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Approximation Methods for Common Fixed Points of Mean Nonexpansive Mapping in Banach Spaces
Fixed Point Theory and Applications volume 2008, Article number: 471532 (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]:
for all , , .
The Ishikawa iteration sequence of and was defined by
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]).
Let be a uniformly convex Banach space and a nonempty closed convex subset of , and are a pair of mean nonexpansive mappings, 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 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
Let , by , it is easy to see that , thus and .
Similarly, we have ,
So
Hence, is bounded.
Second, we show that
We recall that Banach space is called uniformly convex if for every , where the modulus of convexity of is defined by
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
On the other hand, for a common fixed point of and , we have
Thus,
Because
we know is bounded, then there exists , such that . Thus, .
Furthermore, we have
From
and the fact that is uniformly convex Banach space, there exists , such that
Thus,
Using (2.3), we obtain that
So
Let , then we have . It is a contradiction. Hence, .
Third, we show that
Since
we have
Let , then
So
Using (2.4), we get that
Forth, we show that if the Ishikawa sequence converges to some point , then is the common fixed point of and . By
we have . Since is a convergent sequence, we get . It is easy to see that and . On the other hand,
By (1.1), we obtain
Since
we get
So
Let , Since , we have
It is easy to see that
Note that , then we get
So .
Let , then . By (1.1), we have
Let , then we get
Since , it follows that
Similarly, we can prove that . So is the common fixed point of and .
Finally, we show that is a Cauchy sequence. For any ,
Since , thus we get Simplify, then we have
where , and By (2.16) and (2.30), we know that
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 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
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 mappings 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 mappings. Demonstratio Mathematica 1990, 23(3):709-712.
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
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.
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 Mapping 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