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Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups
Fixed Point Theory and Applications volume 2010, Article number: 914702 (2009)
Abstract
We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the corresponding results announced by Saejung (2008) and some others.
1. Introduction
Let be a subset of real Hilbert spaces
with the inner product
and the norm
.
is called a nonexpansive mapping if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ1_HTML.gif)
We denote by the set of fixed points of
, that is,
.
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:
(i)
(ii) for all
(iii)for each the mapping
is continuous;
(iv) for all
and
The Mann's iterative algorithm was introduced by Mann [1] in 1953. This iterative process is now known as Mann's iterative process, which is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ2_HTML.gif)
where the initial guess is taken in
arbitrarily and the sequence
is in the interval
.
In 1967, Halpern [2] first introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ3_HTML.gif)
see also Browder [3]. He pointed out that the conditions and
are necessary in the sence that, if the iteration (1.3) converges to a fixed point of
, then these conditions must be satisfied.
On the other hand, in 2002, Suzuki [4] was the first to introduce the following implicit iteration process in Hilbert spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ4_HTML.gif)
for the nonexpansive semigroup. In 2005, Xu [5] established a Banach space version of the sequence (1.4) of Suzuki [4].
In 2007, Chen and He [6] studied the viscosity approximation process for a nonexpansive semigroup and prove another strong convergence theorem for a nonexpansive semigroup in Banach spaces, which is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ5_HTML.gif)
where is a fixed contractive mapping.
Recently He and Chen [7] is proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in the mathematical programming. Very recently, Saejung [8] proved a convergence theorem by the new iterative method introduced by Takahashi et al. [9] without Bochner integrals for a nonexpansive semigroup with
in Hilbert spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ6_HTML.gif)
where denotes the metric projection from
onto a closed convex subset
of
.
In 1974, Ishikawa [10] introduced a new iterative scheme, which is defined recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ7_HTML.gif)
where the initial guess is taken in
arbitrarily and the sequences
and
are in the interval
.
In this paper, motivated by the iterative sequences (1.6) given by Saejung in [8] and Ishikawa [10], we introduce the modified Ishikawa iterative scheme for two nonexpansive semigroups in Hilbert spaces. Further, we obtain strong convergence theorems by using the hybrid methods. This result extends and improves the result of Saejung [8] and some others.
2. Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the next section.
It is known that every Hilbert space satisfies the Opial's condition [11], that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ8_HTML.gif)
Recall that the metric (nearest point) projection from a Hilbert space
to a closed convex subset
of
is defined as follows. Given
is the only point in
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ9_HTML.gif)
is characterized as follows.
Lemma 2.1.
Let be a real Hilbert space,
a closed convex subset of
. Given
and
. Then
if and only if there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ10_HTML.gif)
Lemma 2.2.
There holds the identity in a Hilbert space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ11_HTML.gif)
for all and
Lemma 2.3 (see [12, Lemma ]).
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
3. Main Results
3.1. The Shrinking Projection Method
In this section, we prove strong convergence of an iterative sequence generated by the shrinking hybrid projection method in mathematical programming.
Theorem 3.1.
Let be a closed convex subset of a real Hilbert space
. Let
and
be nonexpansive semigroups on
with a nonempty common fixed point set
, that is,
. Let
,
and
be the sequences such that
,
and
. Suppose that
is a sequence generated by the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ12_HTML.gif)
then converges strongly to
Proof.
We first show that is closed and convex for each
. From the definition of
it is obvious that
is closed for each
. We show that
is convex for any
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ13_HTML.gif)
and hence is convex. Next we show that
for all
. Let
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ15_HTML.gif)
Substituting (3.3) into (3.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ16_HTML.gif)
This means that for all
. Thus,
is well defined. Since
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ17_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ18_HTML.gif)
for . This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ19_HTML.gif)
Therefore, is nondecreasing. From
, we also have
, for all
Since , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ20_HTML.gif)
Thus, for , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ21_HTML.gif)
Thus, , for all
and
. Then
exists and
is bounded.
Next, we show that as
. From (3.6) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ22_HTML.gif)
Since exists, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ23_HTML.gif)
Further, as in the proof of [8, page 3], we have which is a Cauchy sequence. So, we have
By definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ24_HTML.gif)
Since and (3.12), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ25_HTML.gif)
We now show that .
For , we have
This implies that
and hence
Moreover, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ26_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ27_HTML.gif)
And since is a nonexpansive mapping, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ28_HTML.gif)
Since and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ29_HTML.gif)
As in the proof of [12, Theorem ], by Lemma 2.3, we can choose a sequence
of positive real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ30_HTML.gif)
In similar way, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ31_HTML.gif)
Next, we show that . To see this, we fix
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ32_HTML.gif)
As and (3.19), we obtain
and so
Similarly, we have
Thus
.
Finally, we show that Since
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ33_HTML.gif)
But as
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ34_HTML.gif)
Hence as required. This completes the proof.
Corollary 3.2.
Let be a closed convex subset of a real Hilbert space
. Let
be nonexpansive semigroups on
with a nonempty common fixed point set
, that is,
. Let
,
and
be the sequences such that
,
and
. Suppose that
is a sequence iteratively generated by the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ35_HTML.gif)
then converges strongly to
Proof.
Putting , in Theorem 3.1, we obtain the conclusion immediately.
Corollary 3.3 (see [8, Theorem ]).
Let be a closed convex subset of a real Hilbert space
. Let
be a nonexpansive semigroups on
with a nonempty common fixed point set
, that is,
. Let
and
be the sequences such that
,
and
. Suppose that
is a sequence iteratively generated by the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ36_HTML.gif)
then
Proof.
If for all
and
for every
in Theorem 3.1 then (3.1) reduced to (3.25). By using Theorem 3.1, we get the following conclusion.
3.2. The CQ Hybrid Method
In this section, we consider the modified Ishikawa iterative scheme computing by the CQ hybrid method [13–15]. We use the same idea as Saejung's Theorem in [8] and our Theorem 3.1 to obtain the following result and the proof is omitted.
Theorem 3.4.
Let be a closed convex subset of a real Hilbert space
. Let
and
be nonexpansive semigroups on
with a nonempty common fixed point set
, that is,
. Let
,
and
be the sequences such that
,
and
. Suppose that
is a sequence generated by the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ37_HTML.gif)
then converges strongly to
Proof.
First, we show that both and
are closed and convex, and
for all
. It follows easily from the definition that
and
are just intersection of
and the half-spaces see also [9]. 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.
Next, we show that and
We first claim that Indeed, as
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ38_HTML.gif)
For fixed . It follows from
for all
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ39_HTML.gif)
This implies that sequence is bounded and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ40_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ41_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ42_HTML.gif)
By using the same argument of Saejung [8, Theorem , page 6] and in the proof of Theorem 3.1, we have
and
. And we can choose a subsequence
of
such that
,
,
and
as
.
From (3.21), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ43_HTML.gif)
By the Opial's condition of , we have
and
for all
, that is,
.
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ44_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ45_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ46_HTML.gif)
Hence the whole sequence must converge to , as required. This completes the proof.
Corollary 3.5.
Let be a closed convex subset of a real Hilbert space
. Let
be nonexpansive semigroups on
with a nonempty common fixed point set
, that is,
. Let
,
and
be the sequences such that
,
and
. Suppose that
is a sequence iteratively generated by the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ47_HTML.gif)
then converges strongly to
Proof.
If for all
, in Theorem 3.4 then (3.26) reduced to (3.36). So, we obtain the result immediately.
We also deduce the following corollary.
Corollary 3.6 (see [8, Theorem ]).
Let be a closed convex subset of a real Hilbert space
. Let
be a nonexpansive semigroups on
with a nonempty common fixed point set
, that is,
. Let
and
be the sequences such that
,
and
. Suppose that
is a sequence iteratively generated by the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F914702/MediaObjects/13663_2009_Article_1364_Equ48_HTML.gif)
then
References
Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272
Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proceedings of the American Mathematical Society 2002,131(7):2133–2136.
Xu H-K: A strong convergence theorem for contraction semigroups in Banach spaces. Bulletin of the Australian Mathematical Society 2005,72(3):371–379. 10.1017/S000497270003519X
Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. Applied Mathematics Letters 2007,20(7):751–757. 10.1016/j.aml.2006.09.003
He H, Chen R: Strong convergence theorems of the CQ method for nonexpansive semigroups. Fixed Point Theory and Applications 2007, 2007:-8.
Saejung S: Strong convergence theorems for nonexpansive semigroups without Bochner integrals. Fixed Point Theory and Applications 2008, 2008:-7.
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 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062
Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0
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
Haugazeau Y: Sur les Inéquations variationnelles et la minimisation de fonctionnelles convexes, Ph.D. thesis. Université Paris, Paris, France; 1968.
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
Nakajo K, Shimoji K, Takahashi W: Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces. Taiwanese Journal of Mathematics 2006,10(2):339–360.
Acknowledgments
The authors would like to thank the editors and the anonymous referees for their valuable suggestions which help to improve this paper. This research was supported by the Computational Science and Engineering Research Cluster, King Mongkut's University of Technology Thonburi (KMUTT) (National Research Universities under CSEC Project no. E01008).
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Wattanawitoon, K., Kumam, P. Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups. Fixed Point Theory Appl 2010, 914702 (2009). https://doi.org/10.1155/2010/914702
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DOI: https://doi.org/10.1155/2010/914702