Abstract
We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.
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1. Introduction
Let be a nonempty closed and convex subset of a real Hilbert space
with inner product
and norm
, and let
be a nonlinear mapping. The variational inequality problem is formulated as finding a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ1_HTML.gif)
Variational inequalities were initially studied by Kinderlehrer and Stampacchia in [1] and ever since have been widely investigated, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [1–3]).
It is well known that if is an
-Lipschitz continuous and
-strongly monotone, that is,
satisfies the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ2_HTML.gif)
where and
are fixed positive numbers, then (1.1) has a unique solution. It is also known that (1.1) is equivalent to the fixed-point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ3_HTML.gif)
where denotes the metric projection from
onto
and
is an arbitrarily fixed positive constant.
Let be a finite family of nonexpansive self-mappings of
. For finding an element
, Xu and Ori introduced in [4] the following implicit iteration process. For
and
, the sequence
is generated as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ4_HTML.gif)
The compact expression of the method is the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ5_HTML.gif)
where , for integer
, with the mod function taking values in the set
. They proved the following result.
Theorem 1.1.
Let be a real Hilbert space and
a nonempty closed convex subset of
. Let
be
nonexpansive self-maps of
such that
, where
. Let
and
be a sequence in
such that
. Then, the sequence
defined implicitly by (1.5) converges weakly to a common fixed point of the mappings
.
Further, Zeng and Yao introduced in [5] the following implicit method. For an arbitrary initial point , the sequence
is generated as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ6_HTML.gif)
The scheme is written in a compact form as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ7_HTML.gif)
They proved the following result.
Theorem 1.2.
Let be a real Hilbert space and
a mapping such that for some constants
,
is
-Lipschitz continuous and
-strongly monotone. Let
be
nonexpansive self-maps of
such that
. Let
, and let
and
satisfying the conditions:
and
, for some
. Then, the sequence
defined by (1.7) converges weakly to a common fixed point of the mappings
. The convergence is strong if and only if
.
Recently, Ceng et al. [6] extended the above result to a finite family of asymptotically self-maps.
Clearly, from we have that
as
. To obtain strong convergence without the condition
, in this paper we propose the following implicit algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ8_HTML.gif)
where are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ9_HTML.gif)
denotes the identity mapping of
, and the parameters
for all
satisfy the following conditions:
as
and
.
2. Main Result
We formulate the following facts for the proof of our results.
Lemma 2.1 (see [7]).
(i) and for any fixed
, (ii)
, for all
.
Put ; for any nonexpansive mapping
of
, we have the following lemma.
Lemma 2.2 (see [8]).
and for a fixed number
, where
.
Lemma 2.3 (Demiclosedness Principle [9]).
Assume that is a nonexpansive self-mapping of a closed convex subset
of a Hibert space
. If
has a fixed point, then
is demiclosed; that is, whenever
is a sequence in
weakly converging to some
and the sequence
strongly converges to some
, it follows that
.
Now, we are in a position to prove the following result.
Theorem 2.4.
Let be a real Hilbert space and
a mapping such that for some constants
,
is
-Lipschitz continuous and
-strongly monotone. Let
be
nonexpansive self-maps of
such that
. Let
and let
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ10_HTML.gif)
Then, the net defined by (1.8)-(1.9) converges strongly to the unique element
in (1.1).
Proof..
By using Lemma 2.2 with , that is,
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ11_HTML.gif)
So, is a contraction in
. By Banach's Contraction Principle, there exists a unique element
such that
for all
.
Next, we show that is bounded. Indeed, for a fixed point
, we have that
for
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ12_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ13_HTML.gif)
that implies the boundedness of . So, are the nets
.
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ14_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ15_HTML.gif)
Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ16_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ17_HTML.gif)
Further, for the sake of simplicity, we put and prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ18_HTML.gif)
as for
.
Let be an arbitrary sequence converging to zero as
and
. We have to prove that
, where
are defined by (2.5) with
and
. Let
be a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ19_HTML.gif)
Let be a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ20_HTML.gif)
From (2.6) and Lemma 2.1, it implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ21_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ22_HTML.gif)
By Lemma 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ23_HTML.gif)
Without loss of generality, we can assume that for some
. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ24_HTML.gif)
This together with (2.13) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ25_HTML.gif)
It means that as
for
.
Next, we show that as
. In fact, in the case that
we have
. So,
as
. Further, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ26_HTML.gif)
and , we have that
. Therefore, from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ27_HTML.gif)
it follows that as
. On the other hand, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ28_HTML.gif)
we obtain that as
. Now, from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ29_HTML.gif)
and , it follows that
. Similarly, we obtain that
, for
and
as
.
Let be any sequence of
converging weakly to
as
. Then,
, for
and
also converges weakly to
. By Lemma 2.3, we have
and from (2.8), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ30_HTML.gif)
Since , by replacing
by
in the last inequality, dividing by
and taking
in the just obtained inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ31_HTML.gif)
The uniqueness of in (1.1) guarantees that
. Again, replacing
in (2.8) by
, we obtain the strong convergence for
. This completes the proof.
3. Application
Recall that a mapping is called a
-strictly pseudocontractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ32_HTML.gif)
It is well known [10] that a mapping by
with a fixed
for all
is a nonexpansive mapping and
. Using this fact, we can extend our result to the case
, where
is
-strictly pseudocontractive as follows.
Let be fixed numbers. Then,
with
, a nonexpansive mapping, for
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ33_HTML.gif)
So, we have the following result.
Theorem 3.1 ..
Let be a real Hilbert space and
a mapping such that for some constants
,
is
-Lipschitz continuous and
-strongly monotone. Let
be
-strictly pseudocontractive self-maps of
such that
. Let
and let
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ34_HTML.gif)
Then, the net defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ35_HTML.gif)
where , for
, are defined by (3.2) and
, converges strongly to the unique element
in (1.1).
It is known in [11] that where
with
and
for
-strictly pseudocontractions
. Moreover,
is
-strictly pseudocontractive with
. So, we also have the following result.
Theorem 3.2 ..
Let be a real Hilbert space and
a mapping such that for some constants
,
is
-Lipschitz continuous and
-strongly monotone. Let
be
-strictly pseudocontractive self-maps of
such that
. Let
, where
,
, and let
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ36_HTML.gif)
Then, the net , defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F276859/MediaObjects/13663_2010_Article_1393_Equ37_HTML.gif)
where ,
, and
, converges strongly to the unique element
in (1.1).
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Acknowledgment
This work was supported by the Vietnamese National Foundation of Science and Technology Development.
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Buong, N., Quynh Anh, N. An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2011, 276859 (2011). https://doi.org/10.1155/2011/276859
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DOI: https://doi.org/10.1155/2011/276859