Abstract
Let be a uniformly convex Banach space, and let
be a pair of mean nonexpansive map**s. In this paper, it is proved that the sequence of Ishikawa iterations associated with
and
converges to the common fixed point of
and
.
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1. Introduction and Preliminaries
Let be a Banach space and let
,
be map**s from
to
. The pair of mean nonexpansive map**s was introduced by Bose in [1]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ1_HTML.gif)
for all ,
,
.
The Ishikawa iteration sequence of
and
was defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ2_HTML.gif)
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 map**s. 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 map**s, 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 map**s, and
. Then,
-
(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 map**s. 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ3_HTML.gif)
Let , by
, it is easy to see that
, thus
and
.
Similarly, we have ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ4_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ5_HTML.gif)
Hence, is bounded.
Second, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ6_HTML.gif)
We recall that Banach space is called uniformly convex if
for every
, where the modulus
of convexity of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ8_HTML.gif)
On the other hand, for a common fixed point of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ9_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ10_HTML.gif)
Because
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ11_HTML.gif)
we know is bounded, then there exists
, such that
. Thus,
.
Furthermore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ12_HTML.gif)
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ13_HTML.gif)
and the fact that is uniformly convex Banach space, there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ14_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ15_HTML.gif)
Using (2.3), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ16_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ17_HTML.gif)
Let , then we have
. It is a contradiction. Hence,
.
Third, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ18_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ19_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ20_HTML.gif)
Let , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ21_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ22_HTML.gif)
Using (2.4), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ23_HTML.gif)
Forth, we show that if the Ishikawa sequence converges to some point
, then
is the common fixed point of
and
. By
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ24_HTML.gif)
we have . Since
is a convergent sequence, we get
. It is easy to see that
and
. On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ25_HTML.gif)
By (1.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ26_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ27_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ28_HTML.gif)
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ29_HTML.gif)
Let , Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ30_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ31_HTML.gif)
Note that , then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ32_HTML.gif)
So .
Let , then
. By (1.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ33_HTML.gif)
Let , then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ34_HTML.gif)
Since , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ35_HTML.gif)
Similarly, we can prove that . So
is the common fixed point of
and
.
Finally, we show that is a Cauchy sequence. For any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ36_HTML.gif)
Since , thus we get
Simplify, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ37_HTML.gif)
where , and
By (2.16) and (2.30), we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F471532/MediaObjects/13663_2007_Article_1087_Equ38_HTML.gif)
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.
References
Bose SC: Common fixed points of map**s in a uniformly convex Banach space. Journal of the London Mathematical Society 1978, 18(1):151-156. 10.1112/jlms/s2-18.1.151
Rashwan RA, Saddeek AM: On the Ishikawa iteration process in Hilbert spaces. Collectanea Mathematica 1994, 45(1):45-52.
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.
Maingé P-E: Approximation methods for common fixed points of nonexpansive map**s in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007, 325(1):469-479. 10.1016/j.jmaa.2005.12.066
Rashwan RA: On the convergence of Mann iterates to a common fixed point for a pair of map**s. Demonstratio Mathematica 1990, 23(3):709-712.
Song Y, Chen R: Iterative approximation to common fixed points of nonexpansive map** sequences in reflexive Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(3):591-603. 10.1016/j.na.2005.12.004
Ćirić LjB, Ume JS, Khan MS: On the convergence of the Ishikawa iterates to a common fixed point of two map**s. Archivum Mathematicum 2003, 39(2):123-127.
Acknowledgment
The work was partially supported by the Emphases Natural Science Foundation of Guangdong Institution of Higher Learning, College and University (no. 05Z026).
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Gu, Z., Li, Y. Approximation Methods for Common Fixed Points of Mean Nonexpansive Map** in Banach Spaces. Fixed Point Theory Appl 2008, 471532 (2008). https://doi.org/10.1155/2008/471532
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DOI: https://doi.org/10.1155/2008/471532