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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

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

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

where satisfies

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

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

where ,
, and
are in
,
, for finding
, when
satisfies the uniformly asymptotically regularity condition

uniformly in , and
is any bounded subset of
. Further, Plubtieng and Pupaeng in [7] studied the following algorithm:

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

where and
satisfy

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:

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:

where is a contraction on
, that is,
and
, for all
,
,

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:

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:

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

Lemma 2.5 (see [9]).
Assume that satisfies the conditions (A1)–(A4). For
and
, define a mapping
as follows:

Then, the following statements hold:
(i) is single-valued,
(ii) is firmly nonexpansive, that is, for any
,

(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
,

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

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

Put . Clearly,
. Suppose that
. Then, we have, from the nonexpansive property of
, condition (i) and (2.8), that

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

Then, we observe that

and, hence,

Now, we estimate the value by using
and
. We have from (2.4) that


Putting in (2.13) and
in (2.14), adding the one to the other obtained result and using (A2), we obtain that

and, hence,

Without loss of generality, let us assume that there exists a real number such that
for all
. Then, we have

and, hence,

On the other hand,

So, we get from (2.10), (2.12), (2.18), (2.19), and the nonexpansive property of that

So,

and by Lemma 2.8, we have

Consequently, it follows from (2.10) and condition (iii) that

By (2.18), (2.23), and

we also obtain

We have, for every , from (iii) in Lemma 2.5, that

and, hence,

Therefore, from the convexity of and condition (i), we have

and, hence,

Without loss of generality, we assume that for all
. Then, for sufficiently large
,

So, we have

Further, since , by condition (i), (2.19) and

we obtain that

Then, from (2.25), (2.33) and the conditions on and
, it implies that

and so

Since

we obtain from (2.31) that

Next, we show that

We choose a subsequence of the sequence
such that

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

and, hence, by using condition (A2), we get

Therefore,

This together with condition (A3) and (2.31) imply that

So, for all
. It means that
.
Next we show that . Since
, we have

and, hence, from (2.31) it follows that

Thus, (2.37) together with (2.45) imply

Therefore, also converges weakly to
, as
.
On the other hand, for each , we have that

Let . Since
, we have from (2.33) that

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

for every fixed , and hence, by (2.45)–(2.47), we obtain

for each . By Lemma 2.7,
for all
, because
for any mapping
. It means that
. Therefore,
. Since
, we have from Lemma 2.2 that

So, (2.38) is proved. Further, since , by using Lemma 2.1, we have that

This with (2.8) implies that

where

Using Lemma 2.3, we get

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

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

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

Clearly, in addition, if is a contraction of
into itself and
, then we obtain the algoritm

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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Acknowledgment
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
Keywords
- Convex Subset
- Equilibrium Problem
- Nonexpansive Mapping
- Strong Convergence
- Satisfying Condition