Research Article
Open Access
Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals
- Satit Saejung1Email author
Fixed Point Theory and Applications20082008:745010
https://doi.org/10.1155/2008/745010
© Satit Saejung. 2008
Received: 28 November 2007
Accepted: 30 January 2008
Published: 5 February 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:
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,
such that (2.16) of [3] are satisfied, that is,
(1.1)
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:
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:
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:
(2.1)
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
,
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
,
(2.2)
That is,
as required.
Notice that
(2.3)
is convex since
(2.4)
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
,
is bounded. As
,
(2.5)
In particular, for
for all
, the sequence
is bounded and hence so is
.
Next, we show that
is a Cauchy sequence. As
and
,
is a Cauchy sequence. As
and
,
(2.6)
Moreover, since the sequence
is bounded,
is bounded,
(2.7)
Note that
(2.8)
In particular, since
for all
,
for all
,
(2.9)
It then follows from the existence of
that
is a Cauchy sequence. In fact, for
, there exists a natural number
such that, for all
,
that
is a Cauchy sequence. In fact, for
, there exists a natural number
such that, for all
,
(2.10)
where
. In particular, if
and
, then
. In particular, if
and
, then
(2.11)
Moreover,
(2.12)
We now assume that
for some
. Now since
for all
and
,
for some
. Now since
for all
and
,
(2.13)
The last convergence follows from (2.12). We choose a sequence
of positive real number such that
of positive real number such that
(2.14)
We now show that how such a special subsequence can be constructed. First we fix
such that
such that
(2.15)
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
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
(2.16)
Consequently, (2.14) is satisfied.
We next show that
. To see this, we fix
,
. To see this, we fix
,
(2.17)
As
and (2.14), we have
and so
.
Finally, we show that
. Since
and
,
. Since
and
,
(2.18)
But
; we have
; we have
(2.19)
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:
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:
(2.20)
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,
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,
(2.21)
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
. To see this, we first prove that
(2.22)
As
and
,
and
,
(2.23)
For fixed
. It follows from
for all
that
. It follows from
for all
that
(2.24)
This implies that sequence
is bounded and
is bounded and
(2.25)
Notice that
(2.26)
This implies that
(2.27)
It then follows from
that
and hence
that
and hence
(2.28)
As in Theorem 2.1, we can choose a subsequence
of
such that
of
such that
(2.29)
Consequently, for any
,
,
(2.30)
This implies that
(2.31)
In virtue of Opial's condition of
, we have
for all
, that is,
. Next, we observe that
, we have
for all
, that is,
. Next, we observe that
(2.32)
This implies that
(2.33)
Consequently,
(2.34)
Hence the whole sequence must converge to
, as required.
Declarations
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).
Authors’ Affiliations
(1)
Department of Mathematics, Faculty of Science, Khon Kaen University
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/S0002-9939-02-06844-2MATHMathSciNetView ArticleGoogle Scholar
- 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)00458-4MATHMathSciNetView ArticleGoogle Scholar
- He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups. Fixed Point Theory and Applications 2007, 2007:-8.Google Scholar
- 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.2004.11.017MATHMathSciNetView ArticleGoogle Scholar
- 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.MathSciNetView ArticleGoogle Scholar
Copyright
© Satit Saejung. 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.
