Throughout this paper, we always assume that is a real Hilbert space with the inner product and the norm . Let be a nonlinear mapping. In this paper, we use to denote the fixed point set of
Recall the following definitions.
(1)The mapping is said to becontractive with the coefficient if
(2)The mapping is said to benonexpansive if
(3)The mapping is said to bestrictly pseudocontractive with the coefficient if
(4)The mapping is said to bepseudocontractive if
Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. See, for example, [1–6] and the references therein.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :
where is a linear bounded and strongly positive operator and is a potential function for (i.e., for ).
Recently, Marino and Xu  studied the following iterative scheme:
They proved that the sequence generated in the above iterative scheme converges strongly to the unique solution of the variational inequality:
which is the optimality condition for the minimization problem (1.5).
Next, let be a nonlinear mapping. Recall the following definitions.
(1)The mapping is said to bemonotone if for each , we have
(2) is said to be -strongly monotone if
(3)The mapping is said to be -inverse-strongly monotone if there exists a constant such that
(4)The mapping is said to berelaxed-cocoercive if there exists a constant such that
(5)The mapping is said to berelaxed-cocoercive if there exist two constants such that
(6)Recall also that a set-valued mapping is called monotone if for all , and imply The monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping.
The so-called quasi-variational inclusion problem is to find a for a given element such that
where and are two nonlinear mappings. See, for example, [7–12]. A special case of the problem (1.13) is to find an element such that
In this paper, we use to denote the solution of the problem (1.14). A number of problems arising in structural analysis, mechanics, and economic can be studied in the framework of this class of variational inclusions.
Next, we consider two special cases of the problem (1.14).
If , where is a proper convex lower semicontinuous function and is the subdifferential of , then the variational inclusion problem (1.14) is equivalent to finding such that
which is said to be themixed quasi-variational inequality. See, for example, [7, 8] for more details.
If is the indicator function of then the variational inclusion problem (1.14) is equivalent to the classical variational inequality problem, denoted by , to find such that
For finding a common element of the set of fixed points of a nonexpansive mapping and of the set of solutions to the variational inequality (1.16), Iiduka and Takahashi  proved the following theorem.
be a closed convex subset of a real Hilbert space
-inverse-strongly monotone mapping of
be a nonexpansive mapping of
into itself such that
. Suppose that
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
Recently, Zhang et al.  considered the problem (1.14). To be more precise, they proved the following theorem.
be a real Hilbert space,
-inverse-strongly monotone mapping,
a maximal monotone mapping, and
a nonexpansive mapping. Suppose that the set
is the set of solutions of variational inclusion ( 1.14 ). Suppose that
is the sequence defined by
where and is a sequence in satisfying the following conditions:
Then converges strongly to
In this paper, motivated by the research work going on in this direction, see, for instance, [2, 3, 7–21], we introduce an iterative method for finding a common element of the set of fixed points of a strict pseudocontraction and of the set of solutions to the problem (1.14) with multivalued maximal monotone mapping and relaxed -cocoercive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.
In order to prove our main results, we need the following conceptions and lemmas.
Definition 1.1 (see ).
Let be a multivalued maximal monotone mapping. Then the single-valued mapping defined by for all , is called the resolvent operator associated with , where is any positive number and is the identity mapping.
Lemma 1.2 (see ).
Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that
Lemma 1.3 (see ).
Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all and Then
Lemma 1.4 (see ).
is a solution of variational inclusion (1.14) if and only if for all , that is,
Lemma 1.5 (see ).
The resolvent operator associated with is single-valued and nonexpansive for all .
Lemma 1.6 (see ).
Let be a closed convex subset of a strictly convex Banach space . Let and be two nonexpansive mappings on . Suppose that is nonempty. Then a mapping on defined by , where , for is well defined and nonexpansive and holds.
Lemma 1.7 (see ).
Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be a nonexpansive mapping. Then is demiclosed at zero.
Lemma 1.8 (see ).
Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction. Define by for each . Then, as , is nonexpansive such that .