1Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin (RMUTR), Bangkok 10100, Thailand 2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand
We propose new iterative schemes for finding the common element of the set of common
fixed points of countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for
relaxed cocoercive and Lipschitz continuous, the set of solutions of system of variational inclusions problem, and the set of
solutions of equilibrium problems in a real Hilbert space by using the viscosity approximation method. We prove strong
convergence theorem under some parameters. The results in this paper unify and generalize some well-known results in
Let be a nonempty closed convex subset of a real Hilbert space . A mapping of into itself is called nonexpansive if for all . We denote by the set of fixed points of ; that is, . If is nonempty, closed and convex and let be a nonexpansive mapping, then is closed and convex and , when is bounded; see, for example, [1, 2]. The metric projection, , onto a given nonempty, closed and convex subset , satisfies the nonexpansive with . A mapping is called monotone if for all . A mapping is called -inverse-strongly monotone if there exists a constant such that , for all . A mapping is called relaxed -cocoercive if there exists such that
A mapping is said to be -Lipschitz continuous if there exists such that
Let be a single-valued nonlinear mapping and a multivalued mapping. The variational inclusion problem is to find such that
where is the zero vector in . The set of solutions of problem (1.3) is denoted by . If , where is a nonempty closed convex subset of and is the indicator function of ; that is,
then, the variational inclusion problem (1.3) is equivalent to the variational inequality problems denoted by which is to find such that
In 2003, Takahashi and Toyoda  to find introduced the following iterative scheme:
where is a -inverse-strongly monotone mapping, is a sequence in (0, 1), and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.6) converges weakly to some .
In 2008, Zhang et al.  to find . They introduced the following new iterative scheme:
where is the resolvent operator associated with and a positive number is a sequence in the interval .
Let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that
The set of solutions of (1.8) is denoted by . Many problems in applied sciences, such as monotone inclusion problems, variational inequality problems, saddle point problems, Nash equilibria in noncooperative games, as well as certain fixed-point problems reduce to finding some element to in Hilbert and Banach spaces (see [5–14]).
Given any . The operator defined by
is called the resolvent of (see [5, 6]).
It is shown in  that, under suitable hypotheses on (to be stated precisely in Section 2), is single valued and firmly nonexpansive and satisfies
Using this result, for finding an element of , Su et al.  introduced the following iterative scheme by the viscosity approximation method in Hilbert spaces:
where is a contraction (i.e., and , , and satisfy some appropriate conditions. Furthermore, they prove converges strongly to the same point , where .
In this paper, motivated and inspired by the above facts, we introduce a new iterative scheme for finding a common element of the set of solutions of the variational inequalities for -Lipschitz continuous and relaxed -cocoercive mapping, the set of solutions to the variational inclusion for family of -inverse strongly monotone mappings, the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of an equilibrium problem in a real Hilbert space by using the viscosity approximation method. Strong convergence results are derived under suitable conditions in a real Hilbert space.
In this section, we will recall some basic notations and collect some conclusions that will be used in the next section.
Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. We denote strong convergence of to by and weak convergence by . Let be nonempty closed convex subset of . Recall that for all there exists a unique nearest point in to denoted ; that is, . The mapping is nonexpansive; that is, . The mapping is firmly nonexpansive; that is, . It is well known that
A set-valued mapping is called monotone if, for all , and imply . A monotone mapping is called maximal, if its graph of any Graph of is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping is maximal if and only if for all Graph (the graph of mapping ) implies that .
Definition 2.1. Let be a multivalued maximal monotone mapping; then the set-valued mapping defined by
is called the resolvent operator associated with , where is any positive number and is the identity mapping.
Lemma 2.2 (see ). Let be a maximal monotone mapping and let be a Lipschitz continuous mapping. Then the mapping is a maximal monotone mapping.
Lemma 2.3 (see [16, 17]). (1)The resolvent operator is single valued and nonexpansive for all ; that is,(2)The resolvent operator is 1-inverse-strongly monotone; that is,
Lemma 2.4 (see ). (1)Let is a solution of problem (1.3) if and only if for all ; that is,(2)If , then is a closed convex subset in .
Lemma 2.5 (see ). Each Hilbert space satisfies Opial’s condition; that is, for any sequence with , the inequality
holds for each with .
Lemma 2.6 (see ). Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose for all integers and . Then, .
Lemma 2.7 (see ). Assume is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence in such that(1),(2) or . Then .
Lemma 2.8. Let be a real Hilbert space. Then hold the following identities:(i),
Lemma 2.9 (see ). Let be a nonempty closed subset of a Banach space, and let be a sequence of mappings of into itself. Suppose that . Then, for each , converges strongly to some point of . Moreover, let be a mapping of into itself defined by
For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:(A1) for all ;(A2) is monotone, that is, ;(A3) for each ;(A4) for each is convex and lower semicontinuous.
Lemma 2.10 (see ). Let be a nonempty closed convex subset of , and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that
Lemma 2.11 (see ). Assume that satisfies (A1)–(A4). For and , define a mapping as follows:
for all . Then, the following hold:(i) is single valued;(ii) is firmly nonexpansive; that is, for any ;(iii);(iv) is closed and convex.
Lemma 2.12 (see ). Let be a Hilbert space and a maximal monotone on . Then, the following holds:
where and .
3. Main Results
In this section, we will use the viscosity approximation method to prove a strong convergence theorem for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous mappings, the set of solutions of system of variational inclusions, and the set of solutions of equilibrium problem in a real Hilbert space.
Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space , and let be relaxed -cocoercive and -Lipschitz continuous with , for some . Let be a finite family of -inverse strongly monotone mappings from into , and let be a bifunction from satisfying (A1)–(A4). Let be a contraction with coefficient , and let be a sequence of nonexpansive mappings of into itself such that
Let the sequences and be generated by
where , and satisfy the following conditions:(C1),(C2),
(C4) for some , with and ,(C5) and , for each ,(C6) and .Suppose that for any bounded subset of . Let be a mapping of into itself defined by and suppose that . Then, the sequences and converge strongly to the same point , where .
Proof. First, we prove that the mapping has a unique fixed point. In fact, since is a contraction with and is also a contraction, we obtain
Therefore, there exists a unique element such that , where
Now, we prove that is nonexpansive. Indeed, for any , since is a -Lipschitz continuous and relaxed -cocoercive mappings with and , we obtain
which implies that
Hence is nonexpansive. We divide the proof of Theorem 3.1 into five steps. Step 1. We show that the sequence is bounded. Now, let and if is a sequence of mappings defined as in Lemma 2.11, then , and let . So, we have
For and for any positive integer number , we define the operator as follows:
for all , we get . On the other hand, since is -inverse strongly monotone and , then is nonexpansive. Thus is nonexpansive. From Lemma 2.4(1), we have . It follows that
Setting and is a nonexpansive mapping, we obtain
From (3.2) and (3.12), we deduce that
It follows from induction that
Therefore, is bounded and hence so are , , , , and .Step 2. We claim that . By the definition of , and , we get
Taking in (3.15) and in (3.16), we have
So, from (A2) we have
Without loss of generality, let us assume that there exists a real number such that for all . Then, we have
where . Notice from Lemma 2.12 that
where is an appropriate constant such that
Since is nonexpansive mappings, we have the following estimates:
Substituting (3.23) into (3.25), we obtain
Indeed, define for all . It follows that
Thus, we have
Combining (3.28) and (3.29), we have
It follows that
This together with conditions (C1)–(C6) and implies that
Hence, by Lemma 2.6, we obtain as . It then follows that
By (3.26), we also have
Step 3. We claim that . Since is -inverse strongly monotone mappings, by the choice of for given and , we also have
Form (3.13), we have
It follows that
By condition (C2), (3.33), and , we obtain
From Lemma 2.3(2) and as is nonexpansive, we have
which yields that
Substituting (3.40) into (3.36), we obtain
It follows that
By condition (C2), (3.33), (3.38), and , we obtain
For , we obtain