© H. Zhou and X. Gao. 2009
Received: 26 May 2009
Accepted: 14 September 2009
Published: 29 September 2009
The purpose of this paper is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for a family of quasi- -nonexpansive mappings. The strong convergence theorem is proven in the more general reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of S. Matsushita and W. Takahashi (2005), X. L. Qin and Y. F. Su (2007), and others.
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 [2–7] and the references therein.
They proved the following convergence theorem.
Theorem 1 MT.
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 (1.1), where is the normalized duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto .
They proved the following convergence theorem.
Theorem 1 QS.
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 such that . Assume that and are sequences in such that and . Suppose that is given by (1.2). If is uniformly continuous, then converges strongly to .
Can both Theorems MT and QS be extended to more general reflexive, strictly convex, and smooth Banach spaces with the property (K)?
The purpose of this paper is to give some affirmative answers to the questions mentioned previously, by introducing a modified hybrid projection iteration algorithm and by proving a strong convergence theorem for a family of closed and quasi- -nonexpansive mappings by using new analysis techniques in the setting of reflexive, strictly convex, and smooth Banach spaces with the property (K). The results of this paper improve and extend the results of Matsushita and Takahashi , Qin and Su , and others.
where denotes the generalized duality pairing between and . It is well known that if is reflexive, strictly convex, and smooth, then is single-valued, demi-continuous and strictly monotone (see, e.g., [8, 9]).
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 in turn implies that From the smoothness of , we know that is single valued, and hence we have . Since is strictly convex, is strictly monotone, in particular, is one to one, which implies that one may consult [8, 9] for the details.
Let be a closed convex subset of , and a mapping from into itself. A point in is said to be asymptotic fixed point of  if contains a sequence which converges weakly to such that . The set of asymptotic fixed point of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [5, 14–16] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [14–16].
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- -nonexpansive mapping from onto with .
Recall that a Banach space has the property (K) if for any sequence and , if weakly and , then . For more information concerning property (K) the reader is referred to  and references cited therein.
In order to prove our main result of this paper, we need to the following facts.
Now we are in a proposition to prove the main results of this paper.
3. Main Results
We split the proof into six steps.
From Theorem 3.1, we can obtain the following corollary.
Theorem 3.1 and its corollary improve and extend Theorems MT and QS at several aspects.
(i)From uniformly convex and uniformly smooth Banach spaces extend to reflexive, strictly convex and smooth Banach spaces with the property (K). In Theorem 3.1 and its corollary the hypotheses on are weaker than the usual assumptions of uniform convexity and uniform smoothness. For example, any strictly convex, reflexive and smooth Musielak-Orlicz space satisfies our assumptions  while, in general, these spaces need not to be uniformly convex or uniformly smooth.
Corollary 3.2 presents some affirmative answers to Questions 1 and 2.
In this section, we present some applications of the main results in Section 3.
Now the desired conclusion follows from Theorem 3.1. This completes the proof.
This work was supported by the National Natural Science Foundation of China under Grant (10771050).
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