- Research Article
- Open access
- Published:
A Method for a Solution of Equilibrium Problem and Fixed Point Problem of a Nonexpansive Semigroup in Hilbert's Spaces
Fixed Point Theory and Applications volume 2011, Article number: 208434 (2011)
Abstract
We introduce a viscosity approximation method for finding a common element of the set of solutions for an equilibrium problem involving a bifunction defined on a closed, convex subset and the set of fixed points for a nonexpansive semigroup on another one in Hilbert's spaces.
1. Introduction
Let be a nonempty, closed, and convex subset of a real Hilbert space
. Denote the metric projection from
onto
by
. Let
be a nonexpansive mapping on
, that is,
and
, for all
. We use
to denote the set of fixed points of
, that is,
.
Let be a nonexpansive semigroup on a closed convex subset
, that is,
(1)for each ,
is a nonexpansive mapping on
,
(2) for all
,
(3) for all
,
(4)for each , the mapping
from
into
is continuous.
Denote by . We know [1, 2] that
is a closed, convex subset in
and
if
is bounded.
The equilibrium problem is for a bifunction defined on
to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ1_HTML.gif)
Assume that the bifunction satisfies the following set of standard properties:
(A1), for all
,
(A2) for all
,
(A3)for every ,
is weakly lower semicontinuous and convex,
(A4), for all
.
Denote the set of solutions of (1.1) by . We also know [3] that
is a closed convex subset in
.
The problem studied in this paper is formulated as follows. Let and
be closed convex subsets in
. Let
be a bifunction satisfying conditions (A1)–(A4) with
replaced by
and let
be a nonexpansive semigroup on
. Find an element
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ2_HTML.gif)
where and
denote the set of solutions of an equilibrium problem involving by a bifunction
on
and the fixed point set of a nonexpansive semigroup
on a closed convex subset
, respectively.
In the case that ,
,
, and
, a nonexpansive mapping on
, for all
, (1.2) is the fixed point problem of a nonexpansive mapping. In 2000, Moudafi [4] proved the following strong convergence theorem.
Theorem 1.1.
Let be a nonempty, closed, convex subset of a Hilbert space
and let
be a nonexpansive mapping on
such that
. Let
be a contraction on
and let
be a sequence generated by:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ3_HTML.gif)
where satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ4_HTML.gif)
Then, converges strongly to
, where
.
Such a method for approximation of fixed points is called the viscosity approximation method. It has been developed by Chen and Song [5] to find , the set of fixed points for a semigroup
on
. They proposed the following algorithm:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ5_HTML.gif)
where , is a contraction,
and
are sequences of positive real numbers satisfying the conditions:
,
, and
as
.
Recently, Yao and Noor [6] proposed a new viscosity approximation method
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ6_HTML.gif)
where ,
, and
are in
,
, for finding
, when
satisfies the uniformly asymptotically regularity condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ7_HTML.gif)
uniformly in , and
is any bounded subset of
. Further, Plubtieng and Pupaeng in [7] studied the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ8_HTML.gif)
where and
are in
satisfying the following conditions:
,
,
, and
is a positive divergent real sequence.
There were some methods proposed to solve equilibrium problem (1.1); see for instance [8–12]. In particular, Combettes and Histoaga [3] proposed several methods for solving the equilibrium problem.
In 2007, S. Takahashi and W. Takahashi [13] combinated the Moudafi's method with the Combettes and Histoaga's result in [3] to find an element . They proved the following strong convergence theorem.
Theorem 1.2.
Let be a nonempty, closed, convex subset of a Hilbert space
, let
be a nonexpansive mapping on
and let
be a bifunction from
to
satisfying (A1)–(A4) such that
. Let
be a contraction on
and let
and
be sequences generated by:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ9_HTML.gif)
where and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ10_HTML.gif)
Then, and
converge strongly to
, where
.
Very recently, Ceng and Wong in [14] combined algorithm (1.6) with the result in [3] to propose the following procudure:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ11_HTML.gif)
for finding an element in the case that
under the uniformly asymptotic regularity condition on the nonexpansive semigroup
on
.
In this paper, motivated by the above results, to solve (1.2), we introduce the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ12_HTML.gif)
where is a contraction on
, that is,
and
, for all
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ13_HTML.gif)
for all ,
,
, and
be the sequences in (0,1), and
,
are the sequences in
satisfy the following conditions:
(i),
(ii),
,
(iii),
(iv) with bounded
,
(v) and
.
The strong convergence of (1.12)-(1.13) and its corollaries are showed in the next section.
2. Main Results
We formulate the following facts needed in the proof of our results.
Lemma 2.1.
Let be a real Hilbert space
. There holds the following identity:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ14_HTML.gif)
Lemma 2.2 (see [15]).
Let be a nonempty, closed, convex subset of a real Hilbert space
. For any
, there exists a unique
such that
, for all
, and
if and only if
for all
.
Lemma 2.3 (see [16]).
Let be a sequence of nonnegative real numbers satisfying the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ15_HTML.gif)
where and
are sequences of real numbers such that
,
, and
. Then,
.
Lemma 2.4 (see [9]).
Let be a nonempty, closed, convex subset of
and
be a bifunction of
into
satisfying the conditions (A1)–(A4). Let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ16_HTML.gif)
Lemma 2.5 (see [9]).
Assume that satisfies the conditions (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ17_HTML.gif)
Then, the following statements hold:
(i) is single-valued,
(ii) is firmly nonexpansive, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ18_HTML.gif)
(iii),
(iv) is closed and convex.
Lemma 2.6 (see [17]).
Let be a nonempty bounded closed convex subset in a real Hilbert space
and let
be a nonexpansive semigroup on
. Then, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ19_HTML.gif)
Lemma 2.7 (Demiclosedness Principle [18]).
If is a closed convex subset of
,
is a nonexpansive mapping on
,
is a sequence in
such that
and
, then
.
Lemma 2.8 (see [19]).
Let and
be bounded sequences in a Banach space
and
be a sequence in
with
. Suppose
for all
and
. Then,
.
Now, we are in a position to prove the following result.
Theorem 2.9.
Let and
be two nonempty, closed, convex subsets in a real Hilbert space
. Let
be a bifunction from
to
satisfying conditions (A1)–(A4) with
replaced by
, let
be a nonexpansive semigroup on
such that
and let
be a contraction of
into itself. Then,
and
generated by (1.12)-(1.13) converge strongly to
, where
.
Proof.
Let . Then,
is a contraction of
into itself. In fact, from
for all
and the nonexpansive property of
for a closed convex subset
in
, it implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ20_HTML.gif)
Hence, is a contraction of
into itself. Since
is complete, there exists a unique element
such that
. Such a
is an element of
, because
.
By Lemma 2.4, and
are well defined. For each
, by putting
and using (ii) and (iii) in Lemma 2.5, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ21_HTML.gif)
Put . Clearly,
. Suppose that
. Then, we have, from the nonexpansive property of
, condition (i) and (2.8), that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ22_HTML.gif)
So, for all
and hence
is bounded. Therefore,
,
, and
are also bounded.
Next, we show that as
. For this purpose, we define a sequence
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ23_HTML.gif)
Then, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ24_HTML.gif)
and, hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ25_HTML.gif)
Now, we estimate the value by using
and
. We have from (2.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ27_HTML.gif)
Putting in (2.13) and
in (2.14), adding the one to the other obtained result and using (A2), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ28_HTML.gif)
and, hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ29_HTML.gif)
Without loss of generality, let us assume that there exists a real number such that
for all
. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ30_HTML.gif)
and, hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ31_HTML.gif)
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ32_HTML.gif)
So, we get from (2.10), (2.12), (2.18), (2.19), and the nonexpansive property of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ33_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ34_HTML.gif)
and by Lemma 2.8, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ35_HTML.gif)
Consequently, it follows from (2.10) and condition (iii) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ36_HTML.gif)
By (2.18), (2.23), and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ37_HTML.gif)
we also obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ38_HTML.gif)
We have, for every , from (iii) in Lemma 2.5, that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ39_HTML.gif)
and, hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ40_HTML.gif)
Therefore, from the convexity of and condition (i), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ41_HTML.gif)
and, hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ42_HTML.gif)
Without loss of generality, we assume that for all
. Then, for sufficiently large
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ43_HTML.gif)
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ44_HTML.gif)
Further, since , by condition (i), (2.19) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ45_HTML.gif)
we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ46_HTML.gif)
Then, from (2.25), (2.33) and the conditions on and
, it implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ47_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ48_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ49_HTML.gif)
we obtain from (2.31) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ50_HTML.gif)
Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ51_HTML.gif)
We choose a subsequence of the sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ52_HTML.gif)
As is bounded, there exists a subsequence
of the sequence
which converges weakly to
. From (2.37), we also have that
converges weakly to
. Since
and
and
,
are two closed convex subsets in
, we have that
.
First, we prove that . From (2.4) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ53_HTML.gif)
and, hence, by using condition (A2), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ54_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ55_HTML.gif)
This together with condition (A3) and (2.31) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ56_HTML.gif)
So, for all
. It means that
.
Next we show that . Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ57_HTML.gif)
and, hence, from (2.31) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ58_HTML.gif)
Thus, (2.37) together with (2.45) imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ59_HTML.gif)
Therefore, also converges weakly to
, as
.
On the other hand, for each , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ60_HTML.gif)
Let . Since
, we have from (2.33) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ61_HTML.gif)
So, is a nonempty bounded closed convex subset. It is easy to verify that
is a nonexpansive semigroup on
. By Lemma 2.6, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ62_HTML.gif)
for every fixed , and hence, by (2.45)–(2.47), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ63_HTML.gif)
for each . By Lemma 2.7,
for all
, because
for any mapping
. It means that
. Therefore,
. Since
, we have from Lemma 2.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ64_HTML.gif)
So, (2.38) is proved. Further, since , by using Lemma 2.1, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ65_HTML.gif)
This with (2.8) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ66_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ67_HTML.gif)
Using Lemma 2.3, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ68_HTML.gif)
From (2.33) it follows that as
. This completes the proof.
Remarks 2.
-
(a)
Note that the following parameters
,
,
,
for any fixed number
, and
with
for all
satisfy all conditions in Theorem 2.9.
-
(b)
If
for all
and
, then we have the following corollary.
Corollary 2.10.
Let be a nonempty, closed, convex subsets in a real Hilbert space
. Let
be a bifunction from
to
satisfying conditions (A1)–(A4), let
be a nonexpansive mapping on
such that
and let
be a contraction of
into itself. Let
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ69_HTML.gif)
where ,
,
, and
satisfy conditions (i)–(v). Then,
and
converge strongly to
, where
.
Proof.
From the proof of the theorem, in (2.12).
-
(c)
In the case that
, a closed convex subset in
,
for all
, we have the following result.
Corollary 2.11.
Let be a nonempty, closed, convex subsets in a real Hilbert space
. Let
be a nonexpansive semigroup on
such that
and let
be a contraction of
into itself. Let
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ70_HTML.gif)
where is defined by (1.13) for all
and
,
,
, and
satisfy conditions (i)–(v). Then, the sequences
and
converge strongly to
, where
.
Proof.
By Lemma 2.2, if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ71_HTML.gif)
Clearly, in addition, if is a contraction of
into itself and
, then we obtain the algoritm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F208434/MediaObjects/13663_2010_Article_1386_Equ72_HTML.gif)
where is defined by (1.13) and
,
,
, and
satisfy conditions (i)–(v). This algorithm is different from Yao and Noor's algorithm (1.6), in which
for all
. It likes completely the Plubtieng and Punpaeng's algorithm (1.8), but converges under a new condition on
.
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This work was supported by the Vietnamese National Foundation of Science and Technology Development.
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Buong, N., Duong, N. A Method for a Solution of Equilibrium Problem and Fixed Point Problem of a Nonexpansive Semigroup in Hilbert's Spaces. Fixed Point Theory Appl 2011, 208434 (2011). https://doi.org/10.1155/2011/208434
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DOI: https://doi.org/10.1155/2011/208434