• Research Article
• Open Access

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

Fixed Point Theory and Applications20082008:471532

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

• Accepted: 2 January 2008
• Published:

## 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 .

## Keywords

• Hilbert Space
• Banach Space
• Real Number
• Approximation Method
• Differential Geometry

## 1. Introduction and Preliminaries

Let be a Banach space and let , be mappings from to . The pair of mean nonexpansive mappings was introduced by Bose in [1]:
(1.1)

for all , , .

The Ishikawa iteration sequence of and was defined by
(1.2)

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 [26]) 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]).

Let be a uniformly convex Banach space and a nonempty closed convex subset of , and are a pair of mean nonexpansive mappings, and . Then,
1. (i)

and have a common fixed point ;

2. (ii)
further, if , then
1. (a)

is the unique common fixed point and unique as a fixed point of each and ,

2. (b)

the sequence defined by , for any , converges strongly to .

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.

First, we show that the sequence is bounded. For a common fixed point of and , we have
(2.1)

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

Similarly, we have ,
(2.2)
So
(2.3)

Hence, is bounded.

Second, we show that
(2.4)
We recall that Banach space is called uniformly convex if for every , where the modulus of convexity of is defined by
(2.5)

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

Assume that , then there exist a subsequence of and a real number , such that
(2.6)
On the other hand, for a common fixed point of and , we have
(2.7)
Thus,
(2.8)
Because
(2.9)

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

Furthermore, we have
(2.10)
From
(2.11)
and the fact that is uniformly convex Banach space, there exists , such that
(2.12)
Thus,
(2.13)
Using (2.3), we obtain that
(2.14)
So
(2.15)

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

Third, we show that
(2.16)
Since
(2.17)
we have
(2.18)
Let , then
(2.19)
So
(2.20)
Using (2.4), we get that
(2.21)
Forth, we show that if the Ishikawa sequence converges to some point , then is the common fixed point of and . By
(2.22)
we have . Since is a convergent sequence, we get . It is easy to see that and . On the other hand,
(2.23)
By (1.1), we obtain
(2.24)
Since
(2.25)
we get
(2.26)
So
(2.27)
Let , Since , we have
(2.28)
It is easy to see that
(2.29)
Note that , then we get
(2.30)

So .

Let , then . By (1.1), we have
(2.31)
Let , then we get
(2.32)
Since , it follows that
(2.33)

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

Finally, we show that is a Cauchy sequence. For any ,
(2.34)
Since , thus we get Simplify, then we have
(2.35)
where , and By (2.16) and (2.30), we know that
(2.36)

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

(1)
Department of Foundation, Guangdong Finance and Economics College, Guangzhou, 510420, China
(2)
Institute of Logic and Cognition, Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China

## References

1. 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/s2-18.1.151
2. Rashwan RA, Saddeek AM: On the Ishikawa iteration process in Hilbert spaces. Collectanea Mathematica 1994, 45(1):45-52.
3. 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.
4. 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.2005.12.066
5. 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.
6. 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.2005.12.004
7. 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.