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Convergence Theorems on Generalized Equilibrium Problems and Fixed Point Problems with Applications
Journal of Inequalities and Applications volume 2010, Article number: 189036 (2010)
Abstract
The purpose of this work is to introduce an iterative method for finding a common element of a solution set of a generalized equilibrium problem, of a solution set solutions of a variational inequality problem and of a fixed point set of a strict pseudocontraction. Strong convergence theorems are established in the framework of Hilbert spaces.
1. Introduction and Preliminaries
Let be a real Hilbert space,
a nonempty closed and convex subset of
and
a nonlinear mapping. Recall the following definitions.
(a)The mapping is said to be monotone if

(b) is said to be
-strongly monotone if there exists a constant
such that

(c) is said to be
-inverse-strongly monotone if there exists a constant
such that

The classical variational inequality problem is to find such that

In this paper, we use to denote the solution set of the problem (1.4). One can easily see that the variational inequality problem is equivalent to a fixed point problem.
is a solution to the problem (1.4) if and only if
is a fixed point of the mapping
, where
is a constant and
is the identity mapping.
Let be a nonlinear mapping. In this paper, we use
to denote the fixed point set of
. Recall the following definitions.
(d)The mapping is said to be nonexpansive if

(e) is strictly pseudocontractive with a constant
if

For such a case, is called a
-strict pseudocontraction.
(f) is said to be pseudocontractive if

Clearly, the class of strict pseudocontractions falls into the one between a class of nonexpansive mappings and a class of pseudocontractions.
Recently, many authors considered the problem of finding a common element of the solution set of the variational inequality (1.4) and of fixed point set of a nonexpansive mapping in Hilbert spaces; see, for examples, [1–5] and the references therein.
In 2005, Iiduka and Takahashi [2] obtained the following theorem in a real Hilbert space.
Theorem 1 IT.
Let be a closed convex subset of a real Hilbert space
. Let
be an
-inverse-strongly monotone mapping of
into
and let
be a nonexpansive mapping of
into itself such that
. Suppose
and
is given by

for every where
is a sequence in
and
is a sequence in
. If
and
are chosen so that
for some
with
,

then converges strongly to
Let be an inverse-strongly monotone mapping, and
a bifunction of
into
, where
is the set of real numbers. We consider the following equilibrium problem:

In this paper, the set of such is denoted by
, that is,

In the case of , the zero mapping, the problem (1.10) is reduced to

In this paper, we use to denote the solution set of the problem (1.12), which was studied by many others; see, for examples, [1, 3, 6–23] and the reference therein. In the case of
, the problem (1.10) is reduced to the classical variational inequality (1.4). The problem (1.10) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instances, [15, 24].
To study the problems (1.10) and (1.12), we may assume that the bifunction satisfies the following conditions:
(A1) for all
;
(A2) is monotone, that is,
for all
;
(A3)for each ,

(A4)for each ,
is convex and weakly lower semicontinuous.
Recently, S. Takahashi and W. Takahashi [21] considered the problem (1.12) by introducing an iterative method in a Hilbert space. To be more precise, they proved the following theorem.
Theorem 1 TT1.
Let be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying (A1)–(A4), and let
be a nonexpansive mapping of
into
such that
. Let
be a contraction of
into itself, and let
and
be sequences generated by
and

where and
satisfy

Then and
converge strongly to
where
Very recently, S. Takahashi and W. Takahashi [22] further considered the problem (1.10). Strong convergence theorems of common elements are established. More precisely, they obtained the following result.
Theorem 2 TT2.
Let be a closed convex subset of a real Hilbert space
and let
be a bifunction satisfying (A1), (A2), (A3) and (A4). Let
be an
-inverse-strongly monotone mapping of
into
and let
be a nonexpansive mapping of
into itself such that
. Let
and
and let
and
be sequences generated by

where ,
, and
satisfy

Then, converges strongly to
In this paper, motivated by Theorem IT, Theorem TT1, and Theorem TT2, we introduce a general iterative method for the problem of finding a common element of a solution set of a generalized equilibrium problem (1.10), of a solution set of a variational inequality problem (1.4), and of a fixed point set of a strict pseudocontraction. Strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.
In order to prove our main results, we need the following lemmas.
The following lemmas can be found in [11, 24].
Lemma 1.1.
Let be a nonempty closed convex subset of
and let
be a bifunction satisfying (A1)–(A4). Then, for any
and
, there exists
such that

Further, define a mapping by

for all and
Then, the following hold.
(1) is single-valued;
(2) is firmly nonexpansive, that is, for any
,

(3);
(4) is closed and convex.
Lemma 1.2 (see [25]).
Let be a nonempty closed convex subset of a real Hilbert space
and
a
-strict pseudocontraction. Define
by
for each
. Then, as
,
is nonexpansive and
.
Lemma 1.3 (see [26]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in
with

Suppose that for all integers
and

Then
The following lemma can be deduced from Bruck [8].
Lemma 1.4.
Let be a closed convex subset of a strictly convex Banach space
. Let
,
and
be three nonexpansive mappings on
. Suppose
is nonempty. Let
,
and
be three constant in
Then the mapping
on
defined by

for is well defined, nonexpansive, and
holds.
Lemma 1.5 (see [6]).
Let be a real Hilbert space,
a nonempty closed and convex subset of
and
a nonexpansive mapping. Then
is demiclosed at zero.
Lemma 1.6 (see [14]).
Let be a real Hilbert space,
a nonempty closed and convex subset of
and
a
-strict pseudocontraction. Then
is closed and convex.
Lemma 1.7 (see [27]).
Assume that is a sequence of nonnegative real numbers such that

where is a sequence in
and
is a sequence such that
(a)
(b) or
Then
2. Main Results
Theorem 2.1.
Let be a nonempty closed and convex subset of a real Hilbert space
and
a bifunction from
to
satisfying (A1)–(A4). Let
be an
-inverse-strongly monotone mapping of
into
and
a
-inverse-strongly monotone mapping of
into
. Let
be a
-strict pseudocontraction with a fixed point. Assume that
. Let
be a sequence in
generated by

where is a fixed element in
,
,
,
,
,
and
are sequences in
,
is sequence in
,
and
. Assume that the above control sequences satisfy the following restrictions
(R1)
(R2),
;
(R3)
(R4) and
.
Then the sequence defined by the iterative process (2.1) converges strongly to
.
Proof.
The proof is divided into six steps.
Step 1.
Show that is well defined.
From Lemma 1.6, we see that is closed and convex. On the other hand, we see that the mapping
, where
, is nonexpansive. Indeed, for any
, we have that

This shows that is nonexpansive mapping. Similarly, we can prove that
, where
is nonexpansive. It follows that
,
is closed and convex. From Lemma 1.1, we see that
. Since
is nonexpansive, we obtain that
is closed and convex. This shows that
is well defined.
Step 2.
Show that is bounded.
Put for each
In view of Lemma 1.2 and (R4), we obtain that
is nonexpansive and
. Letting
, we obtain that

Note that for each
It follows that
Putting

we have that

It follows that

This shows that the sequence is bounded. Note that

This proves that the sequences and
are bounded, too.
Step 3.
Show that as
Note that

where is an appropriate constant such that
. On the other hand, we have

It follows from (2.1) and (2.9) that

where is an appropriate constant such that

Put , for each
, that is,

Now, we compute Notice that

It follows that

Substituting (2.10) into (2.14), we arrive at

It follows from the restrictions (R2)–(R4) that

From Lemma 1.3, we obtain that

From (2.12), we see that

In view of (2.17), we get that

Step 4.
Show that as
From the iterative process (2.1), we have

This implies that

It follows from the restrictions (R2) and (R3) that we arrive at

Step 5.
Show that where
To show that, we can choose a sequence of
such that

Since is bounded, we see that there exists a subsequence
of
which converges weakly to
. Without loss of generality, we may assume that
.
Next, we show that . In fact, define a mapping
by

where . Note that
is nonexpansive and
. From Lemma 1.4, we see that
is a nonexpansive mapping such that

On the other hand, we have

It follows from the condition (R4) that Note that

From (2.22), we arrive at

It follows from Lemma 1.5 that

Thanks to (2.23), we arrive at

Step 6.
Show that as
Notice that

which yields that

In view of the restrictions (R2) and (2.30), we from Lemma 1.7 can conclude the desired conclusion easily. This completes the proof.
As corollaries of Theorem 2.1, we have the following results.
Corollary 2.2.
Let be a nonempty closed and convex subset of a real Hilbert space
and
a bifunction from
to
satisfying (A1)–(A4). Let
be a
-inverse-strongly monotone mapping of
into
. Let
be a
-strict pseudocontraction with a fixed point. Assume that
. Let
be a sequence in
generated by

where is a fixed element in
,
,
,
,
,
and
are sequences in
,
is sequence in
,
and
. Assume that the above control sequences satisfy the following restrictions:
(R1)
(R2),
;
(R3)
(R4) and
for all
.
Then the sequence defined by the iterative process (2.1) converges strongly to
.
Proof.
In Theorem 2.1, put , the zero mapping. Then for any
, we see that the the following inequality holds.

Then, we can obtain the desired conclusion easily from Theorem 2.1. This completes the proof.
Corollary 2.3.
Let be a nonempty closed and convex subset of a real Hilbert space
and
a bifunction from
to
satisfying (A1)–(A4). Let
be a
-strict pseudocontraction of
into
and
a
-strict pseudocontraction of
into
. Let
be a
-strict pseudocontraction with a fixed point. Assume that
. Let
be a sequence in
generated by

where is a fixed element in
,
,
,
,
,
and
are sequences in
,
is sequence in
,
and
. Assume that the above control sequences satisfy the following restrictions:
(R1)
(R2),
;
(R3)
(R4) and
.
Then the sequence defined by the above iterative process converges strongly to
.
Proof.
Put and
. Then, we see that
is
-inverse-strongly monotone and
is
-inverse-strongly monotone; see [7]. We have
and

It is easy to obtain the desired conclusion from Theorem 2.1.
Remark 2.4.
If is a contractive mapping and we replace
by
in the recursion formula (2.1), we can obtain the so-called viscosity iteration method. We note that all theorems and corollaries of this paper carry over trivially to the so-called viscosity iteration method; see [28] for more details.
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Acknowledgment
This project is supported by the National Natural Science Foundation of China (no. 10901140).
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Hao, Y., Cho, S. & Qin, X. Convergence Theorems on Generalized Equilibrium Problems and Fixed Point Problems with Applications. J Inequal Appl 2010, 189036 (2010). https://doi.org/10.1155/2010/189036
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DOI: https://doi.org/10.1155/2010/189036
Keywords
- Hilbert Space
- Variational Inequality
- Convex Subset
- Nonexpansive Mapping
- Common Element