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