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New Hybrid Iterative Schemes for an Infinite Family of Nonexpansive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2010, Article number: 572838 (2010)
Abstract
We propose some new iterative schemes for finding common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the proposed schemes. Our results extend and improve ones of Nakajo and Takahashi (2003).
1. Introduction and Preliminaries
Let be a Hilbert space and
a nonempty closed convex subset of
. Let
be a nonlinear mapping of
into itself. We use
and
to denote the set of fixed points of
and the metric projection from
onto
, respectively.
Recall that is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ1_HTML.gif)
for all .
For approximating the fixed point of a nonexpansive mapping in a Hilbert space, Mann [1] in 1953 introduced a famous iterative scheme as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ2_HTML.gif)
where is a nonexpansive mapping of
into itself and
is a sequence in
. It is well known that
defined in (1.2) converges weakly to a fixed point of
.
Attempts to modify the normal Mann iteration method (1.2) for nonexpansive mappings so that strong convergence is guaranteed have recently been made; see, for example, [2–9].
Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.2) for a single nonexpansive mapping in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ3_HTML.gif)
where denotes the metric projection from
onto a closed convex subset
of
. They proved that if the sequence
is bounded above from one, then the sequence
generated by (1.3) converges strongly to
.
In this paper, we introduce some new iterative schemes for infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the algorithms. Our results extend and improve the corresponding one of Nakajo and Takahashi [5].
The following two lemmas will be used for the main results of this paper.
Lemma 1.1.
Let be a closed convex subset of a real Hilbert space
and let
be the metric projection from
onto
(i.e., for
,
is the only point in
such that
. Given
and
, then
if and only if there holds the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ4_HTML.gif)
Lemma 1.2 (see [10]).
Let be a real Hilbert space. Then the following equation holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ5_HTML.gif)
2. Main Results
Theorem 2.1.
Let be a nonempty closed convex subset of a Hilbert space
. Let
be an infinite family of nonexpansive mappings such that
. Let
be a sequence generated by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ6_HTML.gif)
where is a sequence in
satisfying
and
is a sequence in
satisfying
. Then
defined by (2.1) converges strongly to
.
Proof.
We first show that is closed and convex. By Lemma 1.2, one observes that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ7_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ8_HTML.gif)
for all . So,
is closed and convex for all
and hence
is also closed and convex for all
. This implies that
is well defined.
Next, we show that for all
. To end this, we need to prove that
for all
. Indeed, for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ9_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ10_HTML.gif)
Therefore, and
is nonempty for all
. On the other hand, from the definition of
, we see that
for all
.
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ11_HTML.gif)
Since for all
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ12_HTML.gif)
This implies that is bounded. For each fixed
, by (2.1) we have (noting that
)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ13_HTML.gif)
for all . Since
is bounded,
is bounded for each
.
On the other hand, observing that for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ14_HTML.gif)
for all . Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ15_HTML.gif)
for all . It follows from (2.7) and (2.10) that the limit of
exists.
Since and
for all
and
, by Lemma 1.1 one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ16_HTML.gif)
It follows from (2.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ17_HTML.gif)
Since the limit of exists, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ18_HTML.gif)
It follows that is a Cauchy sequence. Since
is a Hilbert space and
is closed and convex, one can assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ19_HTML.gif)
By taking in (2.12), one arrives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ20_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ21_HTML.gif)
Noticing that , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ22_HTML.gif)
This implies that . Since each
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ23_HTML.gif)
From (2.16) and (2.18), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ24_HTML.gif)
By and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ25_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ26_HTML.gif)
Finally, we prove that . From
and
, one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ27_HTML.gif)
Taking the limit in (2.22) and noting that as
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ28_HTML.gif)
In view of Lemma 1.1, one sees that . This completes the proof.
Corollary 2.2.
Let be a nonempty closed convex subset of a Hilbert space
. Let
be a nonexpansive mapping such that
. Let
be a sequence generated by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ29_HTML.gif)
where is a sequence in
satisfying that
. Then
defined by (2.24) converges strongly to
.
Proof.
Set for all
,
and
for all
in Theorem 2.1. By Theorem 2.1, we obtain the desired result.
Theorem 2.3.
Let be a nonempty closed convex subset of a Hilbert space
. Let
be an infinite family of nonexpansive mappings such that
. Let
be a sequence generated by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ30_HTML.gif)
where is a strictly decreasing sequence in
and set
. Then
defined by (2.25) converges strongly to
.
Proof.
Obviously, is closed and convex for all
and hence
is also closed and convex for all
. Next, we prove that
for all
. For any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ31_HTML.gif)
This shows that for all
. Therefore,
for all
. It follows that
for all
.
By using the method of Theorem 2.1, we can conclude that is bounded,
,
and
as
. This implies that
as
.
Next, we show that . To end this, we see a fact. For
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ32_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ33_HTML.gif)
for each
Observe that , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ34_HTML.gif)
It follows from (2.28) and (2.29) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ35_HTML.gif)
Since is strictly decreasing,
and
as
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ36_HTML.gif)
for each Since each
is nonexpansive, one has
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572838/MediaObjects/13663_2009_Article_1305_Equ37_HTML.gif)
Finally, by using the method of Theorem 2.1, we can conclude that . This completes the proof.
Remark 2.4.
In this paper, we extend result of Nakajo and Takahashi [5] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.
Remark 2.5.
The iterative schemes introduced in this paper are new and of independent interest.
Remark 2.6.
It is of interest to extend the algorithm (2.25) to certain Banach spaces.
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Acknowledgment
The work was supported by Youth Foundation of North China Electric Power University.
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Guo, B., Wang, S. New Hybrid Iterative Schemes for an Infinite Family of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2010, 572838 (2010). https://doi.org/10.1155/2010/572838
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DOI: https://doi.org/10.1155/2010/572838