# Approximation Methods for Common Fixed Points of Mean Nonexpansive Mapping in Banach Spaces

- Zhaohui Gu
^{1}and - Yongjin Li
^{2}Email author

**2008**:471532

**DOI: **10.1155/2008/471532

© Z. Gu and Y. Li. 2008

**Received: **17 October 2007

**Accepted: **2 January 2008

**Published: **15 January 2008

## Abstract

Let be a uniformly convex Banach space, and let be a pair of mean nonexpansive mappings. In this paper, it is proved that the sequence of Ishikawa iterations associated with and converges to the common fixed point of and .

## 1. Introduction and Preliminaries

for all , , .

where , . The recursion formulas (1.2) were first introduced in 1994 by Rashwan and Saddeek [2] in the framework of Hilbert spaces.

In recent years, several authors (see [2–6]) have studied the convergence of iterations to a common fixed point for a pair of mappings. Rashwan has studied the convergence of Mann iterations to a common fixed point (see [5]) and proved that the Ishikawa iterations converge to a unique common fixed point in Hilbert spaces (see [2]). Recently, Ćirić has proved that if the sequence of Ishikawa iterations sequence associated with and converges to , then is the common fixed point of and (see [7]). In [4, 6], the authors studied the same problem. In [1], Bose defined the pair of mean nonexpansive mappings, and proved the existence of the fixed point in Banach spaces. In particular, he proved the following theorem.

Theorem 1.1 ([1]).

- (i)
and have a common fixed point ;

- (ii)further, if , then
- (a)
is the unique common fixed point and unique as a fixed point of each and ,

- (b)
the sequence defined by , for any , converges strongly to .

- (a)

It is our purpose in this paper to consider an iterative scheme, which converges to a common fixed point of the pair of mean nonexpansive mappings. Theorem 2.1 extends and improves the corresponding results in [1].

## 2. Main Results

Now we prove the following theorem which is the main result of this paper.

Theorem 2.1.

Let be a uniformly convex Banach space, and are a pair of mean nonexpansive with a nonempty common fixed points set; if , then the Ishikawa sequence converges to the common fixed point of and .

Proof.

Let , by , it is easy to see that , thus and .

Hence, is bounded.

for every with . It is easy to see that Banach space is uniformly convex if and only if for any implies .

we know is bounded, then there exists , such that . Thus, .

Let , then we have . It is a contradiction. Hence, .

So .

Similarly, we can prove that . So is the common fixed point of and .

So it is easy to see that . Thus, , that is is a Cauchy sequence. Hence, there exists , such that . We know that and is the common fixed point of and . This completes the proof of the theorem.

## Declarations

### Acknowledgment

The work was partially supported by the Emphases Natural Science Foundation of Guangdong Institution of Higher Learning, College and University (no. 05Z026).

## Authors’ Affiliations

## References

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## Copyright

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