansive mappings satisfying condition for any bounded subset of by using convex combination of a general sequence of nonexpansive mappings with a common fixed point.
Let be a closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A4), a monotone, -Lipschitz continuous mapping and let be a family of nonnegative numbers with indices with such that
(i) for all ;
(ii) for every ;
where , and satisfy the following conditions:
Then the sequences , and converge strongly to the same point , where .
Setting and in Theorem 3.1, we have the following result.
Corollary 3.3 (see [10, Theorem 3.1]).
where are sequences in satisfying the following conditions:
Then converges strongly to
Put for all and in Theorem 3.1. Thus, we have . Then the sequence generated in Corallary 3.3 converges strongly to .
In this section, we consider the problem of finding a zero of a monotone operator. A multivalued operator with domain and range is said to be monotone if for each and we have . A monotone operator is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. Let denote the identity operator on and let be a maximal monotone operator. Then we can define, for each , a nonexpansive single-valued mapping by . It is called the resolvent (or the proximal mapping) of . We also define the Yosida approximation by . We know that and for all . We also know that for all ; see, for instance, Rockafellar  or Takahashi .
Lemma 4.1 (the resolvent identity).
By using Theorem 3.1 and Lemma 4.1, we may obtain the following improvement.
where , and satisfy the following conditions:
Then converges strongly to .
By the assumption that , we obtain for some . Since , we obtain that for all . Since for all , we have . Therefore, by Theorem 3.1, converges strongly to .
The author would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper. This research was partially supported by the Commission on Higher Education.
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