# Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces

- Feng Gu
^{1}Email author and - Qiuping Fu
^{2}

**2009**:819036

**DOI: **10.1155/2009/819036

© F. Gu and Q. Fu. 2009

**Received: **19 November 2008

**Accepted: **9 April 2009

**Published: **15 April 2009

## Abstract

We establish strong convergence theorem for multi-step iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. Our results extend and improve the recent ones announced by Plubtieng and Wangkeeree (2006), and many others.

## 1. Introduction

From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically nonexpansive in the intermediate sense.

where , are bounded sequences in and , , , are appropriate real sequences in such that for each .

The purpose of this paper is to establish a strong convergence theorem for common fixed points of the multistep iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in a uniformly convex Banach space. The results presented in this paper extend and improve the corresponding ones announced by Plubtieng and Wangkeeree [2], and many others.

## 2. Preliminaries

Definition 2.1 (see [1]).

Lemma 2.2 (see [3]).

where and . Then

(1) exists;

(2)If , then .

Lemma 2.3 (see [4]).

## 3. Main Results

Lemma 3.1.

Let be a uniformly convex Banach space, , are two sequences of , and be a real sequence. If there exists such that

(i) for all ;

(ii) ;

(iii) ;

(iv) ,

then .

Proof.

The proof is clear by Lemma 2.3.

Lemma 3.2.

so that . Let , , and be real sequences in satisfying the following condition:

(i) for all and ;

(ii) for all .

If is the iterative sequence defined by (1.3), then, for each , the limit exists.

Proof.

This together with Lemma 2.2 gives that exists. This completes the proof.

Lemma 3.3.

so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then we have the following:

(1) ;

(2) .

Proof.

This completes the proof of .

This completes the proof.

Theorem 3.4.

so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then converges strongly to a common fixed point of the mappings .

Proof.

for all . This completes the proof.

Remark 3.5.

Theorem 3.4 improves and extends the corresponding results of Plubtieng and Wangkeeree [2] in the following ways.

Remark 3.6.

If and in Theorem 3.4, we obtain strong convergence theorem for Noor iteration scheme with error for asymptotically nonexpansive mapping in the intermediate sense in Banach space, we omit it here.

## Authors’ Affiliations

## References

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.