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

*Journal of Inequalities and Applications*
**volumeÂ 2009**, ArticleÂ number:Â 819036 (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

Let be a subset of real normal linear space . A mapping is said to be asymptotically nonexpansive on if there exists a sequence in with such that for each ,

If , then is known as a nonexpansive mapping. is called asymptotically nonexpansive in the intermediate sense [1] provided is uniformly continuous and

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

Let be a nonempty subset of normed space , and Let be mappings. For a given and a fixed ( denotes the set of all positive integers), compute the iterative sequences defined by

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]).

A Banach space is said to be a uniformly convex if the modulus of convexity of is

Lemma 2.2 (see [3]).

Let , and be three nonnegative real sequences satisfying the following condition:

where and . Then

(1) exists;

(2)If , then .

Lemma 2.3 (see [4]).

Let be a uniformly convex Banach space and for all . Suppose that and are two sequences of such that

for some . Then

## 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.

Let be a uniformly convex Banach space, let be a nonempty closed bounded convex subset of , and let be asymptotically nonexpansive mappings in the intermediate sense such that . Put

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.

For each , we note that

where . Since

we see that

It follows from (3.2) that

where . Since

we see that

It follows from (3.5) that

where , and so

By continuing the above method, there are nonnegative real sequences such that

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

Lemma 3.3.

Let be a uniformly convex Banach space, let be a nonempty closed bounded convex subset of , and let be asymptotically nonexpansive mappings in the intermediate sense such that . Put

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.

Taking each , it follows from Lemma 3.2 that exists. Let

for some . We note that

where is a nonnegative real sequence such that

It follows that

which implies that

Next, we observe that

Thus we have

Also,

gives that

Note that

This together with (3.18), (3.20), and Lemma 3.1, gives

This completes the proof of .

For each ,

Since

we obtain

It follows that

which implies that

On the other hand, we note that

where is a nonnegative real sequence such that

Thus we have

and hence

Next, we observe that

Thus we have

Also,

gives that

Note that

Therefore, it follows from (3.33), (3.35), and Lemma 3.1 that

This completes the proof.

Theorem 3.4.

Let be a uniformly convex Banach space and let be a nonempty closed bounded convex subset of . Let be asymptotically nonexpansive mappings in the intermediate sense such that and there exists one member in which is completely continuous. Put

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.

From Lemma 3.3, it follows that

which implies that

and so

It follows from (3.22), (3.37) that

Let for all . Then we have

Notice that . Thus and the above inequality becomes

and so

Since

we have

and so

Since is bounded and one of is completely continuous, we may assume that is completely continuous, without loss of generality. Then there exists a subsequence of such that as . Moreover, by (3.48), we have

which implies that as . By (3.48) again, we have

It follows that . Since exists, we have

that is,

Moreover, we observe that

for all and

Therefore,

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.

The iterative process defined by (1.3) in [2] is replaced by the new iterative process defined by (1.3) in this paper.

Theorem 3.4 generalizes Theorem of Plubtieng and Wangkeeree [2] from a asymptotically nonexpansive mappings in the intermediate sense to a finite family of asymptotically nonexpansive mappings in the intermediate sense.

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.

## References

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**A fixed point theorem for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1972,**35**(1):171â€“174. 10.1090/S0002-9939-1972-0298500-3Plubtieng S, Wangkeeree R:

**Strong convergence theorems for multi-step Noor iterations with errors in Banach spaces.***Journal of Mathematical Analysis and Applications*2006,**321**(1):10â€“23. 10.1016/j.jmaa.2005.08.029Liu Q:

**Iterative sequences for asymptotically quasi-nonexpansive mappings with error member.***Journal of Mathematical Analysis and Applications*2001,**259**(1):18â€“24. 10.1006/jmaa.2000.7353Schu J:

**Iterative construction of fixed points of strictly pseudocontractive mappings.***Applicable Analysis*1991,**40**(2â€“3):67â€“72. 10.1080/00036819108839994

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Gu, F., Fu, Q. Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces.
*J Inequal Appl* **2009**, 819036 (2009). https://doi.org/10.1155/2009/819036

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DOI: https://doi.org/10.1155/2009/819036