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

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

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
(2)
Institute of Logic and Cognition, Department of Mathematics, Sun Yat-Sen University

References

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Copyright

© Z. Gu and Y. Li. 2008

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.