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A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2010, Article number: 262691 (2010)
Abstract
We introduce a new general iterative method by using the -mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong convergence theorem of the purposed iterative method is established under some certain control conditions. Our results improve and extend the results announced by many others.
1. Introduction
Let be a real Hilbert space, and let
be a nonempty closed convex subset of
. A mapping
of
into itself is called nonexpansive if
for all
A point
is called a fixed point of
provided that
. We denote by
the set of fixed points of
(i.e.,
). Recall that a self-mapping
is a contraction on
, if there exists a constant
such that
for all
A bounded linear operator
on
is called strongly positive with coefficient
if there is a constant
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ1_HTML.gif)
In 1953, Mann [1] introduced a well-known classical iteration to approximate a fixed point of a nonexpansive mapping. This iteration is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ2_HTML.gif)
where the initial guess is taken in
arbitrarily, and the sequence
is in the interval
. But Mann's iteration process has only weak convergence, even in a Hilbert space setting. In general for example, Reich [2] showed that if
is a uniformly convex Banach space and has a Frehet differentiable norm and if the sequence
is such that
, then the sequence
generated by process (1.2) converges weakly to a point in
. Therefore, many authors try to modify Mann's iteration process to have strong convergence.
In 2005, Kim and Xu [3] introduced the following iteration process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ3_HTML.gif)
They proved in a uniformly smooth Banach space that the sequence defined by (1.3) converges strongly to a fixed point of
under some appropriate conditions on
and
.
In 2008, Yao et al. [4] alsomodified Mann's iterative scheme 1.2 to get a strong convergence theorem.
Let be a finite family of nonexpansive mappings with
There are many authors introduced iterative method for finding an element of
which is an optimal point for the minimization problem. For
,
is understood as
with the mod function taking values in
. Let
be a fixed element of
In 2003, Xu [5] proved that the sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ4_HTML.gif)
converges strongly to the solution of the quadratic minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ5_HTML.gif)
under suitable hypotheses on and under the additional hypothesis
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ6_HTML.gif)
In 1999, Atsushiba and Takahashi [6] defined the mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ7_HTML.gif)
where This mapping is called the
mapping generated by
and
.
In 2000, Takahashi and Shimoji [7] proved that if is strictly convex Banach space, then
, where
.
In 2007,Shang et al.[8] introduced a composite iteration scheme as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ8_HTML.gif)
where is a contraction, and
is a linear bounded operator.
Note that the iterative scheme (1.8) is not well-defined, because may not lie in
, so
is not defined. However, if
, the iterative scheme (1.8) is well-defined and Theorem
[8] is obtained. In the case
, we have to modify the iterative scheme (1.8) in order to make it well-defined.
In 2009, Kangtunyakarn and Suantai [9] introduced a new mapping, called -mapping, for finding a common fixed point of a finite family of nonexpansive mappings. For a finite family of nonexpansive mappings
and sequence
in
, the mapping
is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ9_HTML.gif)
The mapping is called the K-mapping generated by
and
.
In this paper, motivated by Kim and Xu [3], Marino and Xu [10], Xu [5], Yao et al. [4], andShang et al. [8], we introduce a composite iterative scheme as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ10_HTML.gif)
where is a contraction, and
is a bounded linear operator. We prove, under certain appropriate conditions on the sequences
and
that
defined by (1.10) converges strongly to a common fixed point of the finite family of nonexpansive mappings
, which solves a variational inequaility problem.
In order to prove our main results, we need the following lemmas.
Lemma 1.1.
For all there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ11_HTML.gif)
Lemma 1.2 (see [11]).
Let and
be bounded sequences in a Banach space
, and let
be a sequence in
with
. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ12_HTML.gif)
for all integer , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ13_HTML.gif)
Then
Lemma 1.3 (see [5]).
Assume that is a sequence of nonnegative real numbers such that
  
, where
and
is a sequence in
such that
(i),
(ii) or
.
Then .
Lemma 1.4 (see [10]).
Let be a strongly positive linear bounded operator on a Hilbert space
with coefficient
and
. Then
.
Lemma 1.5 (see [10]).
Let be a Hilbert space. Let
be a strongly positive linear bounded operator with coefficient
. Assume that
. Let
be a nonexpansive mapping with a fixed point
of the contraction
. Then
converges strongly as
to a fixed point
of
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ14_HTML.gif)
Lemma 1.6 (see [1]).
Demiclosedness principle. Assume that is nonexpansive self-mapping of closed convex subset
of a Hilbert space
. If
has a fixed point, then
is demiclosed. That is, whenever
is a sequence in
weakly converging to some
and the sequence
strongly converges to some
, it follows that
. Here,
is identity mapping of
.
Lemma 1.7 (see [9]).
Let be a nonempty closed convex subset of a strictly convex Banach space. Let
be a finite family of nonexpansive mappings of
into itself with
, and let
be real numbers such that
for every
and
Let
be the
-mapping of
into itself generated by
and
. Then
.
By using the same argument as in [9, Lemma ], we obtain the following lemma.
Lemma 1.8.
Let be a nonempty closed convex subset of Banach space. Let
be a finite family of nonexpanxive mappings of
into itself and
sequences in
such that
Moreover, for every
, let
and
be the K -mappings generated by
and
, and
and
, respectively. Then, for every bounded sequence
, one has
Let be real Hilbert space with inner product
,
a nonempty closed convex subset of
. Recall that the metric (nearest point) projection
from a real Hilbert space
to a closed convex subset
of
is defined as follows. Given that  
,
is the only point in
with the property
. Below Lemma 1.9 can be found in any standard functional analysis book.
Lemma 1.9.
Let be a closed convex subset of a real Hilbert space
. Given that
and
then
(i) if and only if the inequality
for all
,
(ii) is nonexpansive,
(iii) for all
,
(iv) for all
and
.
2. Main Result
In this section, we prove strong convergence of the sequences defined by the iteration scheme (1.10).
Theorem 2.1.
Let be a Hilbert space,
a closed convex nonempty subset of
. Let
be a strongly positive linear bounded operator with coefficient
, and let
Let
be a finite family of nonexpansive mappings of
into itself, and let
be defined by (1.9). Assume that
and
. Let
, given that
and
are sequences in
, and suppose that the following conditions are satisfied:
(C1)
(C2)
(C3)
(C4) and
, where
;
(C5)
(C6).
If is the composite process defined by (1.10), then
converges strongly to
, which also solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ15_HTML.gif)
Proof.
First, we observe that is bounded. Indeed, take a point
, and notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ16_HTML.gif)
Since , we may assume that
for all
. By Lemma 1.4, we have
for all
.
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ17_HTML.gif)
By simple inductions, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ18_HTML.gif)
Therefore is bounded, so are
and
. Since
is nonexpansive and
, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ19_HTML.gif)
By using the inequalities (2.6) and (2.11) of [9, Lemma ], we can conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ20_HTML.gif)
where .
By (2.5) and (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ21_HTML.gif)
where ,
. Since
,
, and
, for all
, by Lemma 1.3, we obtain
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ22_HTML.gif)
Since and
,
are bounded, we have
as
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ23_HTML.gif)
it implies that as
.
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ24_HTML.gif)
which implies that .
From condition and
as
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ25_HTML.gif)
By (C4), we have for all
. Let
be the
-mapping generated by
and
. Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ26_HTML.gif)
where with
being the fixed point of the contraction
. Thus,
solves the fixed point equation
. By Lemma 1.5 and Lemma 1.7, we have
and
for all
. It follows by (2.11) and Lemma 1.8 that
Thus, we have
. It follows from Lemma 1.1 that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ27_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ28_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ29_HTML.gif)
Letting in (2.15) and (2.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ30_HTML.gif)
where is a constant such that
for all
and
. Taking
in (2.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ31_HTML.gif)
On the other hand, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ32_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ33_HTML.gif)
Therefore, from   (2.17) and , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ34_HTML.gif)
Hence (2.12) holds. Finally, we prove that  . By using (2.2) and together with the Schwarz inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ35_HTML.gif)
Since ,
, and
are bounded, we can take a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ36_HTML.gif)
for all . It then follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ37_HTML.gif)
where . By
, we get
. By applying Lemma 1.3 to (2.23), we can conclude that
. This completes the proof.
If and
in Theorem 2.1, we obtain the following result.
Corollary 2.2.
Let be a Hilbert space,
a closed convex nonempty subset of
, and let
. Let
be a finite family of nonexpansive mappings of
into itself, and let
be defined by (1.9). Assume that
. Let
, given that
and
are sequences in
, and suppose that the following conditions are satisfied:
(C1)
(C2)
(C3)
(C4) and
, where
;
(C5)
(C6).
If is the composite process defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ38_HTML.gif)
then converges strongly to
, which also solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ39_HTML.gif)
If ,
,
, and
is a constant in Theorem 2.1, we get the results of Kim and Xu [3].
Corollary 2.3.
Let be a Hilbert space,
a closed convex nonempty subset of
, and let
. Let
be a nonexpansive mapping of
into itself.
. Let
, given that
and
are sequences in
, and suppose that the following conditions are satisfied:
(C1)
(C2)
(C3)
(C4)
(C5).
If is the composite process defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ40_HTML.gif)
then converges strongly to
, which also solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F262691/MediaObjects/13663_2010_Article_1238_Equ41_HTML.gif)
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Acknowledgments
The authors would like to thank the referees for valuable suggestions on the paper and thank the Center of Excellence in Mathematics, the Thailand Research Fund, and the Graduate School of Chiang Mai University for financial support.
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Singthong, U., Suantai, S. A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2010, 262691 (2010). https://doi.org/10.1155/2010/262691
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DOI: https://doi.org/10.1155/2010/262691