- Research Article
- Open access
- Published:
Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping
Fixed Point Theory and Applications volume 2011, Article number: 274820 (2011)
Abstract
The purpose of this paper is to prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method.
1. Introduction
Let be a closed convex subset of a real Hilbert space
, and let
be a bifunction. Recall that the equilibrium problem for a bifunction
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by . Given a mapping
, let
for all
. Then,
if and only if
for all
; that is,
is a solution of the variational inequality. Let
be a nonlinear mapping. The variational inequality problem is to find a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ2_HTML.gif)
for all . The set of solutions of the variational inequality is denoted by
. Now, we consider the following generalized equilibrium problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ3_HTML.gif)
The set of is denoted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ4_HTML.gif)
In the case of ,
is denoted by
. In the case of
,
is also denoted by
. Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics are reduced to find a solution of (1.3); see, for instance, [1–3].
A mapping of
into
is called inverse strongly monotone mapping, see [4], if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ5_HTML.gif)
for all . The following definition is well known.
Definition 1.1.
A mapping is said to be a
-strict pseudocontraction if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ6_HTML.gif)
A mapping is called nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ7_HTML.gif)
for all .
We know that -strict pseudocontraction includes a class of nonexpansive mappings. If
,
is said to be a pseudocontractivemapping.
is strong pseudocontraction if there exists a positive constant
such that
is pseudocontraction. In a real Hilbert space
, (1.6) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ8_HTML.gif)
is pseudocontraction if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ9_HTML.gif)
Then, is strong pseudocontraction if there exists positive constant
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ10_HTML.gif)
The class of -strict pseudocontractions falls into the one between classes of nonexpansive mappings, and the pseudocontraction mappings, and the class of strong pseudocontraction mappings is independent of the class of
-strict pseudocontraction.
We denote by the set of fixed points of
. If
is bounded, closed, and convex, and
is a nonexpansive mapping of
into itself, then
is nonempty; for instance, see [5]. Browder and Petryshyn [6] show that if a
-strict pseudocontraction
has a fixed point in
, then starting with an initial
, the sequence
generated by the recursive formula:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ11_HTML.gif)
where is a constant such that
, converges weakly to a fixed point of
. Marino and Xu [7] have extended Browder and Petryshyns above-mentioned result by proving that the sequence
generated by the following Manns algorithm [8]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ12_HTML.gif)
converges weakly to a fixed point of provided the control sequence
satisfies the conditions that
for all
and
. In 1974, S. Ishikawa proved the following strong convergence theorem of pseudocontractive mapping.
Theorem 1.2 (see [9]).
Let be a convex compact subset of a Hilbert space
, and let
be a Lipschitzian pseudocontractive mapping. For any
, suppose that the sequence
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ13_HTML.gif)
where ,
are two real sequences in
satisfying
(i),
(ii),
(iii).
Then converges strongly to a fixed point of
.
In order to prove a strong convergence theorem of Mann algorithm (1.12) associated with strictly pseudocontractive mapping, in 2006, Marino and Xu [7] proved the following theorem for strict pseudocontractive mapping in Hilbert space by using method.
Theorem 1.3 (see [7]).
Let be a closed convex subset of a Hilbert space
. Let
be a
-strict pseudocontraction for some
, and assume that the fixed point set
of
is nonempty. Let
be the sequence generated by the following
algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ14_HTML.gif)
Assume that the control sequence is chosen so that
for all
. Then
converges strongly to
. Very recently, in 2010, [10] established the hybrid algorithm for Lipschitz pseudocontractive mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ15_HTML.gif)
Under suitable conditions of and
, they proved that the sequence
defined by (1.15) converges strongly to
.
Many authors study the problem for finding a common element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2, 3, 11–15]. The motivation of (1.14), (1.15), and the research in this direction, we prove the strong convergence theorem for finding solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method.
2. Preliminaries
In order to prove our main results, we need the following lemmas. Let be closed convex subset of a real Hilbert space
, and let
be the metric projection of
onto
; that is, for
,
satisfies the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ16_HTML.gif)
The following characterizes the projection .
Lemma 2.1 (see [5]).
Given that and
, then
if and only if the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ17_HTML.gif)
The following lemma is well known.
Lemma 2.2.
Let be Hilbert space, and let
be a nonempty closed convex subset of
. Let
be
-strictly pseudocontractive, then the fixed point set
of
is closed and convex so that the projection
is well defined.
Lemma 2.3 ((demiclosedness principle) (see [16]).
If is a
-strict pseudocontraction on closed convex subset
of a real Hilbert space
, then
is demiclosed at any point
.
To solve the equilibrium problem for a bifunction , assume that
satisfies the following conditions:
() for all
,
() is monotone, that is,
,
()for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ18_HTML.gif)
()for all is convex and lower semicontinuous.
The following lemma appears implicitly in [1].
Lemma 2.4 (see [1]).
Let be a nonempty closed convex subset of
, and let
be a bifunction of
into
satisfying
. Let
, and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ19_HTML.gif)
for all .
Lemma 2.5 (see [11]).
Assume that satisfies
. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ20_HTML.gif)
for all . Then, the following hold:
(1) is single-valued;
(2) is firmly nonexpansive, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ21_HTML.gif)
(3);
(4) is closed and convex.
Lemma 2.6 (see [17]).
Let be a closed convex subset of
. Let
be a sequence in
and
. Let
; if
is such that
and satisfy the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ22_HTML.gif)
then , as
.
Lemma 2.7 (see [7]).
For a real Hilbert space H, the following identities hold: if is a sequence in H weak convergence to
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ23_HTML.gif)
for all .
3. Main Result
Theorem 3.1.
Let be a nonempty closed convex subset of a Hilbert space
. Let
and
be bifunctions from
into
satisfying
, respectively. Let
be an α-inverse strongly monotone mapping, and let
be a
inverse strongly monotone mapping. Let
be a
strict pseudocontraction mapping with
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ24_HTML.gif)
where is sequence in
,
, and
satisfy the following conditions:
(i),
(ii).
Then converges strongly to
.
Proof.
First, we show that is nonexpansive. Let
. Since
is
inverse strongly monotone mapping and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ25_HTML.gif)
Thus is nonexpansive, so are
,
, and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ26_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ27_HTML.gif)
By Lemma 2.5, we have . By the same argument as above, we conclude that
.
Let . Then
and
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ28_HTML.gif)
Again by Lemma 2.5, we have . By nonexpansiveness of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ29_HTML.gif)
By (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ30_HTML.gif)
Next, we show that is closed and convex for every
. It is obvious that
is closed. In fact, we know that, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ31_HTML.gif)
So, we have that for all and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ32_HTML.gif)
Then, we have that is convex. By Lemmas 2.5 and 2.2, we conclude that
is closed and convex. This implies that
is well defined. Next, we show that
for every
.
Taking , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ33_HTML.gif)
It follows that . Then, we have
,
. Since
, for every
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ34_HTML.gif)
In particular, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ35_HTML.gif)
By (3.11), we have that is bounded, so are
. Since
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ36_HTML.gif)
It is implied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ37_HTML.gif)
Hence, we have that exists. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ38_HTML.gif)
it is implied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ39_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ40_HTML.gif)
And by (3.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ41_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ42_HTML.gif)
by (3.16) and (3.18), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ43_HTML.gif)
Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ44_HTML.gif)
Let , by (3.10) and (3.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ45_HTML.gif)
Since ,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ46_HTML.gif)
Since ,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ47_HTML.gif)
Substituting (3.23) and (3.24) into (3.22),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ48_HTML.gif)
It is implied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ49_HTML.gif)
By (3.20) and condition , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ50_HTML.gif)
By using the same method as (3.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ51_HTML.gif)
By Lemma 2.5 and firm nonexpansiveness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ52_HTML.gif)
By (3.29), it is implied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ53_HTML.gif)
Again, by Lemma 2.5 and firm nonexpansiveness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ54_HTML.gif)
By (3.31), it is implied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ55_HTML.gif)
Substituting (3.30) and (3.32) into (3.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ56_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ57_HTML.gif)
and by (3.27), (3.28), (3.20), and conditions ,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ58_HTML.gif)
By using the same method as (3.35), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ59_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ60_HTML.gif)
from (3.35), (3.36), and condition , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ61_HTML.gif)
By (3.20) and (3.38), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ62_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ63_HTML.gif)
from (3.39) and condition , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ64_HTML.gif)
Let be the set of all weaks
-limit of
. We will show that
. Since
is bounded, then
. Letting
, there exists a subsequence
of
converging to
. By (3.35), we have
as
. Since
, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ65_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ66_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ67_HTML.gif)
Put for all
and
. Then, we have
. So, from (3.44), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ68_HTML.gif)
Since , we have
. Further, from monotonicity of
, we have
. So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ69_HTML.gif)
From ,
, and (3.46), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ70_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ71_HTML.gif)
Letting , we have, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ72_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ73_HTML.gif)
From (3.36), we have . Since
, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ74_HTML.gif)
By using the same method as (3.50), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ75_HTML.gif)
Since and (3.38), we have
. By Lemma 2.3,
is demiclosed at zero, and by (3.41), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ76_HTML.gif)
From (3.50), (3.52), and (3.53), we have . Hence
. Therefore, by (3.12) and Lemma 2.6, we have that
converges strongly to
. The proof is completed.
4. Applications
By using our main result, we have the following results in Hilbert spaces.
Theorem 4.1.
Let be a nonempty closed convex subset of a Hilbert space
. Let
and
be bifunctions from
into
satisfying
, respectively. Let
be a
-strict pseudocontraction mapping with
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ77_HTML.gif)
where is sequence in
,
, and
satisfy the following conditions:
(i),
(ii).
Then converges strongly to
.
Proof.
Putting in Theorem 3.1, we have the desired conclusions.
Theorem 4.2.
Let be a nonempty closed convex subset of a Hilbert space
. Let
be bifunctions from
into
satisfying
, respectively. Let
be an α-inverse strongly monotone mapping, and let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F274820/MediaObjects/13663_2010_Article_1392_Equ78_HTML.gif)
where is sequence in
,
, and
, for all
. Then
converges strongly to
.
Proof.
Putting ,
, and
, for all
, in Theorem 3.1, we have the desired conclusions.
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Kangtunyakarn, A. Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping. Fixed Point Theory Appl 2011, 274820 (2011). https://doi.org/10.1155/2011/274820
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DOI: https://doi.org/10.1155/2011/274820