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Strong Convergence of Monotone Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansive Mappings
Fixed Point Theory and Applications volume 2009, Article number: 261932 (2009)
Abstract
We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping in a Banach space by using monotone hybrid iteration method. By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Banach space.
1. Introduction
Let be a real Banach space and let
be the dual space of
Let
be a maximal monotone operator from
to
. It is well known that many problems in nonlinear analysis and optimization can be formulated as follows. Find a point
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ1_HTML.gif)
We denote by the set of all points
such that
Such a problem contains numerous problems in economics, optimization, and physics and is connected with a variational inequality problem. It is well known that the variational inequalities are equivalent to the fixed point problems. There are many authors who studied the problem of finding a common element of the fixed point of nonlinear mappings and the set of solutions of a variational inequality in the framework of Hilbert spaces see; for instance, [1–11] and the reference therein.
A well-known method to solve problem (1.1) is called the proximal point algorithm: and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ2_HTML.gif)
where and
are the resovents of
. Many researchers have studies this algorithm in a Hilbert space; see, for instance, [12–15] and in a Banach space; see, for instance, [16, 17].
In 2005, Matsushita and Takahashi [18] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space
:
chosen arbitrarily,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ3_HTML.gif)
where is the duality mapping on
,
. They proved that
generated by (1.3) converges strongly to a fixed point of
under condition that
.
In 2008, Su et al. [19] modified the CQ method (1.3) for approximation a fixed point of a closed hemi-relatively nonexpansive mapping in a Banach space. Their method is known as the monotone hybrid method defined as the following. chosen arbitrarily, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ4_HTML.gif)
where is the duality mapping on
,
They proved that
generated by (1.4) converges strongly to a fixed point of
under condition that
.
Note that the hybrid method iteration method presented by Matsushita and Takahashi [18] can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping.
Very recently, Inoue et al. [20] proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method.
Theorem 1.1 (Inoue et al. [20]).
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty closed convex subset of
Let
be a monotone operator satisfying
and let
for all
. Let
be a relatively nonexpansive mapping such that
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ5_HTML.gif)
for all , where
is the duality mapping on
,
and
for some
. If
, then
converges strongly to
, where
is the generalized projection from
onto
.
Employing the ideas of Inoue et al. [20] and Su et al. [19], we modify iterations (1.4) and (1.5) to obtain strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemi-relatively nonexpansive mappings in a Banach space. The results of this paper modify and improve the results of Inoue et al. [20], and some others.
2. Preliminaries
Throughout this paper, all linear spaces are real. Let and
be the sets of all positive integers and real numbers, respectively. Let
be a Banach space and let
be the dual space of
. For a sequence
of
and a point
the weak convergence of
to
and the strong convergence of
to
are denoted by
and
, respectively.
Let be a Banach space. Then the duality mapping
from
into
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ6_HTML.gif)
Let be the unit sphere centered at the origin of
. Then the space
is said to be smooth if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ7_HTML.gif)
exists for all . It is also said to be uniformly smooth if the limit exists uniformly in
. A Banach space
is said to be strictly convex if
whenever
and
. It is said to be uniformly convex if for each
, there exists
such that
whenever
and
. We know the following (see, [21]):
(i)if in smooth, then
is single valued;
-
(ii)
if
is reflexive, then
is onto;
-
(iii)
if
is strictly convex, then
is one to one;
(iv)if is strictly convex, then
is strictly monotone;
(v)if is uniformly smooth, then
is uniformly norm-to-norm continuous on each bounded subset of
.
Let be a smooth strictly convex and reflexive Banach space and let
be a closed convex subset of
Throughout this paper, define the function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ8_HTML.gif)
Observe that, in a Hilbert space , (2.3) reduces to
, for all
. It is obvious from the definition of the function
that for all
,
(1)
(2),
(3)
Following Alber [22], the generalized projection from
onto
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%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ9_HTML.gif)
Existence and uniqueness of the operator follow from the properties of the functional
and strict monotonicity of the mapping
In a Hilbert space,
is the metric projection of
onto
.
Let be a closed convex subset of a Banach space
and let
be a mapping from
into itself. We use
to denote the set of fixed points of
that is,
Recall that a self-mapping
is hemi-relatively nonexpansive if
and
for all
and
.
A point is said to be an asymptotic fixed point of
if
contains a sequence
which converges weakly to
and
. We denote the set of all asymptotic fixed points of
by
. A hemi-relative nonexpansive mapping
is said to be relatively nonexpansive if
. The asymptotic behavior of a relatively nonexpansive mapping was studied in [23].
Recall that an operator in a Banach space is call closed, if
and
, then
.
We need the following lemmas for the proof of our main results.
Lemma 2.1 (Kamimura and Takahashi [13]).
Let be a uniformly convex and smooth Banach space and let
and
be two sequences in
such that either
or
is bounded. If
, then
.
Lemma 2.2 (Matsushita and Takahashi [18]).
Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space
and let
be a relatively hemi-nonexpansive mapping from
into itself. Then
is closed and convex.
Lemma 2.3 (Alber [22], Kamimura and Takahashi [13]).
Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space,
and let
. Then,
if and only if
for all
.
Lemma 2.4 (Alber [22], Kamimura and Takahashi [13]).
Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ10_HTML.gif)
Let be a smooth, strictly convex, and reflexive Banach space, and let
be a set-valued mapping from
to
with graph
, domain
and range
We denote a set-valued operator
from
to
by
is said to be monotone of
A monotone operator
is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. It is known that a monotone mapping
is maximal if and only if for
for every
implies that
. We know that if
is a maximal monotone operator, then
is closed and convex; see [19] for more details. The following result is well known.
Lemma 2.5 (Rockafellar [24]).
Let be a smooth, strictly convex, and reflexive Banach space and let
be a monotone operator. Then
is maximal if and only if
for all
Let be a smooth, strictly convex, and reflexive Banach space, let
be a nonempty closed convex subset of
and let
be a monotone operator satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ11_HTML.gif)
Then we can define the resolvent by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ12_HTML.gif)
We know that consists of one point. For
, the Yosida approximation
is defined by
for all
.
Lemma 2.6 (Kohsaka and Takahashi [25]).
Let be a smooth, strictly convex, and reflexive Banach space, let
be a nonempty closed convex subset of
and let
be a monotone operator satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ13_HTML.gif)
Let and let
and
be the resolvent and the Yosida approximation of
, respectively. Then, the following hold:
(i)
(ii);
(iii)
Lemma 2.7 (Kamimura and Takahashi [13]).
Let be a uniformly convex and smooth Banach space and let
. Then there exists a strictly increasing, continuous and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ14_HTML.gif)
for all , where
.
3. Main Results
In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemi-relatively nonexpansive mapping in a Banach space by using the monotone hybrid iteration method.
Theorem 3.1.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty closed convex subset of
. Let
be a monotone operator satisfying
and let
for all
. Let
be a closed hemi-relatively nonexpansive mapping such that
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ15_HTML.gif)
for all , where
is the duality mapping on
and
for some
. If
, 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
. Next, we prove that
is convex.
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ16_HTML.gif)
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ17_HTML.gif)
which is affine in , and hence
is convex. So,
is a closed and convex subset of
for all
. Let
Put
for all
Since
and
are hemi-relatively nonexpansive mappings, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ18_HTML.gif)
So, for all
, which implies that
. Next, we show that
for all
. We prove that by induction. For
, we have
. Assume that
for some
. Because
is the projection of
onto
by Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ19_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ20_HTML.gif)
This together with definition of implies that
and hence
for all
. So, we have that
for all
. This implies that
is well defined. From definition of
we have
. So, from
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ21_HTML.gif)
Therefore, is nondecreasing. It follows from Lemma 2.4 and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ22_HTML.gif)
for all . Therefore,
is bounded. Moreover, by definition of
, we know that
and
are bounded. So, the limit of
exists. From
we have that for any positive integer,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ23_HTML.gif)
This implies that . Since
is bounded, there exists
such that
. Using Lemma 2.7, we have, for
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ24_HTML.gif)
where is a continuous, strictly increasing, and convex function with
. Then the properties of the function
yield that
is a Cauchy sequence in
. So there exists
such that
. In view of
and definition of
, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ25_HTML.gif)
It follows that Since
is uniformly convex and smooth, we have from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ26_HTML.gif)
So, we have Since
is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ27_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ28_HTML.gif)
This follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ29_HTML.gif)
From (3.13) and we obtain that
Since is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ30_HTML.gif)
From
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ31_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ32_HTML.gif)
From (3.4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ33_HTML.gif)
Using and Lemma 2.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ34_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ35_HTML.gif)
From (3.13) and we have
Since is uniformly convex and smooth, we have from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ36_HTML.gif)
From we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ37_HTML.gif)
Since and
we have
. Since
is a closed operator and
,
is a fixed point of
. Next, we show
. Since
is uniformly norm-to-norm continuous on bounded sets, from (3.22) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ38_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ39_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ40_HTML.gif)
For , from the monotonicity of
, we have
for all
. Letting
, we get
. From the maximality of
, we have
. Finally, we prove that
. From Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ41_HTML.gif)
Since and
we get from Lemma 2.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ42_HTML.gif)
By the definition of , it follows that
and
, whence
. Therefore, it follows from the uniqueness of the
that
.
As direct consequences of Theorem 3.1, we can obtain the following corollaries.
Corollary 3.2.
Let be a uniformly convex and uniformly smooth Banach space. Let
be a maximal monotone operator with
and let
for all
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ43_HTML.gif)
for all where
is the duality mapping on
,
and
for some
. Then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Putting ,
and
in Theorem 3.1, we obtain Corollary 3.2.
Let be a Banach space and let
be a proper lower semicontinuous convex function. Define the subdifferential of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ44_HTML.gif)
for each . Then, we know that
is a maximal monotone operator; see [21] for more details.
Corollary 3.3 (Su et al. [19, Theorem  3.1]).
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
and let
be a closed hemi-relatively nonexpansive mapping from
into itself such that
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ45_HTML.gif)
for all , where
is the duality mapping on
and
. If  
then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Set in Theorem 3.1, where
is the indicator function; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ46_HTML.gif)
Then, we have that is a maximal monotone operator and
for
, in fact, for any
and
, we have from Lemma 2.3 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ47_HTML.gif)
So, we obtain the desired result by using Theorem 3.1.
Since every relatively nonexpansive mapping is a hemi-relatively one, the following theorem is obtained directly from Theorem 3.1.
Theorem 3.4.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty closed convex subset of
Let
be a monotone operator satisfying
and let
for all
Let
be a closed relatively nonexpansive mapping such that
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ48_HTML.gif)
for all , where
is the duality mapping on
and
for some
. If
then
converges strongly to
, where
is the generalized projection from
onto
.
Corollary 3.5 (Su et al. [19, Theorem  3.2]).
Let be a uniformly convex and uniformly smooth Banach space, let
be a nonempty closed convex subset of
and let
be a closed relatively nonexpansive mapping from
into itself such that
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F261932/MediaObjects/13663_2009_Article_1127_Equ49_HTML.gif)
for all , where
is the duality mapping on
and
. If  
, then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Set in Theorem 3.4, where
is the indicator function. So, from Theorem 3.4, we obtain the desired result.
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The authors would like to thank the referee for valuable suggestions that improve this manuscript and the Thailand Research Fund (RGJ Project) and Commission on Higher Education for their financial support during the preparation of this paper. The first author was supported by the Royal Golden Jubilee Grant PHD/0018/2550 and by the Graduate School, Chiang Mai University, Thailand.
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Klin-eam, C., Suantai, S. Strong Convergence of Monotone Hybrid Method for Maximal Monotone Operators and Hemirelatively Nonexpansive Mappings. Fixed Point Theory Appl 2009, 261932 (2009). https://doi.org/10.1155/2009/261932
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DOI: https://doi.org/10.1155/2009/261932