© Min Liu et al. 2010
Received: 18 September 2009
Accepted: 18 February 2010
Published: 3 March 2010
We prove a strong convergence theorem for finding a common element of the set of solutions for generalized equilibrium problems, the set of fixed points of a relatively nonexpansive mapping, and the set of fixed points of a quasi- -nonexpansive mapping in a Banach space by using the shrinking Projection method. Our results improve the main results in S. Takahashi and W. Takahashi (2008) and Takahashi and Zembayashi (2008). Moreover, the method of proof adopted in the paper is different from that of S. Takahashi and W. Zembayashi (2008).
Let be a Banach space and let be a closed convex subsets of . Let be an equilibrium bifunction from into and let be a nonlinear mapping. Then, we consider the following generalized equilibrium problem: find such that
Recently, in Hilbert space, Tada and Takahashi , and S. Takahashi and W. Takahashi  considered iterative methods for finding an element of . Very recently, S. Takahashi and W. Takahashi  introduced an iterative method for finding an element of , where is an inverse-strongly monotone mapping and then proved a strong convergence theorem. On the other hand, Takahashi and Zembayashi  prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using the shrinking Projection method which is different from S. Takahashi and W. Takahashi's hybrid method .
In this paper, motivated by Takahashi and Zembayashi , in Banach space, we prove a strong convergence theorem for finding an element of , where is a continuous and monotone operator, is a relatively nonexpansive mapping, and is quasi- -nonexpansive mapping. Moreover, the method of proof adopted in the paper is different from that of .
Throughout this paper, all the Banach spaces are real. We denote by and the sets of positive integers and real numbers, respectively. Let be a Banach space and let be the topological dual of . For all and , we denote the value of at by . Then, the duality mapping on is defined by
for every . By the Hahn-Banach theorem, is nonempty; see  for more details. We denote the weak convergence and the strong convergence of a sequence to in by and , respectively. We also denote the convergence of a sequence to in by . A Banach space is said to be strictly convex if for with and . It is also said to be uniformly convex if for each , there exists such that for with and . A uniformly convex Banach space has the Kadec-Klee property, that is, and imply . The space is said to be smooth if the limit
exists for all . It is also said to be uniformly smooth if the limit exists uniformly in . We know that if is smooth, strictly convex, and reflexive, then the duality mapping is single valued, one to one, and onto; see  for more details.
Following Alber , the generalized projection from onto is defined by , where is the solution to the following minimization problem:
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , and let be a mapping from into itself. We denoted by the set of fixed points of . A point is said to be an asymptotic fixed point of [9, 10] if there exists in which converges weakly to and . We denote the set of all asymptotic fixed point of by . Following Matsushita and Takahashi , a mapping is said to be relatively nonexpansive if the following conditions are satisfied:
The following lemma is due to Matsushita and Takahashi .
Lemma 2.3 (Matsushita and Takahashi ).
We also know the following lemmas.
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.7 (see ).
Lemma 2.8 (see ).
3. The Main Results
In this section, we prove a strong convergence theorem which is the main result in the paper.
We divide the proof of Theorem 3.1 into six steps.
This completes the proof of Theorem 3.1.
The authors would like to express their thanks to the referees for their helpful suggestions and comments.
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