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Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals
Fixed Point Theory and Applications volume 2008, Article number: 745010 (2008)
Abstract
We prove a convergence theorem by the new iterative method introduced by Takahashi et al. (2007). Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also correct the strong convergence theorem recently proved by He and Chen (2007).
1. Introduction
Let be a real Hilbert space with the inner product
and the norm
. Let
be a family of mappings from a subset
of
into itself. We call it a nonexpansive semigroup on
if the following conditions are satisfied:
-
(1)
for all
;
-
(2)
for all
;
-
(3)
for each
the mapping
is continuous;
-
(4)
for all
and
.
Motivated by Suzuki's result [1] and Nakajo-Takahashi's results [2], He and Chen [3] recently proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. However, their proof of the main result ([3, Theorem 2.3]) is very questionable. Indeed, the existence of the subsequence such that (2.16) of [3] are satisfied, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ1_HTML.gif)
needs to be proved precisely. So, the aim of this short paper is to correct He-Chen's result and also to give a new result by using the method recently introduced by Takahashi et al.
We need the following lemma proved by Suzuki [4, Lemma 1].
Lemma 1.1.
Let be a real sequence and let
be a real number such that
. Suppose that either of the following holds:
-
(i)
, or
-
(ii)
.
Then is a cluster point of
. Moreover, for
,
, there exists
such that
for every integer
with
.
2. Results
2.1. The Shrinking Projection Method
The following method is introduced by Takahashi et al. in [5]. We use this method to approximate a common fixed point of a nonexpansive semigroup without Bochner integrals as was the case in [5, Theorem 4.4].
Theorem 2.1.
Let be a closed convex subset of a real Hilbert space
. Let
be a nonexpansive semigroup on
with a nonempty common fixed point
, that is,
. Suppose that
is a sequence iteratively generated by the following scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ2_HTML.gif)
where ,
,
, and
. Then
Proof.
It is well known that is closed and convex. We first show that the iterative scheme is well defined. To see that each
is nonempty, it suffices to show that
. The proof is by induction. Clearly,
. Suppose that
. Then, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ3_HTML.gif)
That is, as required.
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ4_HTML.gif)
is convex since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ5_HTML.gif)
This implies that each subset is convex. It is also clear that
is closed. Hence the first claim is proved.
Next, we prove that is bounded. As
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ6_HTML.gif)
In particular, for for all
, the sequence
is bounded and hence so is
.
Next, we show that is a Cauchy sequence. As
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ7_HTML.gif)
Moreover, since the sequence is bounded,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ8_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ9_HTML.gif)
In particular, since for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ10_HTML.gif)
It then follows from the existence of that
is a Cauchy sequence. In fact, for
, there exists a natural number
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ11_HTML.gif)
where . In particular, if
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ12_HTML.gif)
Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ13_HTML.gif)
We now assume that for some
. Now since
for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ14_HTML.gif)
The last convergence follows from (2.12). We choose a sequence of positive real number such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ15_HTML.gif)
We now show that how such a special subsequence can be constructed. First we fix such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ16_HTML.gif)
From (2.13), there exists such that
for all
. By Lemma 1.1,
is a cluster point of
. In particular, there exists
such that
. Next, we choose
such that
for all
. Again, by Lemma 1.1,
is a cluster point of
and this implies that there exists
such that
. Continuing in this way, we obtain a subsequence
of
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ17_HTML.gif)
Consequently, (2.14) is satisfied.
We next show that . To see this, we fix
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ18_HTML.gif)
As and (2.14), we have
and so
.
Finally, we show that . Since
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ19_HTML.gif)
But ; we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ20_HTML.gif)
Hence as required. This completes the proof.
2.2. The Hybrid Method
We consider the iterative scheme computing by the hybrid method (some authors call the CQ-method). The following result is proved by He and Chen [3]. However, the important part of the proof seems to be overlooked. Here we present the correction under some additional restriction on the parameter .
Theorem 2.2.
Let be a closed convex subset of a real Hilbert space
. Let
be a nonexpansive semigroup on
with a nonempty common fixed point
, that is,
. Suppose that
is a sequence iteratively generated by the following scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ21_HTML.gif)
where ,
,
, and
. Then
.
Proof.
For the sake of clarity, we give the whole sketch proof even though some parts of the proof are the same as [3]. To see that the scheme is well defined, it suffices to show that both and
are closed and convex, and
for all
. It follows easily from the definition that
and
are just the intersection of
and the half-spaces, respectively,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ22_HTML.gif)
As in the proof of the preceding theorem, we have for all
. Clearly,
. Suppose that
for some
, we have
. In particular,
, that is,
. It follows from the induction that
for all
. This proves the claim.
We next show that . To see this, we first prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ23_HTML.gif)
As and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ24_HTML.gif)
For fixed . It follows from
for all
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ25_HTML.gif)
This implies that sequence is bounded and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ26_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ27_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ28_HTML.gif)
It then follows from that
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ29_HTML.gif)
As in Theorem 2.1, we can choose a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ30_HTML.gif)
Consequently, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ31_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ32_HTML.gif)
In virtue of Opial's condition of , we have
for all
, that is,
. Next, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ33_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ34_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F745010/MediaObjects/13663_2007_Article_1105_Equ35_HTML.gif)
Hence the whole sequence must converge to , as required.
References
Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proceedings of the American Mathematical Society 2003, 131(7):2133-2136. 10.1090/S0002-9939-02-06844-2
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003, 279(2):372-379. 10.1016/S0022-247X(02)00458-4
He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups. Fixed Point Theory and Applications 2007, 2007:-8.
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005, 305(1):227-239. 10.1016/j.jmaa.2004.11.017
Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007, 341(1):276-286.
Acknowledgments
The author would like to thank the referee(s) for his comments and suggestions on the manuscript. This work is supported by the Commission on Higher Education and the Thailand Research Fund (Grant MRG4980022).
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Saejung, S. Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals. Fixed Point Theory Appl 2008, 745010 (2008). https://doi.org/10.1155/2008/745010
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DOI: https://doi.org/10.1155/2008/745010