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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 572326, 18 pages
Hybrid Iterative Scheme by a Relaxed Extragradient Method for Equilibrium Problems, a General System of Variational Inequalities and Fixed-Point Problems of a Countable Family of Nonexpansive Mappings
1College of Science, Civil Aviation University of China, Tianjin 300300, China
2College of Science, National University of Defense Technology, Changsha 410073, China
Received 29 November 2011; Accepted 5 December 2011
Academic Editor: Yonghong Yao
Copyright © 2012 Qiao-Li Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 and hence
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 .
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.
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 Therefore, 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 and hence, 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 which implies that From , it follows that . It is clear that Therefore, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that . This completes the proof.
Corollary 3.2. 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 nonexpansive mapping such that .Let be a contraction with coefficient . For given arbitrarily, let the sequences , , and, be generated By where , , and sequences , , , and , , are such that (i), (ii)and ,(iii), (iv). Then the sequence generated by (3.50) converges strongly to