An New Iterative Scheme for Variational Inequalities and Nonexpansive Mappings in Hilbert Spaces
Qiqiong Chen1, *, Congjun Zhang2
1Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, Jiangsu, China
2College of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu, China
Abstract
In this paper, a new three-step iterative scheme is introduced for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality for -inverse-strongly monotone mappings. The result reveals that the proposed iterative sequence converges strongly to the common element of this two. And our studies can be regarded as an extension of the existing results, which we illustrate one by one in our remarks.
Keywords
Variational Inequality, Nonexpansive Mapping, Fixed Point
Received:May 18, 2015
Accepted: May 27, 2015
Published online: July 9, 2015
@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/
1. Introduction
Variational inequalities first studied by Stampacchia [1] in 1960s have played an important role in the development of pure and applied mathematics. They have also witnessed an explosive growth in theoretical progression, algorithmic development, etc.; see e.g. [2-14]. Let be a real Hilbert space, whose inner product and norm are denoted by
and ‖.‖, respectively. Let
be a nonempty closed convex subset of
and
a mapping from
to
. The classical variational inequality problem is to find a vector
such that
,
For all or The set of solutions of the variational inequality is denoted by
. A mapping
of
to
is called α-inverse-strongly monotone [6] if there exists a positive real number α such that
for any . A mapping
of
into itself is called nonexpansive [6] if
for all . We denote the set of fixed points of
by
.
In order to seek for an element of Takahashi and Toyoda [4] introduced the following iterative scheme
(1)
for every , where
is a sequence in
and
is a sequence in (0,2
is the metric projection of
onto
They proved that the iterative consequences
generated by (1) converge weakly to an element
For convenience, we will use
through the whole paper.
On the other hand, Iiduka and Takahashi [5] put forward another iterative scheme:
(2)
for every , where
is a sequence in
is a sequence in (0, 2
is the metric projection of
onto
It was proved that the iterative consequences
generated by (2) converge strongly to an element
.
Furthermore, Yao and Yao [6] proposed the following mixed gradient method:
(3)
for every , where
,
are sequences in
satisfied
and
is a sequence in (0, 2
is the metric projection of
onto
They proved that the iterative consequences defined by (3) converge strongly to
, where
was the metric projection of
onto
In recent years, many authors have studied some different iterative schemes both in Hilbert spaces and Banach spaces, see e.g. [2-14]. Inspired and motivated by those previous researches, we suggest and analyze a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of a variational inequality for -inverse-strongly monotone mapping in real Hilbert spaces. Strong convergence theorems are established and the iterative methods considered by [4,5,7,10] are included in our results.
2. Preliminary
For convenience, we would like to list some definitions and fundamental lemmas which are useful in the following consequent analysis. They can be found in any standard functional analysis books such as [15,16].
Definition 2.1 A mapping is a contraction on
if there exists a constant
such that
Definition 2.2 A set-valued mapping is called monotone if for all
and
imply
Definition 2. 3 A monotone mapping is maximal if its graph
is not properly contained in the graph of any other monotone mapping.
It is known that a monotone mapping is maximal if and only if, for
for every
implies
Let
be a monotone mapping of
into
and let
be the normal cone operator to
defined by
. Define
(4)
Then is maximal monotone and
if and only if
(see [11]).
Definition 2.4 For every point , there exists a unique nearest point
in
denoted by
, such that
for all
is called the metric projection of
onto
It is well known that is a nonexpansive mapping of
onto
and satisifies
for every
Moreover, is characterized by the following properties:
for all
(5)
It is easy to see that the following is true:
(6)
Note that satisfies Opial's condition [17], i.e., for any sequence
with
the inequality
holds for every
with
Next we present some useful lemmas.
The following lemma is an immediate consequence of equality:
Lemma 2. 1 Let be a real Hilbert space. Then the following inequality holds:
Lemma 2. 2 (Osilike [14]) Let be an inner space. Then for all
and
with
we have
=
Lemma 2. 3 (Xu [10]) Assume is a sequence of nonnegative real numbers such that
)
+
, where
is a sequence in
and
is a sequence such that
(i). =
(ii). or
Then
For convenience, we use for strong convergence and
for weak convergence in the following analysis.
3. Main Results
In this section, we suggest and analyze a new iterative scheme for finding the common element of the fixed points of a nonexpansive mapping and the solution set of variational inequalities for an -inverse-strongly monotone mapping in a real Hilbert space. Strong convergence theorems are established and several special cases are also discussed.
Theorem 3. 1 Let be a nonempty closed convex subset of a real Hilbert space
Let
be an
-inverse-strongly monotone mapping of
into
and let
be a nonexpansive mapping of
into itself such that
be a contraction mapping with coefficient
Suppose
and
are given by
(7)
where ,{
} are three sequences in
and
is a sequence in
. Assume that
,{
} are chosen so that
for some
with
and
(C1) =
,
(C2) =
,
(C3)
Then the sequence converges strongly to
where
or equivalently
satisfies the following variational inequality:
Proof: We first show that is a nonexpansive mapping. For all
and
, we have
which implies that is nonexpansive .
For convenience, we set
Then the iterative scheme (7) can be written as:
(8)
Let Then we have
by (6) and
Since the proof of the theorem is rather long, it will be more convenient to divide the process into several steps.
Step 1. We claim that is bounded.
Since both and
are nonexpansive mappings, we have
(9)
Similarly, we obtain that
(10)
Combining (8) and (9), together with that is nonexpansive mapping, we see that
(11)
By (10) and (11), we get
Hence
(12)
From (12), we arrive at
(13)
By the method of induction, we have
(14)
Therefore is bounded. Consequently, all those sequences
are bounded.
Step 2. We now in the position to prove that
Since both and
are nonexpansive mappings, we first have
(15)
By similar method, we have
(16)
In view of (15), after simple calculation, we see that
(
). (17)
By (16) and (17), we get
(
)
+(
)
+ (18)
In view of (18), we have
(19)
Where
=
[
]+
(
)
+(
)
+. (20)
By the conditions (C1), (C2) and (C3), we see that and
which combining with Lemma 2.3, yields
. (21)
Since together with (21) and the condition (C1) imply that
. (22)
Since =(1-
and
is bounded, we have
. (23)
Furthermore, combining Lemma 2.2 with that is nonexpansive,
is
-inverse-strongly monotone mapping,
, and
we obtain that
(24)
From (24), together with Lemma 2.2, we see that
(25)
which implies that
(26)
It follows from conditions (C1), (C2) and (21) that
(27)
Step 3. We show that
Since ,
is bounded, and
we have
(28)
We now show that
Since
= {
}
[
], (29)
we get that
+
(30)
Hence
(31)
It follows from (30) that
(32)
Hence
(33)
Since
and
are bounded,
we have
(34)
It follows from (28), (22),(23) and (34) , together with
that
(35)
Step 4. We prove that
As is bounded, there exists a subsequence
of
converges weakly to
Since
combining (23) and (34) we know that
Then
Next we show that
Let
where Then
is maximal monotone. Let
where
Since and
we have
On the other hand, from (5) and
we see that Then
Thus
(36)
Putting , we have
Since is maximal, we have
Hence
Now let us show that Assume that
From Opial’s condition, we have
. (37)
This is a contradiction. Thus we obtain that
Since is a contraction mapping, by Banach's contraction theorem, there exists a unique fixed point
of
, that's
Step 5. We prove that
From (5), we know
(38)
Step 6. We claim that From Lemma 2.1 and Lemma 2.2, we obtain that
=
. (39)
Then we have
. (40)
That is
(41)
where , and
{
}.
From (38) and conditions (C1), (C2) and (C3), letting yields
(42)
Let
Then +
It is easy to check that
By Lemma 2.3, we see that
(43)
The proof is finished.
As an implication of Theorem 3.1, we have the following corollary:
Corollary 3.1 Let be a nonempty closed convex subset of a real Hilbert space
Let
be an
-inverse-strongly monotone mapping of
into
and let
be a be a nonexpansive mapping of
into itself such that
be a contraction mapping with coefficient k
Suppose that
and
are given by
where ,{
} are three sequences in
and
is a sequence in
. Assume that
,{
} are chosen so that
for some
with
and
(C1) =
,
(C2) =
,
(C3)
Then the sequence converges strongly to
where
or equivalently
satisfies the following variational inequality:
Proof: The conclusion follows from Theorem 3.1 by setting
Theorem 3.1 extends the corresponding results of [4,5,7,10].
Remark 3.1 Putting in Theorem 3.1, we can get the iterative scheme provided by [4].
Remark 3.2 Putting in Theorem 3.1, we can get the iterative scheme provided by [5].
Remark 3.3 The proposition 3.1 of [7] is a special case of our result.
In fact, letting in Theorem 3.1, we get
Then
by Theorem 3.1.
Remark 3.4 Putting
in Theorem 3.1, we can get the iterative scheme provided by [10].
Remark 3.5 The conditions in Theorem 3.1 can be easily satisfied, for example
4. Conclusion
By introducing a new iterative scheme for variational inequalities and nonexpansive mappings in Hilbert spaces, we proved that the sequences generated by the iterative scheme strongly converge to a common element of the fixed points of a nonexpansive mapping and the solution set of variational inequality for -inverse-strongly monotone mapping.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11071109) and the China Scholarship Council (No. 201406840039). The authors are so grateful for Professor Yuanguo Zhu’s valuable suggestions to improve this paper. It is accomplished during the first author’s visit to Professor Jinlu Li at Shawnee State University, USA. The authors also would like to express their deep gratitude for the warm hospitality from Shawnee State University.
References