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Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups
Fixed Point Theory and Applications volume 2010, Article number: 301868 (2010)
Abstract
We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi (2003) and of Zegeye and Shahzad (2008) from the class of nonexpansive mappings to asymptotically nonexpansive mappings.
1. Introduction
Throughout this paper, Let be a real Hilbert space with inner product
and norm
, and we write
to indicate that the sequence
converges strongly to
. Let
be a nonempty closed convex subset of
, and let
be a mapping. Recall that
is nonexpansive if
, for all
. We denote the set of fixed points of
by
, that is,
. A mapping
is said to be asymptotically nonexpansive if there exists a sequence
with
for all
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ1_HTML.gif)
Mann's iterative algorithm was introduced by Mann [1] in 1953. This iteration process is now known as Mann's iteration process, which is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_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 iteration scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ3_HTML.gif)
for all , where
and
is a sequence in
. This iteration process is called a Halpern-type iteration.
Recall also that a one-parameter family of self-mappings of a nonempty closed convex subset
of a Hilbert space
is said to be a (continuous) Lipschitzian semigroup on
if the following conditions are satisfied:
(a),
(b), for all
,
;
(c)for each , the map
is continuous on
(d)there exists a bounded measurable function such that, for each
,
, for all
A Lipschitzian semigroup is called nonexpansive if
for all
, and asymptotically nonexpansive if
. We denote by
the set of fixed points of the semigroup
, that is,
.
In 2003, Nakajo and Takahashi [3] proposed the following modification of the Mann iteration method for a nonexpansive mapping in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ4_HTML.gif)
where denotes the metric projection from
onto a closed convex subset
of
. They proved that the sequence
converges weakly to a fixed point of
. Moreover, they introduced and studied an iteration process of a nonexpansive semigroup
in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ5_HTML.gif)
In 2006, Kim and Xu [4] adapted iteration (1.4) to an asymptotically nonexpansive mapping in a Hilbert space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ6_HTML.gif)
where as
. They also proved that if
for all
and for some
, then the sequence
converges weakly to a fixed point of
. Moreover, they modified an iterative method (1.5) to the case of an asymptotically nonexpansive semigroup
in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ7_HTML.gif)
where as
.
In 2007, Zegeye and Shahzad [5] developed the iteration process for a finite family of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups with a closed convex bounded subset of a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ8_HTML.gif)
where as
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ9_HTML.gif)
where as
, with
, for each
Recently, Su and Qin [6] modified the hybrid iteration method of Nakajo and Takahashi through the monotone hybrid method, and to prove strong convergence theorems.
In 2008, Takahashi et al. [7] proved strong convergence theorems by the new hybrid methods for a family of nonexpansive mappings and nonexpansive semigroups in Hilbert spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ10_HTML.gif)
where , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ11_HTML.gif)
where ,
and
.
In this paper, motivated and inspired by the above results, we modify iteration process (1.4)–(1.11) by the new hybrid methods for countable families of asymptotically nonexpansive mappings and semigroups in a Hilbert space, and to prove strong convergence theorems. Our results presented are improvement and extension of the corresponding results in [3, 5–8] and many authors.
2. Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the next section.
Lemma 2.1.
Here holds the identity in a Hilbert space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ12_HTML.gif)
for all and
.
Using this Lemma 2.1, we can prove that the set of fixed points of
is closed and convex. Let
be a nonempty closed convex subset of
. Then, for any
, there exists a unique nearest point in
, denoted by
, such that
for all
, where
is called the metric projection of
onto
. We know that for
and
,
is equivalent to
for all
. We know that a Hilbert space
satisfies Opial's condition, that is, for any sequence
with
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ13_HTML.gif)
hold for every with
. We also know that
has the Kadec-Klee property, that is,
and
imply
. In fact, from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ14_HTML.gif)
we get that a Hilbert space has the Kadec-Klee property.
Let be a nonempty closed convex subset of a Hilbert space
. Motivated by Nakajo et al. [9], we give the following definitions: Let
and
be families of nonexpansive mappings of
into itself such that
, where
is the set of all fixed points of
and
is the set of all common fixed points of
. We consider the following conditions of
and
(see [9]):
(i)NST-condition (I). For each bounded sequence ,
implies that
for all
.
(ii)NST-condition (II). For each bounded sequence ,
implies that
for all
.
(iii)NST-condition (III). There exists with
such that for every bounded subset
of
, there exists
such that
holds for all
and
Lemma 2.2.
Let be a nonempty closed convex subset of
and let
be a nonexpansive mapping of
into itself with
. Then, the following hold:
(i) with
and
satisfy the condition (I) with
.
(ii) with
and
satisfy the condition (I) with
Lemma 2.3 (Opial [10]).
Let be a closed convex subset of a real Hilbert space
and let
be a nonexpansive mapping such that
. If
is a sequence in
such that
and
, then
.
Lemma 2.4 (Lin et al. [11]).
Let be an asymptotically nonexpansive mapping defined on a bounded closed convex subset of a bounded closed convex subset
of a Hilbert space
. If
is a sequence in
such that
and
, then
.
Lemma 2.5 (Nakajo and Takahashi [3]).
Let be a real Hilbert space. Given a closed convex subset
and points
. Given also a real number
. The set
is convex and closed.
Lemma 2.6 (Kim and Xu [4]).
Let be a nonempty bounded closed convex subset of
and
be an asymptotically nonexpansive semigroup on
. If
is a sequence in
satisfying the properties
(a);
(b),
then .
Lemma 2.7 (Kim and Xu [4]).
Let be a nonempty bounded closed convex subset of
and
be an asymtotically nonexpansive semigroup on
. Then it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ15_HTML.gif)
3. Strong Convergence for a Family of Asymptotically Nonexpansive Mappings
Theorem 3.1.
Let be a nonempty bounded closed convex subset of a Hilbert space
and let
for
be a countable family of asymptotically nonexpansive mapping with sequence
for
, respectively. Assume
such that
for all
and
as
. Let
. Further, suppose that
satisfies NST-condition (I) and (III) with T. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ16_HTML.gif)
where as
. Then
converges in norm to
.
Proof.
We first show that is closed and convex for all
. From the Lemma 2.5, it is observed that
is closed and convex for each
.
Next, we show that for all
. Indeed, let
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ17_HTML.gif)
Thus and hence
for all
. Thus
is well defined.
From and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ18_HTML.gif)
So, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ19_HTML.gif)
for all . This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ20_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ21_HTML.gif)
for all Therefore
is nondecreasing.
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ22_HTML.gif)
Using , we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ23_HTML.gif)
So, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ24_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ25_HTML.gif)
Thus, is bounded. So,
exists.
Next, we show that . From (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ26_HTML.gif)
Since exists, we conclude that
.
Since , we have
which implies that
. Now we claim that
as
for all
. We first show that
as
. Indeed, by the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ27_HTML.gif)
for all and it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ28_HTML.gif)
Since as
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ29_HTML.gif)
for all .
Let . Now, for
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ30_HTML.gif)
from (3.14) and as
, yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ31_HTML.gif)
for each Let
and take
with
. By NST-condition (III), there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ32_HTML.gif)
By (3.16) and , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ33_HTML.gif)
By the assumption of and NST-condition (I), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ34_HTML.gif)
Put . Since
for all
,
is bounded. Let
be a subsequence of
such that
. Since
is closed and convex,
is weakly closed and hence
. From (3.19), we have that
. If not, since
satisfies Opial's condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ35_HTML.gif)
This is a contradiction. So, we have that . Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ36_HTML.gif)
and hence . From
, we have
. This implies that
converges weakly to
, and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ37_HTML.gif)
and hence . From
, we also have
. Since
satisfies the Kadec-Klee property, it follows that
. So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ38_HTML.gif)
and hence . This completes the proof.
Corollary 3.2.
Let be a nonempty bounded closed convex subset of a Hilbert space
and let
be an asymptotically nonexpansive mapping with sequence
. Assume
such that
for all
and
as
. Let
. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ39_HTML.gif)
where as
. Then
converges in norm to
.
Proof.
Setting for all
from Lemma 2.2(i) and Theorem 3.1, we immediately obtain the corollary.
Since every family's nonexpansive mapping is family's asymptotically nonexpansive mapping we obtain the following result.
Corollary 3.3.
Let be a nonempty bounded closed convex subset of a Hilbert space
and let
be a family of nonexpansive mappings with sequence
. Assume
such that
for all
and
as
. Let
. Further, suppose that
satisfies NST-condition (I) with T. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ40_HTML.gif)
Assume that if for each bounded sequence ,
for all
implies that
. Then
converges in norm to
.
We have the following corollary for nonexpansive mappings by Lemma 2.2(i) and Theorem 3.1.
Corollary 3.4 (Takahashi et al. [7, Theorem ]).
Let be a bounded closed convex subset of a Hilbert space
and let
be a nonexpansive mapping such that
. Assume that
for all
. Then the sequence
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ41_HTML.gif)
converges in norm to
4. Strong Convergence for a Family of Asymptotically Nonexpansive Semigroups
Theorem 4.1.
Let be a nonempty bounded closed convex subset of a Hilbert space
and let
,
be a countable family of asymptotically nonexpansive semigroups. Assume
such that
for all
and
as
. Let
be a countable positive and divergent real sequence. Let
. Further, suppose that
satisfies NST-condition (I) with T. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ42_HTML.gif)
where as
with
. Then
converges in norm to
.
Proof.
First observe that for all
. Indeed, we have for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ43_HTML.gif)
So, . Hence
for all
. By the same argument as in the proof of Theorem 3.1,
is closed and convex,
is well defined. Also, similar to the proof of Theorem 3.1
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ44_HTML.gif)
We next claim that Indeed, by definition of
and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ45_HTML.gif)
and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ46_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ47_HTML.gif)
which in turn implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ48_HTML.gif)
It follows from (4.5) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ49_HTML.gif)
Let and for each
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ50_HTML.gif)
By (4.8) and Lemma 2.7, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ51_HTML.gif)
Furthermore, from (4.9) and Lemma 2.6 and the boundedness of we obtain that
. By the fact that
for any
, where
and the weak lower semi-continuity of the norm, we have
for all
. However, since
, we must have
for all
. Thus
and then
converges weakly to
. Moreover, following the method of Theorem 3.1,
. This completes the proof.
Corollary 4.2.
Let be a bounded closed convex subset of a Hilbert space
and
be an asymptotically nonexpansive semigroup on
. Assume also that
for all
and
is a positive real divergent sequence. Then, the sequence
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ52_HTML.gif)
converges in norm to , where
as
.
Proof.
By Theorem 4.1, if the semigroup , then
for all
and for all
. Hence
for all
and
then, (4.1) reduces to (4.11).
Corollary 4.3 (Takahashi et al. [7, Theorem ]).
Let be a nonempty closed convex subset of a Hilbert space
and
be a nonexpansive semigroup on
. Assume that
for all
and
is a positive real divergent sequence. If
, then the sequence
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F301868/MediaObjects/13663_2010_Article_1254_Equ53_HTML.gif)
converges in norm to
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Acknowledgments
The authors would like to thank professor Somyot Plubtieng for drawing my attention to the subject and for many useful discussions and the referees for helpful suggestions that improved the contents of the paper. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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Wattanawitoon, K., Kumam, P. Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups. Fixed Point Theory Appl 2010, 301868 (2010). https://doi.org/10.1155/2010/301868
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DOI: https://doi.org/10.1155/2010/301868