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 