Based on the relaxed extragradient method and viscosity method, we introduce a new iterative method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the studied iterative method. The results of this paper extend and improve the results of Ceng et al., (2008), W. Kumam and P. Kumam, (2009), Yao et al., (2010) and many others.
Let be a real Hilbert space with the inner product and the norm . Let be a closed convex subset of . 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 (1.1) is denoted by . The equilibrium problems covers, as special cases, monotone inclusion problems, saddle point problems, minimization problems, optimization problems, variational inequality problems, Nash equilibria in noncooperative games, and various forms of feasibility problems (see [1–4] and the references therein).
A mapping is called -inverse-strongly monotone if there exists a positive real number such that
It is obvious that any -inverse-strongly monotone mapping is monotone and Lipschitz continuous. A mapping is said to be nonexpansive if
We denote by the set of fixed points of . Recently, Wang and Guo  introduced an iterative scheme for a countable family of nonexpansive mappings.
Let be a nonempty closed convex subset of a real Hilbert space . For a given nonlinear operator , consider the following variational inequality problem of finding such that
The set of solutions of the variational inequality (1.4) is denoted by (see [6–9] and the references therein).
Let be two mappings. Consider the following problem of finding such that
which is called a general system of variational inequalities, where and are two constants. The set of solutions of (1.5) is denoted by GSVI(). In particular, if , then problem (1.5) reduces to finding such that
which is defined by Verma  (see also ) and is called the new system of variational inequalities. Further, if we add up the requirement that , then problem (1.6) reduces to the classical variational inequality problem (1.4). Recently, Yao et al.  presented system of variational inequalities in Banach space. For solving problem (1.5), recently, Ceng et al.  introduced and studied a relaxed extragradient method. Based on the relaxed extragradient method and the viscosity approximation method, W. Kumam and P. Kumam  constructed a new viscosity-relaxed extragradient approximation method. Very recently, based on the extragradient method, Yao et al.  proposed an iterative method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space.
Motivated and inspired by the above works, in this paper, we introduce an iterative method based on the extragradient method and viscosity method for finding a common element of solution of equilibrium problems, the solution set of a general system of variational inequalities, and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove the strong convergence theorem of the proposed iterative method.
Let be a closed convex subset of , and let be nonexpansive such that . For all and all , we have
Remark 2.1. Let be -inverse-strongly monotone. For all , , we have
So, if , then is a nonexpansive mapping from to .
Recall that the (nearest point) projection from onto assigns to each the unique point satisfying the property
The following characterizes the projection .
Lemma 2.2. Given that and , then if and only if there holds the inequality
Lemma 2.3 (see ). Let be a Hilbert space, a closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then.
Lemma 2.4 (see ). Assume that is a sequence of nonnegative real numbers such that
where two sequences and satisfy(1);
(2). Then .
Lemma 2.5 (see ). For given , is a solution of problem (1.5) if and only if is a fixed point of the mapping defined by
Note that the mapping is nonexpansive provided that and .
Throughout this paper, the set of fixed points of the mapping is denoted by .
Lemma 2.6 (see ). Let be a nonempty closed convex subset of and satisfy following conditions: (A1), ; (A2) is monotone, that is, , ;(A3), ,,;(A4)for each , is convex and lower semicontinuous.For and , set to be
Then is well defined and the following holds: (1) is single valued;(2) is firmly nonexpansive , that is, for any ,
(3); (4) is closed and convex.
By the proof of Lemma 5 in  (also see ), we have following lemma.
Lemma 2.7. Let be a nonempty closed convex subset of a Hilbert space and be a bifunction. Let and . Then
3. Main Results
Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let the mappings be -inverse strongly monotone and -inverse strongly monotone, respectively. Let be a bifunction from to satisfying (A1)–(A4) and be a countable family of nonexpansive mappings such that . Let be a contraction with coefficient . Set . For given arbitrarily, let the sequences , , , and be generated by
where , , and sequences , , , and , , are such that (i) is strictly decreasing, (ii), (iii) and ,(iv),
(v)Then the sequence generated by (3.1) converges strongly to , and is a solution of the general system of variational inequalities (1.5), where .
Proof. The proof is divided into several steps. Step 1. The sequence defined by (3.1) is bounded. For each , from Lemma 2.6, we have and hence
Since is a -contraction mapping, using (2.3) and (3.2), we have
Set . Since is nonexpansive, from (2.3), we have
Hence we get
From (3.1) and (3.3)–(3.5), we get
where . Hence, is bounded and therefore , , , and are also bounded.Step 2. . Since and , using Lemma 2.7, we have
where . From (3.1) and (3.7), it follows that
From (3.1) and (3.8), we have
By definition of scheme (3.1), we have
Thus, from (3.8)–(3.10),
where . Since is strictly decreasing, we have . Further, by assumption conditions (iv)-(v), we have
Thus, using Lemma 2.4, we have .Step 3. . For any , it follows from Lemma 2.6 that
From (2.3), (3.1)–(3.4), we can get
From (3.1), (3.3)-(3.4), and (3.13)-(3.14), it follows that
which implies that
Since is both bounded, using Step 2 and conditions (ii)–(iv), we conclude the result.Step 4. and .Using (3.3), (3.14), and (3.15), we have
From Step 2, using condition (iii), we get and . From the fact that the is -inverse strongly monotone operator, it follows that
Applying Step 3, we have .Step 5. . Noting that is firmly nonexpansive, we have
It follows that
By (3.3), (3.14)-(3.15), and (3.22), we have
It follows that
From conditions (ii)–(iv), Steps 2 and 4, we get the following:
On the other hand, from (3.14)–(3.21), we have
which implies that
From (ii)-(iii), Steps 2 and 4, we get the following:
Combining (3.25) and (3.28), we get the following:
Using Step 3, we obtain the following:
This together with implies that
For any , we have from (3.1) that
Since each is nonexpansive, from (2.2), we have
Hence, combining this inequality with (3.32), we get the following:
that is (noting that is strictly decreasing),
Now from (3.30)-(3.31) and Step 2, we conclude that
which completes the proof.Step 6. , where . As is bounded, there exists a subsequence of such that weakly. First, it is clear from Step 5 and Lemma 2.3 that . Next, we prove that . From (iii), Step 3 and (3.31), we note that
According to (3.31) and Lemma 2.3, we obtain that . Next, we show that . Indeed, by , we have
From (A2), we have also
According to and , we conclude that and . From (A4), we obtain the following:
For with and , let . Since and (due to as ), we have and hence . So, from (A1) and (A4), we have
and , for all and . From (A3), we obtain the following:
and hence . Therefore, we obtain that . Hence, it follows from Lemma 2.2 that
Step 7. . From (3.1), (3.3)-(3.4), and the convexity of , we have
We can also get
By (3.45), we have