© Yongchun Xu et al. 2010
Received: 19 March 2010
Accepted: 16 August 2010
Published: 19 August 2010
The purpose of this paper is to propose a modified hybrid projection algorithm and prove strong convergence theorems for a family of quasi- -asymptotically nonexpansive mappings. The method of the proof is different from the original one. Our results improve and extend the corresponding results announced by Zhou et al. (2010), Kimura and Takahashi (2009), and some others.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk  in 1972. They proved that if is a nonempty bounded closed convex subset of a uniformly convex Banach space , then every asymptotically nonexpansive self-mapping of has a fixed point. Further, the set of fixed points of is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [1–3] and the references therein).
It is well known that in an infinite-dimensional Hilbert space, the normal Mann's iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann's iteration algorithm; the so-called hybrid projection iteration method is such a modification.
The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau  in 1968. For 40 years, (HPIA) has received rapid developments. For details, the readers are referred to papers in [5–11] and the references therein.
where is a closed convex subset of , denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded above from one then the sequence generated by (1.2) converges strongly to , where denote the fixed points set of .
They proved the following convergence theorem.
Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by (1.10), where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .
They proved the following convergence theorem.
Let be a nonempty bounded closed convex subset of a uniformly convex and uniformly smooth Banach space , and let be a family of -asymptotically nonexpansive mappings such that . Assume that every , is asymptotically regular on . Let be a real sequence in such that . Define a sequence as given by ( 1 ), then converges strongly to , where , for all , , and is the generalized projection from onto .
Very recently, Kimura and Takahashi  established strong convergence theorems by the hybrid method for a family of relatively nonexpansive mappings as follows.
Motivated by these results above, the purpose of this paper is to propose a Modified hybrid projection algorithm and prove strong convergence theorems for a family of - -asymptotically nonexpansive mappings which are asymptotically regular on . In order to get the strong convergence theorems for such a family of mappings, the classical hybrid projection iteration algorithm is modified and then is used to approximate the common fixed points of such a family of mappings. In the meantime, the method of the proof is different from the original one. Our results improve and extend the corresponding results announced by Zhou et al. , and Kimura and Takahashi , and some others.
It is also very well known that if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces , and consequently, 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.
If is a reflexive strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (2.4), we have . This implies that From the definitions of , we have . That is, see [17, 18] for more details.
Let be a uniformly smooth and strictly convex Banach space, and is a maximal monotone mapping such that is nonempty. Then, is a closed and quasi- -asymptotically nonexpansive mapping from onto , and .
Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed and quasi- -asymptotically nonexpansive mapping from onto with .
Let be a sequence of nonempty closed convex subsets of a reflexive Banach space . We denote 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 that for all . We define the Mosco convergence  of as follows. If satisfies that , it is said that converges to in the sense of Mosco, and we write . For more details, see .
The following theorem plays an important role in our results.
Theorem 2.5 (see Ibaraki et al. ).
Let be a smooth, reflexive, and strictly convex Banach space 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 .
We also need the following lemmas for the proof of our main results.
Lemma 2.6 (Kamimura and Takahashi ).
Lemma 2.7 (Alber ).
3. A Modified Algorithm and Strong Convergence Theorems
With above observations, we have the following convergence result.
In the spirit of Theorem 3.1, we can prove the following strong convergence theorem.
This project is supported by the Zhangjiakou city technology research and development projects foundation (0911008B-3), Hebei education department research projects foundation (2006103) and Hebei north university research projects foundation (2009008).
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