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Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space
Journal of Inequalities and Applications volume 2010, Article number: 474813 (2010)
Abstract
We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of strict pseudocontractions in Hilbert Space. Then we study the weak and strong convergence of the sequences.
1. Introduction
Let be a nonempty closed convex subset of a Hilbert space
and let
be a self-mapping of
. Then
is said to be a strict pseudocontraction mappings if for all
, there exists a constant
such that

(if (1.1) holds, we also say that is a
-strict pseudocontraction). We use
to denote the set of fixed points of
,
to denote weak(strong) convergence, and
to denote the
set of
.
Let be a bifunction where
is the set of real numbers. Then, we consider the following equilibrium problem:

The set of such is denoted by
. Numerous problems in physics, optimization, and economics can be reduced to find a solution of (1.2). Some methods have been proposed to solve the equilibrium problem (see [1–3]). Recently, S. Takahashi and W. Takahashi [4] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also studied the strong convergence of the sequences generated by their algorithm for a solution of the EP which is also a fixed point of a nonexpansive mapping defined on a closed convex subset of a Hilbert space.
In this paper, thanks to the condition introduced by Aoyama et al. [5], We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problems and the set of fixed points of a countable family of strict pseudocontractions mappings in Hilbert Space. Then we study the weak and strong convergence of the sequences. The additional condition is inspired by Marino and Xu [6] and Kim and Xu [7].
2. Preliminaries
For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions (see [3]):
(A1)
(A2)
(A3)is upper-hemicontinuous, that is, for each
,
(A4) is convex and lower semicontinuous for each
.
Let be a real Hilbert space. Then there hold the following well-known results:

If is a sequence in
weakly convergent to
, then

Recall that the nearest point projection from
onto
assigns to each
its nearest point denoted by
in
; that is,
is the unique point in
with the property

Given and
, then
if and only if there holds the following relation:

Lemma (see [6]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
:
be a
-strict pseudocontraction such that
.
()(Demi-closed principle) is demi-closed on
, that is, if
and
, then
.
() satisfies the Lipschitz condition

()The fixed point set of
is closed and convex so that the projection
is well defined.
Lemma (see [5]).
Let be a nonempty closed convex subset of a Banach space and let
be a sequence of mapping of
into itself. Suppose
Then, for each
,
converges strongly to some point of
. Moreover, let
be a mapping of
into itself defined by

Then
Lemma (see [8]).
Let be a closed convex subset of
. Let
be a sequence in
and
. Let
. If
is such that
and satisfies the condition

then .
Lemma (see [9]).
Let be a nonempty closed convex subset of
. Let
be a bifunction from
satisfying (A1), (A2), (A3), and(A4). Then, for any
and
, there exists
such that

Further, if , then the following holds:
() is single-valued;
() is firmly nonexpansive, that is,

()
() is closed and convex.
3. Weak Convergence Theorems
Theorem 3.1.
Let be a nonempty closed convex subset of a Hilbert space
and let
be a sequence of
-strict pseudocontractions mappings on
into itself with
. Assume that
Let
be a bifunction satisfying (A1), (A2), (A3), (A4), and
. Let
and
be sequence generated by
and

Assume that for all
, where
is a small enough constant, and
is a sequence in
with
Let
for any bounded subset
of
and let
be a mapping of
into itself defined by
and suppose that
Then the sequences
and
converge weakly to an element of
.
Proof.
Pick . Then from the definition of
in Lemma 2.4, we have
, and therefore
. It follows from (3.1) that

Since for all
, we get
that is, the sequence
is decreasing. Hence
exists. In particular,
is bounded. Since
is firmly nonexpensive,
is also bounded. Also (3.2) implies that

Taking the limit as yields that

Since is bounded, it follows that

We apply Lemma 2.2 to get

Next, we claim that . Indeed, let
be an arbitrary element of
. Then as above

and hence

Therefore, from (3.2), we have

and hence

So, from the existence of , we have

Next, we claim that . since
is bounded and
is reflexive,
is nonempty. Let
be an arbitrary element. Then a subsequence
of
converges weakly to
. Hence, from (3.11) we know that
As
, we obtain that
. Let us show
. Since
, we have

By (A2), we have

and hence

From (A4), we have

Then, for and
, from (A1), and (A4), we also have

Taking and using (A3), we get

and hence . Since
is a strict pseudocontraction mapping, by Lemma 2.1(
) we know that the mapping
is demiclosed at zero. Note that
and
. Thus,
. Consequently, we deduce that
. Since
is an arbitrary element, we conclude that
.
To see that and
are actually weakly convergent, we take
Since
exist for every
, by (2.2), we have

Hence and proof is completed.
4. Strong Convergence Theorems
Theorem 4.1.
Let be a closed convex subset of a real Hilbert space
. Let
be a sequence of
-strict pseudocontractions mappings on
into itself with
. Assume that
Let
be a bifunction satisfying (A1), (A2), (A3), (A4) and
. For
and
, let
and
be sequence generated by
and

Assume that for all
, where
is a small enough constant, and
is a sequence in
with
and
. Let
for any bounded subset
of
and let
be a mapping of
into itself defined by
Suppose that
Then,
converges strongly to
.
Proof.
First, we show that is closed and convex. It is obvious that
is closed and convex. Suppose that
is closed and convex for some
. For
, we know that
is equivalent to

So is closed and convex. Then,
is closed and convex.
Next, we show by induction that for all
.
is obvious. Suppose that
for some
. Let
. Putting
for all
, we know from (4.1) that

and hence . This implies that
for all
.
This implied that is well defined.
From , we have

Using , we have

Then, is bounded. So are
and
. In particular,

From and
we have

Since is bounded,
exists. From
and
we also have

In fact, from (4.8), we have

Since exists, we have that
. On the other hand
implies that

Further, we have

From (4.3), we have

On the other hand, we have

Then, we have

Therefore, we have

We apply Lemma 2.2 to get

Lastly, we show that the sequence converges to
. Since
is bounded and
is reflexive,
is nonempty. Let
be an arbitrary element. Then a subsequence
of
converges weakly to
. From Lemma 2.1 and (4.16), we obtain that
. Next, we show
. Let
be an arbitrary element of
. From
and
, we have

and hence

Therefore, we have

As in the proof of Theorem 3.1, we have

By (A2), we have

and hence

From (A4), we have

Then, for and
, from (A1) and (A4), we also have

Taking and using (A3), we get

and hence . Lemma 2.3 and (4.6) ensure the strong convergence of
to
. This completes the proof.
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Acknowledgment
This work is supported by the National Science Foundation of China, Grant 10771050.
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Chen, R., Shen, X. & Cui, S. Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space. J Inequal Appl 2010, 474813 (2010). https://doi.org/10.1155/2010/474813
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DOI: https://doi.org/10.1155/2010/474813
Keywords
- Hilbert Space
- Banach Space
- Convergence Theorem
- Equilibrium Problem
- Weak Convergence