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Regularization Inertial Proximal Point Algorithm for Monotone Hemicontinuous Mapping and Inverse Strongly Monotone Mappings in Hilbert Spaces
Journal of Inequalities and Applications volume 2010, Article number: 451916 (2010)
Abstract
The purpose of this paper is to present a regularization variant of the inertial proximal point algorithm for finding a common element of the set of solutions for a variational inequality problem involving a hemicontinuous monotone mapping and for a finite family of
-inverse strongly monotone mappings
from a closed convex subset
of a Hilbert space
into
.
1. Introduction
Let be a real Hilbert space with inner product
and norm
. Let
be a closed convex subset of
. Denote the metric projection of
onto
by
, that is

for all
Definition 1.1.
A set is called monotone on
if
has the following property:

A monotone mapping in
is said to be maximal monotone if it is not properly contained in any other monotone mapping on
. Equivalently, a monotone mapping
is maximal monotone if
for all
, where
denotes the range of
.
Definition 1.2.
A mapping is called hemicontinuous at a point
in
if

Definition 1.3.
A mapping of
into
is called
-inverse strongly monotone if there exists a positive number
such that

for all .
Definition 1.4.
A mapping of
into
is called Lipschitz continuous on
if there exists a positive number
, named Lipschitz constant, such that

for all .
It is easy to see that any -inverse strongly monotone mapping
is monotone and Lipschitz continuous with Lipschitz constant
. When
,
is said to be nonexpansive mapping. Note that a nonexpansive mapping in Hilbert space is
-inverse strongly monotone [1].
Definition 1.5.
A mapping is said to be strictly pseudocontractive, if there exists a constant
such that

Clearly, when ,
is nonexpansive. Therefore, the class of
-strictly pseudocontractive mappings includes the class of nonexpansive mappings.
Definition 1.6.
A mapping from
into
is said to be demiclosed at a point
if whenever
is a sequence in
such that
and
, then
where the symbols
and
denote the strong and weak convergences of any sequence, respectively.
The variational inequality problem is to find such that

for all . The set of solutions of the variational inequality problem (1.7) is denoted by
.
Let be a finite family of
-inverse strongly monotone mappings from
into
with the set of solutions denoted by
And set

The problem which will be studied in this paper is to find an element

with assumption .
A following example shows the fact that Consider the following case:


, where
denote the metric projections of
onto
, and the matrix
has the elements
,
, and
. Then,
is
-inverse strongly monotone. Clearly,
if and only if
. It means that
. Consequently,
. It is easy to see that
is monotone, and for
,
if and only if
. Therefore,
.
Since for a nonexpansive mapping , the mapping
is
-inverse strongly monotone, the problem of finding an element of
, where
denotes the set of fixed points of the nonexpansive mapping
, is equivalent to that of finding an element of
, where
denotes the set of solutions of the mapping
, and contained in the class of problem (1.9).
The case, when is
-inverse strongly monotone and
, where
is nonexpansive, is studied in [2].
Theorem 1.7 (see [2]).
Let be a nonempty closed convex subset of a Hilbert space
. Let
be a
-inverse strongly monotone mapping of
into
for
, and let
be a nonexpansive mapping of
into itself such that
. Let
be a sequence generated by

where for some
and
for some
. Then
converges weakly to
, where

For finding an element of the set , one can use the extragradient method proposed in [3] for the case of finite-dimensional spaces. In the infinite-dimensional Hilbert spaces, the weak convergence result of the extragradient method was proved [1] and it was improved to the strong convergence in [4].
On the other hand, when , (1.7) is equivalent to the operator equation

involving a maximal monotone , since the domain of
is the whole space
, and
is hemicontinuous ([5, 6]). A zero element of (1.13) can be approximated by the inertial proximal point algorithm

where and
are two sequences of positive numbers.
Note that the inertial proximal algorithm was proposed by Alvarez [7] in the context of convex minimization. Afterwards, Alvarez and Attouch [8] considered its extension to maximal monotone operators. Recently, Moudafi [9] applied this algorithm for variational inequalities; Moudafi and Elisabeth [10] studied the algorithm by using enlargement of a maximal monotone operator; Moudafi and Oliny [11] considered convergence of a splitting inertial proximal method. The main results in these papers are the weak convergence of the algorithm in Hilbert spaces.
In this paper, by introducing a regularization process we shall show that by adding the regularization term to the inertial proximal point algorithm, called regularization inertial proximal point algorithm, we obtain the strong convergence of the algorithm, and the strong convergence is proved for the general case are
-inverse strongly monotone nonself mappings of
into
;
may not be
is monotone and hemicontinuous at each point
.
2. Main Results
Let be an equilibrium bifunction from
to
, that is
for every
In addition, assume that
is convex and lower semicontinuous in the variable
for each fixed
.
The equilibrium problem for is to find
such that

First, we recall several well-known facts in [12, 13] which are necessary in the proof of our results.
The equilibrium bifunction is said to be
(i)monotone, if for all , we have

(ii)strongly monotone with constant , if, for all
, we have

(iii)hemicontinuous in the variable for each fixed
, if

We can get the following proposition from the above definitions.
Proposition 2.1.
-
(i)
If
is hemicontinuous in the first variable for each fixed
and
is monotone, then
, where
is the solution set of (2.1),
is the solution set of
for all
, and they are closed and convex.
-
(ii)
If
is hemicontinuous in the first variable for each
and
is strongly monotone, then
is a nonempty singleton.
Lemma 2.2 (see [14]).
Let be the sequences of positive numbers satisfying the conditions:
(i),
(ii),
Then,
Lemma 2.3 (see [15]).
Let be a nonempty closed convex subset of a Hilbert space
and
a strictly pseudocontractive mapping. Then
is demiclosed at zero.
We construct a regularization solution for (1.9) by solving the following variational inequality problem: find
such that

where , is the regularization parameter.
We have the following result.
Theorem 2.4.
Let be a nonempty closed convex subset of
. Let
be a
-inverse strongly monotone mapping of
into
, and let
be a monotone hemicontinuous mapping of
into
such that
. Then, we have
(i)For each , the problem (2.5) has a unique solution
;
(ii)If then
and
for all
-
(iii)
(2.6)
where is a positive constant.
Proof.
-
(i)
Let
(2.7)
Then, problem (2.5) has the following form: find such that

where

It is not difficult to verify that are the monotone bifunctions, and for each fixed
, they are hemicontinuous in the variable
. Therefore,
also is monotone hemicontinuous in the variable
for each fixed
. Moreover, it is strongly monotone with constant
. Hence, (2.8) (consequently (2.5)) has a unique solution
for each
.
-
(ii)
Now we prove that

Since and
,
and

By adding the last inequality to (2.8) in which is replaced by
and using the properties of
, we obtain

that implies (2.10). It means that is bounded. Let
, as
. Since
is closed and convex,
is weakly closed. Hence
. We prove that
. From the monotone property of
and (2.8), it follows

Letting , we obtain
for any
By virtue of Proposition 2.1, we have
Now we show that for all
From (2.8),
for any
, and the monotone property of
, it implies that

On the base of -inverse strongly monotone property of
, the monotone property of
,
,
, for all
,
. From the last inequality, we have

Tending in the last inequality, we obtain

Since is
-inverse strongly monotone, the mapping
satisfies (1.6), where
Because
, we have
. When
, this inequality will not be changed if
is replaced by
. Thus,
is strictly pseudocontractive. Applying Lemma 2.2, we can conclude that
It means that
It is well known that the sets
are closed and convex. Therefore,
is also closed and convex. Then, from (2.10) it implies that
is the unique element in
having a minimal norm. Consequently, we have

(iii) From (2.8) and the properties of , for each
, it follows

or

For each is bounded since the operator
is Lipschitzian with Lipschitz constants
. Using (2.10), the boundedness of
and the Lagrange's mean-value theorem for the function
,
,
, on
if
or
if
, we have conclusion (iii). This completes the proof.
Remark 2.5.
Obviously, if , where
is the solution of (2.8) with
, as
, then
Now, we consider the regularization inertial proximal point algorithm

Clearly,

is a bifunction. Moreover, it is strongly monotone with By Proposition 2.1, there exists a unique element
satisfying (2.20).
Theorem 2.6.
Let be a nonempty closed convex subset of a Hilbert space
. Let
be a
-inverse strongly monotone mapping of
into
, and let
be a monotone hemicontinuous mapping of
into
such that
. Assume that the parameters
, and
are chosen such that
(i),
,
(ii),
(iii)
-
(iv)
(2.22)
Then the sequence defined by (2.20) converges strongly to the element
as
.
Proof.
From (2.20) it follows

By the similar argument, from (2.5) it implies

where is the solution of (2.5) when
is replaced by
. By setting
in (2.23) and
in (2.24) and adding one obtained result to the other, we have,

Therefore, from the monotone property of the mappings ,
, it follows

From (2.23), (2.5) with and
, we have

where

Since the seris in (iii) is convergent, Consequently,
From Lemma 2.2, it follows that
as
.
On the other hand, as
. Therefore, we have
as
This completes the proof.
Remark 2.7.
The sequences and
which are defined by

with satisfy all conditions in Theorem 2.6.
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Acknowledgment
This work was supported by the Kyungnam University Research Fund 2009.
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Kim, J., Buong, N. Regularization Inertial Proximal Point Algorithm for Monotone Hemicontinuous Mapping and Inverse Strongly Monotone Mappings in Hilbert Spaces. J Inequal Appl 2010, 451916 (2010). https://doi.org/10.1155/2010/451916
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DOI: https://doi.org/10.1155/2010/451916
Keywords
- Hilbert Space
- Variational Inequality
- Weak Convergence
- Monotone Mapping
- Nonexpansive Mapping