- Yongchun Xu
^{1}, - Xin Zhang
^{2}, - Jinlong Kang
^{2}and - Yongfu Su
^{2}Email author

**2010**:170701

https://doi.org/10.1155/2010/170701

© Yongchun Xu et al. 2010

**Received: **19 March 2010

**Accepted: **16 August 2010

**Published: **19 August 2010

## Abstract

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.

## 1. Introduction

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] 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 [4] 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 that if the sequence is bounded above from one, then the sequence generated by (1.3) converges strongly to .

They proved that if the sequence is bounded above from one, then the sequence generated by (1.5) converges strongly to , where denote the common fixed points set of .

where is a closed convex subset of . They proved that if the sequence is bounded above from one and , then the sequence generated by (1.8) converges strongly to .

where is a closed convex subset of . They proved that if the sequence , then the sequence generated by (1.9) converges strongly to .

They proved the following convergence theorem.

Theorem 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 (1.10), 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.

Theorem ZGT.

*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 [13] established strong convergence theorems by the hybrid method for a family of relatively nonexpansive mappings as follows.

Theorem KT.

*Let*

*be a strictly convex reflexive Banach space having the Kadec-Klee property and a Fréchet differentiable norm, and let*

*be a nonempty and closed convex subset of*

*and*

*a family of relatively nonexpensive mappings of*

*into itself having a common fixed point. Let*

*be a sequence in*

*such that*

*. For an arbitrarily chosen point*

*, generate a sequence*

*by the following iterative scheme:*

*, and*

for every , then converges strongly to , where is the set of common fixed points of and is the metric projection of onto a nonempty closed convex subset of .

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. [11], and Kimura and Takahashi [13], and some others.

## 2. Preliminaries

for all , where denotes the dual space of and the generalized duality pairing between and . It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of .

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 [14] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

for all . Observe that, in a Hilbert space , (2.2) reduces to , , .

Remark 2.1.

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.

We remark that a -asymptotically nonexpansive mapping with a nonempty fixed point set is a quasi- -asymptotically nonexpansive mapping, but the converse may be not true.

We present some examples which are closed and quasi- -asymptotically nonexpansive.

Example 2.2.

Then is continuous quasi-nonexpansive, and hence it is closed and nonexpansive with the constant sequence but not asymptotically nonexpansive.

Example 2.3.

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 .

Example 2.4.

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 [19] of as follows. If satisfies that , it is said that converges to in the sense of Mosco, and we write . For more details, see [20].

The following theorem plays an important role in our results.

Theorem 2.5 (see Ibaraki et al. [21]).

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 [16]).

Let be a uniformly convex and smooth Banach space, and let , be two sequences of if and either or is bounded, then .

Lemma 2.7 (Alber [14]).

Lemma 2.8.

Let be a uniformly convex and smooth Banach space, let be a closed convex subset of , and let be a closed and -asympotically nonexpansive mapping from into itself. Then is a closed convex subset of .

## 3. A Modified Algorithm and Strong Convergence Theorems

Now we are in a proposition to prove the main results of this paper. In the sequel, we use the letter to denote an index set.

Theorem 3.1.

Then converges strongly to , where is the generalized projection from onto .

Proof.

Firstly, we show that is closed and convex for each .

which infers that , so we get that is convex for each . Thus is closed and convex for every .

Secondly, we prove that , for all .

which infers that , for all and , and hence . This proves that , for all and .

By Theorem 2.5, converges strongly to .

as , that is, . Now the closedness property of gives that is a common fixed point of the family , thus .

Finally, since and is a nonempty closed convex subset of , we conclude that . This completes the proof.

Remark 3.2.

The boundedness assumption on in Theorem ZGT can be dropped.

Remark 3.3.

The asymptotic regularity assumption on in Theorem 3.1 can be weakened to the assumption that as .

Remark 3.4.

The assumption that as can be replaced by the uniform Lipschitz continuity of .

With above observations, we have the following convergence result.

Corollary 3.5.

Then converges strongly to , where is the generalized projection from onto .

Proof.

so that as . By Theorem 3.1, we have the desired conclusion. This completes the proof.

When in Theorem 3.1, we obtain the following result.

Corollary 3.6.

Then converges strongly to , where is the generalized projection from onto .

When in Theorem 3.1, we obtain the following result.

Corollary 3.7.

Then converges strongly to , where is the generalized projection from onto .

In the spirit of Theorem 3.1, we can prove the following strong convergence theorem.

Theorem 3.8.

Then converges strongly to , where is the generalized projection from onto .

Proof.

- (1)
- (2)
- (3)
- (4)
- (5)

The closedness property of together with (4) and (5) implies that converges strongly to a common fixed point of the family . As shown in Theorem 3.1, . This completes the proof.

When in Theorem 3.8, we obtain the following result.

Corollary 3.9.

Then converges strongly to , where is the generalized projection from onto .

When in Theorem 3.8, we obtain the following result.

Corollary 3.10.

Then converges strongly to , where is the generalized projection from onto .

## Declarations

### Acknowledgments

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).

## Authors’ Affiliations

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