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

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

**2008**:471532

https://doi.org/10.1155/2008/471532

© Z. Gu and Y. Li. 2008

**Received: **17 October 2007

**Accepted: **2 January 2008

**Published: **15 January 2008

## Abstract

## 1. Introduction and Preliminaries

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)
- (ii)

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 .

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, .

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

- Bose SC: Common fixed points of mappings in a uniformly convex Banach space.
*Journal of the London Mathematical Society*1978, 18(1):151-156. 10.1112/jlms/s2-18.1.151MATHMathSciNetView ArticleGoogle Scholar - Rashwan RA, Saddeek AM: On the Ishikawa iteration process in Hilbert spaces.
*Collectanea Mathematica*1994, 45(1):45-52.MATHMathSciNetGoogle Scholar - Berinde V: On the convergence of the Ishikawa iteration in the class of quasi contractive operators.
*Acta Mathematica Universitatis Comenianae*2004, 73(1):119-126.MATHMathSciNetGoogle Scholar - Maingé P-E: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces.
*Journal of Mathematical Analysis and Applications*2007, 325(1):469-479. 10.1016/j.jmaa.2005.12.066MATHMathSciNetView ArticleGoogle Scholar - Rashwan RA: On the convergence of Mann iterates to a common fixed point for a pair of mappings.
*Demonstratio Mathematica*1990, 23(3):709-712.MATHMathSciNetGoogle Scholar - Song Y, Chen R: Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 66(3):591-603. 10.1016/j.na.2005.12.004MATHMathSciNetView ArticleGoogle Scholar - Ćirić LjB, Ume JS, Khan MS: On the convergence of the Ishikawa iterates to a common fixed point of two mappings.
*Archivum Mathematicum*2003, 39(2):123-127.MATHMathSciNetGoogle Scholar

## 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.