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A Hybrid Method for Monotone Variational Inequalities Involving Pseudocontractions
Fixed Point Theory and Applications volume 2011, Article number: 180534 (2011)
Abstract
We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point with the property
such that
,
where
,
are two pseudocontractive self-mappings of a closed convex subset
of a Hilbert space with the set of fixed points
. Assume the solution set
of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element
. Our results improve and extend a recent result of (Lu et al. 2009).
1. Introduction
Let be a real Hilbert space with inner product
and norm
, respectively, and let
be a nonempty closed convex subset of
. Let
be a nonlinear mapping. A variational inequality problem, denoted
, is to find a point
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ1_HTML.gif)
If the mapping is a monotone operator, then we say that
is monotone. It is well known that if
is Lipschitzian and strongly monotone, then for small enough
, the mapping
is a contraction on
and so the sequence
of Picard iterates, given by
(
) converges strongly to the unique solution of the
. Hybrid methods for solving the variational inequality
were studied by Yamada [1], where he assumed that
is Lipschitzian and strongly monotone.
In this paper, we devote to consider the following monotone variational inequality: finding a point with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ2_HTML.gif)
where are two nonexpansive mappings with the set of fixed points
. Let
denote the set of solutions of VI (1.2) and assume that
is nonempty.
We next briefly review some literatures in which the involved mappings and
are all nonexpansive.
First, we note that Yamada's methods do not apply to VI (1.2) since the mapping fails, in general, to be strongly monotone, though it is Lipschitzian. As a matter of fact, the variational inequality (1.2) is, in general, ill-posed, and thus regularization is needed. Recently, Moudafi and Maingé [2] studied the VI (1.2) by regularizing the mapping
and defined
as the unique fixed point of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ3_HTML.gif)
Since Moudafi and Maingé's regularization depends on , the convergence of the scheme (1.3) is more complicated. Very recently, Lu et al. [3] studied the VI (1.2) by regularizing the mapping
and defined
as the unique fixed point of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ4_HTML.gif)
Note that Lu et al.'s regularization (1.4) does no longer depend on . Related work can also be found in [4–9].
In this paper, we will extend Lu et al.'s result to a general case. We will further study the strong convergence of the algorithm (1.4) for solving VI (1.2) under the assumption that the mappings are all pseudocontractive. As far as we know, this appears to be the first time in the literature that the solutions of the monotone variational inequalities of kind (1.2) are investigated in the framework that feasible solutions are fixed points of a pseudocontractive mapping
.
2. Preliminaries
Let be a nonempty closed convex subset of a real Hilbert space
. Recall that a mapping
is called strongly pseudocontractive if there exists a constant
such that
, for all
. A mapping
is a pseudocontraction if it satisfies the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ5_HTML.gif)
We denote by the set of fixed points of
; that is,
. Note that
is always closed and convex (but may be empty). However, for VI (1.2), we always assume
. It is not hard to find that
is a pseudocontraction if and only if
satisfies one of the following two equivalent properties:
(a) for all
, or
(b) is monotone on
:
for all
.
Below is the so-called demiclosedness principle for pseudocontractive mappings.
Lemma 2.1 (see [10]).
Let be a closed convex subset of a Hilbert space
. Let
be a Lipschitz pseudocontraction. Then,
is a closed convex subset of
, and the mapping
is demiclosed at 0; that is, whenever
is such that
and
, then
.
We also need the following lemma.
Lemma 2.2 (see [3]).
Let be a nonempty closed convex subset of a real Hilbert space
. Assume that the mapping
is monotone and weakly continuous along segments; that is,
weakly as
. Then, the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ6_HTML.gif)
is equivalent to the dual variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ7_HTML.gif)
3. Main Results
In this section, we introduce an implicit algorithm and prove this algorithm converges strongly to which solves the VI (1.2). Let
be a nonempty closed convex subset of a real Hilbert space
. Let
be a strongly pseudocontraction. Let
be two Lipschitz pseudocontractions. For
, we define the following mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ8_HTML.gif)
It easy to see that the mapping is strongly pseudocontractive; that is,
, for all
. So, by Deimling [11],
has a unique fixed point which is denoted
; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ9_HTML.gif)
Below is our main result of this paper which displays the behavior of the net as
and
successively.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a strongly pseudocontraction. Let
be two Lipschitz pseudocontractions with
. Suppose that the solution set
of VI (1.2) is nonempty. Let, for each
,
be defined implicitly by (3.2). Then, for each fixed
, the net
converges in norm, as
, to a point
. Moreover, as
, the net
converges in norm to the unique solution
of the following VI:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ10_HTML.gif)
Hence, for each null sequence in
, there exists another null sequence
in
, such that the sequence
in norm as
.
We divide our details proofs into several lemmas as follows. Throughout, we assume all conditions of Theorem 3.1 are satisfied.
Lemma 3.2.
For each fixed , the net
is bounded.
Proof.
Take any to derive that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ11_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ12_HTML.gif)
It follows that for each fixed ,
is bounded, so are the nets
,
, and
.
We will use to denote possible constant appearing in the following.
Lemma 3.3.
as
.
Proof.
From (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ13_HTML.gif)
Next, we show that, for each fixed , the net
is relatively norm compact as
. It follows from (3.2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ14_HTML.gif)
It turns out that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ15_HTML.gif)
Assume that is such that
as
. By (3.8), we obtain immediately that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ16_HTML.gif)
Since is bounded, without loss of generality, we may assume that as
,
converges weakly to a point
. From (3.6), we get
. So, Lemma 2.1 implies that
. We can then substitute
for
in (3.9) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ17_HTML.gif)
Consequently, the weak convergence of to
actually implies that
strongly. This has proved the relative norm compactness of the net
as
.
Now, we return to (3.9) and take the limit as to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ18_HTML.gif)
In particular, solves the following variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ19_HTML.gif)
or the equivalent dual variational inequality (see Lemma 2.2)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ20_HTML.gif)
Next, we show that as , the entire net
converges in norm to
. We assume
where
. Similarly, by the above proof, we deduce
which solves the following variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ21_HTML.gif)
In (3.13), we take to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ22_HTML.gif)
In (3.14), we take to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ23_HTML.gif)
Adding up (3.15) and (3.16) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ24_HTML.gif)
At the same time, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ25_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ26_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ27_HTML.gif)
Hence, we conclude that the entire net converges in norm to
as
.
Lemma 3.4.
The net is bounded.
Proof.
In (3.13), we take any to deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ28_HTML.gif)
By virtue of the monotonicity of and the fact that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ29_HTML.gif)
It follows from (3.21) and (3.22) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ30_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ31_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ32_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ33_HTML.gif)
Lemma 3.5.
The net which solves the variational inequality (3.3).
Proof.
First, we note that the solution of the variational inequality VI (3.3) is unique.
We next prove that ; namely, if
is a null sequence in
such that
weakly as
, then
. To see this, we use (3.13) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ34_HTML.gif)
However, since is monotone,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ35_HTML.gif)
Combining the last two relations yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ36_HTML.gif)
Letting as
in (3.29), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ37_HTML.gif)
which is equivalent to its dual variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ38_HTML.gif)
Namely, is a solution of VI (1.2); hence,
. We further prove that
, the unique solution of VI (3.3). As a matter of fact, we have by (3.25),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ39_HTML.gif)
Therefore, the weak convergence to of
right implies that that
in norm. Now, we can let
in (3.23) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F180534/MediaObjects/13663_2010_Article_1383_Equ40_HTML.gif)
It turns out that solves VI (3.3). By uniqueness, we have
. This is sufficient to guarantee that
in norm, as
. The proof is complete.
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Yao, Y., Marino, G. & Liou, YC. A Hybrid Method for Monotone Variational Inequalities Involving Pseudocontractions. Fixed Point Theory Appl 2011, 180534 (2011). https://doi.org/10.1155/2011/180534
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DOI: https://doi.org/10.1155/2011/180534