© Min Liu et al. 2010
Received: 30 November 2009
Accepted: 19 January 2010
Published: 14 February 2010
We prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for generalized mixed equilibrium problems, the set of fixed points of a family of quasi- -asymptotically nonexpansive mappings in strictly convex reflexive Banach space with the Kadec-Klee property and, a Fréchet differentiable norm under weaker conditions. The method of the proof is different from, S. Takahashi and W. Takahashi that by (2008) and that by Takahashi and Zembayashi (2008) and see references. It also shows that the type of projection used in the iterative method is independent of the properties of the mappings. The results presented in the paper improve and extend some recent results.
Let be a Banach space and let be a closed convex subsets of . Let be an equilibrium bifunction from into , let be a real-valued function, and let be a nonlinear mapping. The "so-called" generalized mixed equilibrium problem is to find such that
which is called the mixed equilibrium problem; see . The set of solutions of (1.3) is denoted by MEP.
which is called the generalized equilibrium problem; see . The set of solutions of (1.5) is denoted by EP.
which is called the equilibrium problem. The set of solutions of (1.6) is denoted by EP(F).
Recently, Tada and Takahashi  and S. Takahashi and W. Takahashi  considered iterative methods for finding an element of in Hilbert space. Very recently, S.Takahashi and W.Takahashi  introduced an iterative method for finding an element of , where is an inverse-strongly monotone mapping and is nonexpansive mapping and then proved a strong convergence theorem in Hilbert space. 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. Very recently, Kimura and Takahashi  prove a strong convergence theorem for a family of relatively nonexpansive mapping in a Banach space by using a hybrid method.
In this paper, motivated by Kimura and Takahashi , we prove a strong convergence theorem for finding an element of in Banach space by using a hybrid method, where is a continuous and monotone operator and is a family of quasi- -asymptotically nonexpansive mapping. Moreover, the method of proof adopted in the paper is different from that of [2, 5].
Throughout this paper, we assume that 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 . The duality mapping is defined by
By 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. A Banach space is said to be strictly convex if for and . It is also said to be uniformly convex if for each there exists such that for and . is said to have the Kadec-Klee property, that is, for any sequence , if and , then .
for and . A norm of is said to be differentiable if has a limit for each . In this case, is said to be smooth. A norm of is said to be Fréchet differentiable if is attained uniformly for for each . It is known that has a Fréchet differentiable norm if and only if is strictly convex and reflexive, and has the Kadec-Klee property. We know that if is smooth, strictly convex, and reflexive, then the duality mapping is single valued, one to one, and onto. In this case, the inverse mapping coincides with the duality mapping on . See  for more details.
Let be a sequence of nonempty closed convex subset of a reflexive Banach space . We define two subsets and as follows: if and only if there exists such that converges strongly to and that for all . Similarly, if and only if there exists a subsequence of and a sequence such that converges weakly to and for all . We define the Mosco convergence  of as follows: if satisfies that , then it is said that converges to in the sense of Mosco and we write . For more details, see .
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space . Then, for arbitrarily fixed , a function has a unique minimizer . Using such a point, we define the metric projection by for every . In a similar fashion, we can see that a function has a unique minimizer . The generalized projection of onto is defined by for every ; see .
It is well-known that the following conclusions hold.
The following theorem proved by Tsukada  plays an important role in our results.
Let be a smooth, reflexive, and strictly convex Banach space having the Kadec-Klee property. Let be a sequence of nonempty closed convex subset of . If exists and is nonempty, then converges strongly to for each .
Theorem 2.4 is still valid if we replace the metric projections with the generalized projections as follows:
Let be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let be a sequence of nonempty closed convex subsets of . If exists and is nonempty, then converges strongly to for each .
Let be a nonempty closed convex subsets of , and let be a mapping from into itself. We denoted by the set of fixed points of . is said to be -asymptotically nonexpansive, if there exists some real sequence with and such that for all and . is said to be quasi- -asymptotically nonexpansive , if there exists some real sequence with and and such that for all , and . is said to be uniformly Lipschitzian continuous if there exists some such that for all and . A point is said to be an asymptotic fixed point of [15, 16] 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:
Let be a strictly convex reflexive Banach space having the Kadec-Klee property and a Fréchet differentiable norm, be a nonempty closed convex subset of , and let be a uniformly Lipschitzian continuous and quasi- -asymptotically nonexpansive mapping from into itself. Then is closed and convex.
that is, . This implies that . Thus from (2.8) we have . Since and has the Kadec-Klee property, we have . Note that is hemicontinuous, it yields that . Again since , by using the Kadec-Klee property of , we have . Hence as . Since is uniformly Lipschitzian continuous, we have . This completes the proof.
Lemma 2.7 (see).
Lemma 2.8 (see).
Lemma 2.9 (see).
3. The Main Results
Let be a strictly convex reflexive Banach space having a Fréchet differentiable norm, a nonempty closed convex subset of and a sequence of mappings of into itself. Let be a strongly convergent sequence in with a limit and a sequence in defined by for each , where is a convergent sequence in with a limit . Suppose that for all and that converges weakly to , where . Then converges strongly to 0. Moreover, if has the Kadec-Klee property, then converges strongly to .
which is the desired result.
We divide the proof of Theorem 3.2 into five steps.
Since for every , it follows that for every . Fix arbitrarily. From the assumption that , we may take subsequences of and of such that with and converges weakly to a point . Then, by Lemma 3.1, we have that
hence it follows from (3.38) and (3.40) that
This completes the proof of Theorem 3.2.
The proof of Theorem 3.2 shows that the properties of projections used in the iterative scheme do not interact with the properties of mappings . Therefore, we may prove similar results as follows by replacing Theorem 2.4 with Theorem 2.5 in the proof.
The authors would like to express their thanks to the referees for their helpful suggestions and comments.
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