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Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings
Fixed Point Theory and Applications volume 2008, Article number: 284613 (2007)
Abstract
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others.
1. Introduction
In 2005, Shin-ya Matsushita and Wataru Takahashi [1] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ1_HTML.gif)
They proved the following convergence theorem.
Theorem 1.1 (MT).
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty, closed, and convex subset of
, let
be a relatively nonexpansive mapping from
into itself, and let
be a sequence of real numbers such that
and
. Suppose that
is given by (1.1), where
is the duality mapping on
. If the set
of fixed points of
is nonempty, then
converges strongly to
, where
is the generalized projection from
onto
.
The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S.Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S.Matsushita and W. Takahashi [1], and some others.
2. Preliminaries
Let be a real Banach space with dual
. We denote by
the normalized duality mapping from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ2_HTML.gif)
where denotes the generalized duality pairing. It is well known that if
is uniformly convex, then
is uniformly continuous on bounded subsets of
. In this case,
is singe valued and also one to one.
Recall that if is a nonempty, closed, and convex subset of a Hilbert space
and
is the metric projection of
onto
, then
is nonexpansive. This is true only when
is a real Hilbert space. In this connection, Alber [2] has recently introduced a generalized projection operator
in a Banach space
which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that is a smooth Banach space. Consider the functional defined as [2, 3] by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ3_HTML.gif)
Observe that, in a Hilbert space , (2.2) reduces to
,
The generalized projection is a map that assigns to an arbitrary point
the minimum point of the functional
that is,
where
is the solution to the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ4_HTML.gif)
existence and uniqueness of the operator follow from the properties of the functional
and strict monotonicity of the mapping
(see, e.g., [2–4]). In Hilbert space,
It is obvious from the definition of the function
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ5_HTML.gif)
Remark 2.1.
If is a reflexive strict convex and smooth Banach space, then for
,
if and only if
. It is sufficient to show that if
, then
. From (2.4), we have
. This implies
From the definition of
we have
, that is,
see [5] for more details.
We refer the interested reader to the [6], where additional information on the duality mapping may be found.
Let be a closed convex subset of
, and Let
be a mapping from
into itself. We denote by
the set of fixed points of
.
is called hemi-relatively nonexpansive if
for all
and
.
A point in
is said to be an asymptotic fixed point of
[7] if
contains a sequence
which converges weakly to
such that the strong
The set of asymptotic fixed points of
will be denoted by
. A hemi-relatively nonexpansive mapping
from
into itself is called relatively nonexpansive [1, 7, 8] if
.
We need the following lemmas for the proof of our main results.
Lemma 2.2 ({Kamimura and Takahashi [4], [1, Proposition 2.1]}).
Let be a uniformly convex and smooth real Banach space and let
be two sequences of
. If
and either
or
is bounded, then
Lemma 2.3 ({Alber [2], [1, Proposition 2.2]}).
Let be a nonempty closed convex subset of a smooth real Banach space
and
. Then,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ6_HTML.gif)
Lemma 2.4 ({Alber [2], [1, Proposition 2.3]}).
Let be a reflexive, strict convex, and smooth real Banach space, let
be a nonempty closed convex subset of
and let
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ7_HTML.gif)
By using the similar method as [1, Proposition 2.4], the following lemma is not hard to prove.
Lemma 2.5.
Let be a strictly convex and smooth real Banach space, let
be a closed convex subset of
, and let
be a hemi-relatively nonexpansive mapping from
into itself. Then
is closed and convex.
Recall that an operator in a Banach space is called closed, if
, then
.
3. Strong Convergence for Hemi-Relatively Nonexpansive Mappings
Theorem 3.1.
Theorem 3.1 Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty closed convex subset of
, let
be a closed hemi-relatively nonexpansive mapping such that
. Assume that
is a sequence in
such that
. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ8_HTML.gif)
where is the duality mapping on
. Then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
We first show that and
are closed and convex for each
. From the definition of
and
, it is obvious that
is closed and
is closed and convex for each
. We show that
is convex for any
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ9_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ10_HTML.gif)
it follows that is convex.
Next, we show that for all
. Indeed, we have for all
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ11_HTML.gif)
That is, for all
.
Next, we show that for all
, we prove this by induction. For
we have
Assume that
Since
is the projection of
onto
, by Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ12_HTML.gif)
As by the induction assumptions, the last inequality holds, in particular, for all
. This together with the definition of
implies that
.
Since and
for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ13_HTML.gif)
for all . Therefore,
is nondecreasing. In addition, it follows from the definition of
and Lemma 2.3 that
. Therefore, by Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ14_HTML.gif)
for each for all
Therefore,
is bounded, this together with (3.6) implies that the limit of
exists. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ15_HTML.gif)
From Lemma 2.4, we have, for any positive integer , that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ16_HTML.gif)
for all Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ17_HTML.gif)
We claim that is a Cauchy sequence. If not, there exists a positive real number
and subsequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ18_HTML.gif)
for all .
On the other hand, from (3.8) and (3.9) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ19_HTML.gif)
Because from (3.8) we know that is bounded, this and (2.4) imply that
is also bounded, so by Lemma 2.2 we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ20_HTML.gif)
This is a contradiction, so that is a Cauchy sequence, therefore there exists a point
such that
converges strongly to
.
Since , from the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ21_HTML.gif)
It follows from (3.10), (3.14) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ22_HTML.gif)
By using Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ23_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ24_HTML.gif)
Noticing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ25_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ26_HTML.gif)
This together with (3.17) and implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ27_HTML.gif)
Since is also uniformly norm-to-norm continuous on any bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ28_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ29_HTML.gif)
It follows from (3.16) and (3.21) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ30_HTML.gif)
Since is a closed operator and
, then
is a fixed point of
.
Finally, we prove that . From Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ31_HTML.gif)
On the other hand, since and
, for all
, we get from Lemma 2.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ32_HTML.gif)
By the definition of , it follows that both
and
, whence
. Therefore, it follows from the uniqueness of
that
. This completes the proof.
Theorem 3.2.
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty, closed, and convex subset of
, and let
be a closed relative nonexpansive mapping such that
. Assume that
is a sequences in
such that
. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F284613/MediaObjects/13663_2007_Article_1071_Equ33_HTML.gif)
where is the duality mapping on
. Then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Since every relatively nonexpansive mapping is a hemi-relatively one, Theorem 3.2 is implied by Theorem 3.1.
Remaek 3.3.
In recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings have been introduced and studied by various authors [1, 8–11]. In fact, all hybrid iteration methods can be replaced (or modified) by monotone hybrid iteration methods, respectively. In addition, by using the monotone hybrid method we can easily show that the iteration sequence is a Cauchy sequence, without the use of the Kadec-Klee property, demiclosedness principle, and Opial's condition or other methods which make use of the weak topology.
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Acknowledgments
The authors would like to thank the referee for valuable suggestions which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China under Grant no. 10771050.
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Su, Y., Wang, D. & Shang, M. Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings. Fixed Point Theory Appl 2008, 284613 (2007). https://doi.org/10.1155/2008/284613
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DOI: https://doi.org/10.1155/2008/284613