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Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problems
Fixed Point Theory and Applications volume 2010, Article number: 264628 (2010)
Abstract
We introduce an iterative scheme by the viscosity iterative method for finding a common element of the solution set of an equilibrium problem, the solution set of the variational inequality, and the fixed points set of infinitely many nonexpansive mappings in a Hilbert space. Then we prove our main result under some suitable conditions.
1. Introduction
Let be a real Hilbert space with the inner product and the norm being denoted by
and
, respectively. Let
be a nonempty, closed, and convex subset of
and let
be a bifunction of
into
, where
denotes the real numbers. The equilibrium problem for
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ1_HTML.gif)
The solution set of (1.1) is denoted by .
Let be a mapping. The classical variational inequality, denoted by
, is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ2_HTML.gif)
The variational inequality has been extensively studied in the literature (see, e.g., [1–3]). The mapping is called
-inverse-strongly monotone if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ3_HTML.gif)
where is a positive real number.
A mapping is called strictly pseudocontractive if there exists
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ4_HTML.gif)
It is easy to know that is (
)-inverse-strongly-monotone. If
, then
is nonexpansive. We denote by
the fixed points set of
.
In 2003, for , Takahashi and Toyoda [4] introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ5_HTML.gif)
where is a sequence in
,
is an
-inverse-strongly monotone mapping,
is a sequence in
, and
is the metric projection. They proved that if
, then
converges weakly to some
Recently, S. Takahashi and W. Takahashi [5] introduced an iterative scheme for finding a common element of the solution set of (1.1) and the fixed points set of a nonexpansive mapping in a Hilbert space. If is bifunction which satisfies the following conditions:
() for all
() is monotone, that is,
for all
() for each
() for each is convex and lower semicontinuous,
then they proved the following strong convergence theorem.
Theorem A (see [5]).
Let be a closed and convex subset of a real Hilbert space
. Let
be a bifunction which satisfies conditions
.
Let be a nonexpansive mapping such that
and let
be a contraction; that is, there is a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ6_HTML.gif)
and let and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ7_HTML.gif)
where and
satisfy
, and
Then, and
converge strongly to
where
Let be a sequence of nonexpansive mappings of
into itself and
a sequence of nonnegative numbers in
. For each
, define a mapping
of
into itself as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ8_HTML.gif)
Such a mapping is called the
-mapping generated by
and
(see [6]).
In this paper, we introduced a new iterative scheme generated by and find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ9_HTML.gif)
where and
are sequences in
and
are sequences in
,
is a fixed contractive mapping with contractive coefficient
,
is an
-inverse-strongly monotone mapping of
to
,
is a bifunction which satisfies conditions
, and
is generated by (1.8). Then we proved that the sequences
and
converge strongly to
, where
.
2. Preliminaries
Let be a real Hilbert space and let
be a closed and convex subset of
is the metric projection from
onto
, that is, for any
,
for all
It is easy to see that
is nonexpansive and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ10_HTML.gif)
If is an
-inverse-strongly monotone mapping of
to
, then it is obvious that
is (
)-Lipschitz continuous. We also have that for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ11_HTML.gif)
So, if , then
is nonexpansive.
Lemma 2.1 (see [7]).
Let and
be bounded sequences in a Banach space
, and let
be a sequence in
with
. Suppose
for all
and
. Then,
Lemma 2.2 (see [8]).
Assume that is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ12_HTML.gif)
where is a sequence in
and
is a sequence in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ13_HTML.gif)
Then
Lemma 2.3 (see [9]).
Let be a nonempty, closed, and convex subset of
and
a bifunction of
into
that satisfies conditions
. Let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ14_HTML.gif)
Lemma 2.4 (see [9]).
Assume that satisfies conditions
. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ15_HTML.gif)
Then, the following holds:
(i) is single-valued;
(ii) is firmly nonexpansive, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ16_HTML.gif)
(iii)
(iv) is closed and convex.
Lemma 2.5 (Opial's theorem [10]).
Each Hilbert space satisfies Opial's condition; that is, for any sequence
with
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ17_HTML.gif)
holds for each with
Let be a sequence of nonexpansive self-mappings on
, where
is a nonempty, closed and convex subset of a real Hilbert space
. Given a sequence
in
, one defines a sequence
of self-mappings on
generated by (1.8). Then one has the following results.
Lemma 2.6 (see [6]).
Let be a nonempty, closed, and convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive self-mappings on
such that
and
is a sequence in
for some
. Then, for every
and
the limit
exists.
Remark 2.7.
It can be shown from Lemma 2.6 that if is a nonempty and bounded subset of
, then for
there exists
such that
for all
.
Remark 2.8.
Using Lemma 2.6, we can define a mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ18_HTML.gif)
for all Such a
is called the
-mapping generated by
and
Since
is nonexpansive,
is also nonexpansive. Indeed, observe that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ19_HTML.gif)
Let be a bounded sequence in
and
. Then, it is clear from Remark 2.7 that for
there exists
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ20_HTML.gif)
This implies that
Lemma 2.9 (see [6]).
Let be a nonempty, closed, and convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive self-mappings on
such that
and
is a sequence in
for some
. Then,
.
3. Strong Convergence Theorem
Theorem 3.1.
Let be a Hilbert space. Let
be a nonempty, closed, and convex subset of
. Let
be a bifunction which satisfies conditions
,
an
-inverse-strongly monotone mapping of
to
,
a contraction of
into itself, and
a sequence of nonexpansive self-mappings on
such that
. Suppose that
, and
are sequences in
, and
and
are sequences in
which satisfies the following conditions:
(i)
(ii)
(iii)
(iv)
(v).
Then and
generated by (1.9) converge strongly to
, where
.
Proof.
Let . It follows from Lemma 2.4 and (1.9) that
, and hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ21_HTML.gif)
for all . Let
. Since
is nonexpansive and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ23_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ24_HTML.gif)
Hence is bounded. So
, and
are also bounded.
Next, we claim that Indeed, assume that
where
,
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ26_HTML.gif)
Using (1.8) and the nonexpansivity of , we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ27_HTML.gif)
for some constant On the other hand, from
and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ29_HTML.gif)
Setting in (3.8) and
in (3.9), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ30_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ31_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ32_HTML.gif)
Without loss of generality, we may assume that there exists a real number such that
for all
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ33_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ34_HTML.gif)
where . It follows from (3.5), (3.6), (3.7), and (3.14) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ35_HTML.gif)
Therefore, .
Since and
, hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ36_HTML.gif)
Lemma 2.1 yields that . Consequently,
For , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ37_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ38_HTML.gif)
This together with (3.2) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ39_HTML.gif)
and hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ40_HTML.gif)
So (note that
and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ41_HTML.gif)
we obtain and hence
. Thus,
Let Then
is a contraction of
into itself. In fact, there exists
such that
for all
. So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ42_HTML.gif)
for all So
is a contraction by Banach contraction principle [11]. Since
is a complete space, there exists a unique element
such that
.
Next we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ43_HTML.gif)
where To show this inequality, we choose a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ44_HTML.gif)
Since is bounded, there exists a subsequence of
which converges weakly to some
, that is,
. From
, we obtain that
. Now we will show that
. First, we will show
. From
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ45_HTML.gif)
By , we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ46_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ47_HTML.gif)
Since and
, it follows from
that
for all
. For any
and
, let
Since
and
, then we have
and hence
. This together with
and
yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ48_HTML.gif)
and thus . From
, we have
for all
and hence
. Now, we show that
. Indeed, we assume that
; from Opial's condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ49_HTML.gif)
This is a contradiction. Thus, we obtain that . Finally, by the same argument as in the proof of [3, Theorem 3.1], we can show that
. Hence
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ50_HTML.gif)
Now we show that
From (1.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ51_HTML.gif)
and hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ52_HTML.gif)
Using (3.23) and Lemma 2.2, we conclude that converges strongly to
Consequently,
converges strongly to
This completes the proof.
Using Theorem 3.1, we prove the following theorem.
Theorem 3.2.
Let , and
be given as in Theorem 3.1 and let
be an
-strictly pseudocontractive mapping such that
. Suppose that
and
Let
and
be the sequences and find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F264628/MediaObjects/13663_2009_Article_1239_Equ53_HTML.gif)
where , and
are given as in Theorem 3.1. Then
and
converge strongly to
, where
.
Proof.
Put . Then
is (
)-inverse-strongly-monotone. We have
and put
. So by Theorem 3.1 we obtain the desired result.
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Acknowledgments
The author would like to express his thanks to Professor Simeon Reich, Technion-Israel Institute of Technology, Israel, and the anonymous referees for their valuable comments and suggestions on a previous draft, which resulted in the present version of the paper. This work was supported by the Natural Science Foundation of China (10871217) and Grant KJ080725 of the Chongqing Municipal Education Commission.
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Chen, YA., Zhang, YP. Hybrid Viscosity Iterative Method for Fixed Point, Variational Inequality and Equilibrium Problems. Fixed Point Theory Appl 2010, 264628 (2010). https://doi.org/10.1155/2010/264628
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DOI: https://doi.org/10.1155/2010/264628