Báo cáo hóa học: " Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems" ppt

RESEARC H Open Access Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems Yonghong Yao 1 , Yeong-Cheng Liou 2 , Mu-Ming Wong 3* and Jen-Chih Yao 4 * Correspondence: mmwong@cycu.edu.tw 3 Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 32023, Taiwan Full list of author information is available at the end of the article Abstract In this paper, we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method. Mathematics Subject Classification (2000): 47H05; 47H09; 47H10; 47J05; 47J25. Keywords: variational inequality problem, fixed point problems; monotone mapping, nonexpansive mapping, extragradient method, CQ method, projection 1 Introduction Let H be a real Hilbert space with inner product 〈·,·〉 and induced norm || · ||. Let C be a nonempty closed convex subset of H.LetA : C ® H be a nonlinear operator. It is well known that the variational inequality problem VI(C, A) is to find u Î C such that  Au, v − u  ≥ 0, ∀v ∈ C . The set of solutions of the variational inequality is denoted by Ω. Variational inequality theory has emergedasanimportanttoolinstudyingawide class of obstacle, unilateral and equilibrium problems, which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have be en developed fo r solving variational inequalities and related optimiza- tion problems, see [1,1-25] and the references therein. Let us start with Korpelevich’s extragradient method which was introduced by Korpelevich [6] in 1976 and which generates a sequence {x n } via the recursion:  y n = P C [x n − λAx n ], x n+1 = P C [x n − λAy n ], n ≥ 0 , (1:1) where P C is the metric projection from R n onto C, A : C ® H is a monotone opera- tor and l is a con stant. Korpelevich [6] proved that the sequence {x n }converges strongly to a solution of VI(C, A). Note that the setting of the space is Euclid space R n . Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 © 2011 Yao et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permi ts unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Korpelevich’s extragradient method has extensively been studied in the literature for solving a more general problem that consists of finding a co mmon point that lies in the solution set of a variational inequality and the set of fixed points of a nonexpansive mapping. This type of problem aries in various theoret ical and modeli ng contexts, see e.g., [16-22,26] and refe rences therein. Especially, Nadezhkina and Taka hashi [23] introduced the fo llowing iterative method which combines Korpelevich’s extragradient method and a CQ method: x 0 = x ∈ C, y n = P C [x n − λ n Ax n ], z n = α n x n +(1− α n )SP C [x n − λ n Ay n ], C n = {z ∈ C :  z n − z ≤x n − z } , Q n = {z ∈ C : x n − z, x − x n ≥0}, x n+1 = P C n ∩ Q n x, n ≥ 0, n ≥ 0, where P C is the met ric projection from H onto C, A : C ® H is a monotone k-Lipschitz-continuous mapping, S : C ® C is a nonexpansive mapping, {l n }and{a n } are two real number sequences. They p roved the strong convergence of the sequences {x n }, {y n }and{z n } to the same element in Fix(S) ∩ Ω. Ceng et al. [25] suggested a new iterative method as follows: y n = P C [x n − λ n Ax n ], z n = α n x n +(1− α n )S n P C [x n − λ n Ay n ], C n = {z ∈ C :  z n − z ≤x n − z }, find x n+1 ∈ C n such that  x n − x n+1 + e n − σ n Ax n+1 , x n+1 − x  ≥−ε n , ∀x ∈ C n , where A : C ® H is a pseudomonotone, k-lipschitz-continuous and ( w, s)-sequen- tially-continuous mapping, {S i } N i =1 : C → C are N nonexpansive mappings. Under some mild conditions, they proved that the sequences {x n }, {y n } and {z n } converge weakly to thesameelementof  N i =1 Fix(S i ) ∩  if and only if lim inf n 〈Ax n , x - x n 〉 ≥ 0, ∀x Î C. Note that Ceng, Teboulle and Yao’s method has only weak convergence. Very recently, Ceng, H adjisavvas and Wong further introduced the following hybrid extragradient- like approximation method x 0 ∈ C, y n =(1− γ n )x n + γ n P C [x n − λ n Ax n ], z n =(1− α n − β n )x n + α n y n + β n SP C [x n − λ n Ay n ], C n = {z ∈ C :  z n − z 2 ≤x n − z 2 +(3− 3γ n + α n )b 2  Ax n  2 } , Q n = {z ∈ C : x n − z, x 0 − x n ≥0}, x n+1 = P C n ∩ Q n x 0 , for all n ≥ 0. It is shown that the se quences {x n }, {y n }, {z n } generated by the above hybrid extragradient-like approximation method are well defined and converge strongly to P F(S)∩Ω . Moti vated and inspired by the works of Nadezhkina and Takaha shi [23], Ceng et al. [25], and Ceng et al. [27], in this paper we suggest a hybrid method for finding a com- mon e lement of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 2 of 10 nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method. 2 Preliminaries In this section, we will recall some basic no tations and c ollect some conclusions that will be used in the next section. Let C be a nonempty closed convex subset of a r eal H ilbert space H. A mapping A : C ® H is called monotone if  Au − Av, u − v  ≥ 0, ∀u, v ∈ C . Recall that a mapping S : C ® C is said to be nonexpansive if  Sx − S y ≤x − y , ∀x, y ∈ C . Denote by Fix(S) the set of fixed points of S; that is, Fix(S)={x Î C : Sx = x}. It is well known that, for any u Î H, there exists a unique u 0 Î C such that  u − u 0  =inf { u − x  : x ∈ C }. We de note u 0 by P C [u], where P C is called t he metric projection of H onto C.The metric projection P C of H onto C has the following basic properties: (i) ||P C [x]-P C [y]||≤ ||x-y|| for all x, y Î H. (ii) 〈x-P C [x], y-P C [x]〉 ≤ 0 for all x Î H, y Î C. (iii) The property (ii) is equivalent to  x − P C [ x ]  2 +  y − P C [ x ]  2 ≤x − y , ∀x ∈ H, y ∈ C . (iv) In the context of the variational inequality problem, the chara cterization of the projection implies that u ∈  ⇔ u = P C [ u − λAu ] , ∀λ>0 . Recall that H satisfies the Opial’s condition [28]; i.e., for any sequence {x n }withx n converges weakly to x, the inequality lim inf n →∞  x n − x  < lim inf n →∞  x n − y  holds for every y Î H with y ≠ x. Let C be a nonempty closed convex subset of a real Hilbert space H.Let {S i } ∞ i = 1 be infinite family of nonexpansive mappings of C into itself and let {ξ i } ∞ i = 1 be real number sequences such that 0 ≤ ξ i ≤ 1 for every i Î N . For any n Î N, define a m apping W n of C into itself as follows: U n,n+1 = I , U n,n = ξ n S n U n,n+1 +(1− ξ n )I, U n,n−1 = ξ n−1 S n−1 U n,n +(1− ξ n−1 )I , . . . U n,k = ξ k S k U n,k+1 +(1− ξ k )I, U n,k−1 = ξ k−1 S k−1 U n,k +(1− ξ k−1 )I, . . . U n,2 = ξ 2 S 2 U n,3 +(1− ξ 2 )I, W n = U n,1 = ξ 1 S 1 U n,2 + ( 1 − ξ 1 ) I. (2:1) Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 3 of 10 Such W n is called the W -mapping generated by {S i } ∞ i =1 and {ξ i } ∞ i = 1 . We have the following crucial Lemmas 3.1 and 3.2 concerning W n which can be found in [29]. Now we only need the following similar version in Hilbert spaces. Lemma 2.1. Let C be a nonempty closed convex subset o f a real Hilbe rt space H. Let S 1 , S 2 , be nonexpans ive mappings of C into itself such that  ∞ n =1 Fix(S n ) is nonempty, and let ξ 1 , ξ 2 , be real numbers such that 0 < ξ i ≤ b<1 for any i Î N. Then, for every x Î C and k Î N, the limit lim n®∞ U n,k x exists. Lemma 2.2. Let C be a nonempty closed convex subset o f a real Hilbe rt space H. Let S 1 , S 2 , be nonexpans ive mappings of C into itself such that  ∞ n =1 Fix(S n ) is nonempty, and let ξ 1 , ξ 2 , be real numbers such that 0 < ξ i ≤ b<1 for any i Î N. Then, Fix(W)=  ∞ n =1 Fix(S n ) . Lemma 2.3. (see [30]) Using Lemmas 2.1 and 2.2, one can define a mapping W of C into itself as: Wx =lim n®∞ W n x =lim n®∞ U n,1 x, for every x Î C. If {x n } is a bounded sequence in C, then we have lim n → ∞  Wx n − W n x n =0 . We also need the following well-known lemmas for proving our main results. Lemma 2.4. ([31]) LetCbeanonemptyclosedconvexsubsetofarealHilbertspace H. Let S : C ® C be a nonexpansive mapping with Fix(S) ≠ ∅. Then S is demiclosed on C, i.e., if y n ® z Î C weakly and y n -Sy n ® y strongly, then (I-S)z = y. Lemma 2.5. ([32]) LetCbeaclosedconvexsubsetofH.Let{x n } be a sequence in H and u Î H. Let q = P C [u]. If {x n } is such that ω w (x n ) ⊂ C and satisfies the condition  x n − u ≤u − q  f or all n . Then x n ® q. We adopt the following notation: • For a given sequence {x n } ⊂ H, ω w (x n ) denotes the weak ω-limit set of {x n }; that is, ω w (x n ):={x ∈ H : {x n j } converges weakly to x for some subsequence { n j }of{n}}. • x n ⇀ x stands for the weak convergence of ( x n )tox; • x n ® x stands for the strong convergence of ( x n )tox. 3 Main results In this section we will state and prove our main results. Theorem 3.1. LetCbeanonemptyclosedconvexsubsetofarealHilbertspaceH. Let A : C ® H be a monotone, k-Lipschitz-continuous mapping and let { S n } ∞ n = 1 be an infinite family of nonexpansive mappings of C into itself such that  ∞ n =1 Fix(S n ) ∩  = ∅ . Let x 1 = x 0 Î C. For C 1 = C, let {x n }, {y n } and {z n } be sequences generated by y n = P C n [x n − λ n Ax n ], z n = α n x n +(1− α n )W n P C n [x n − λ n Ay n ], C n+1 = {z ∈ C n :  z n − z ≤x n − z } , x n+1 = P C n +1 [x 0 ], n ≥ 1, (3:1) where W n is W -mapping defined by (2.1). Assume the following conditions hold: Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 4 of 10 (i){l n } ⊂ [a, b] for some a, b Î (0, 1/k); (ii){a n } ⊂ [0, c] for some c Î [0, 1). Then the sequences { x n }, {y n } and {z n } generated by (3.1) converge strongly to the same point P  ∞ n =1 Fix(S n )∩ [x 0 ] . Next, we will divide ou r detail proofs into several conclusions. In the sequel, we assume that all assumptions of Theorem 3.1 are satisfied. Conclusion 3.2. (1) Every C n is closed and convex, n ≥ 1; (2)  ∞ n =1 Fix(S n ) ∩  ⊂ C n+1 , ∀ n ≥ 1 , (3) {x n+1 } is well defined. Proof. Fi rst we note that C 1 = C is closed and convex. Assum e that C k is closed and convex. From (3.1), we can rewrite C k+1 as C k+1 = {z ∈ C k : z − x k + z k 2 , z k − x k ≥0} . It is clear that C k+1 is a half space . Hence, C k+1 is closed and convex. By induction, we deduce that C n is closed and convex for all n ≥ 1. Next we show that  ∞ n =1 Fix(S n ) ∩  ⊂ C n+1 , ∀ n ≥ 1 . Set t n = P C n [x n − λ n Ay n ] for a ll n ≥ 1. Pick up u ∈  ∞ n =1 Fix(S n ) ∩  . From property (iii) of P C , we have  t n − u 2 ≤x n − λ n Ay n − u 2 −x n − λ n Ay n − t n  2 =  x n − u 2 −x n − t n  2 +2λ n Ay n , u − t n  =  x n − u 2 −x n − t n  2 +2λ n A y n , u − y n  +2λ n A y n , y n − t n  . (3:2) Since u Î Ω and y n Î C n ⊂ C, we get Au, y n − u≥0 . This together with the monotonicity of A imply that  Ay n , y n − u≥0 . (3:3) Combine (3.2) with (3.3) to deduce  t n − u 2 ≤x n − u 2 −x n − t n  2 +2λ n Ay n , y n − t n  =  x n − u 2 −x n − y n  2 − 2x n − y n , y n − t n −  y n − t n  2 +2λ n Ay n , y n − t n  =  x n − u 2 −x n − y n  2 −y n − t n  2 +2x n − λ n A y n − y n , t n − y n . (3:4) Note that y n = P C n [x n − λ n Ax n ] and t n Î C n . Then, using the property (ii) of P C ,we have x n − λ n Ax n − y n , t n − y n ≤0 . Hence, x n − λ n Ay n − y n , t n − y n  = x n − λ n Ax n − y n , t n − y n  + λ n Ax n − λ n Ay n , t n − y n  ≤λ n Ax n − λ n Ay n , t n − y n  ≤ λ n k  x n − y n  t n − y n  . (3:5) Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 5 of 10 From (3.4) and (3.5), we get  t n − u 2 ≤x n − u 2 −x n − y n  2 −y n − t n  2 +2λ n k  x n − y n  t n − y n  ≤x n − u 2 −x n − y n  2 −y n − t n  2 + λ 2 n k 2  x n − y n  2 +  y n − t n  2 =  x n − u 2 +(λ 2 n k 2 − 1)  x n − y n  2 ≤  x n − u  2 . (3:6) Therefore, from (3.6), together with z n = a n x n +(1a n )W n t n and u = W n u, we get  z n − u 2 =  α n (x n − u)+(1− α n )(W n t n − u) 2 ≤ α n  x n − u 2 +(1− α n )  W n t n − u 2 ≤ α n  x n − u 2 +(1− α n )  t n − u 2 ≤x n − u 2 +(1− α n )(λ 2 n k 2 − 1)  x n − y n  2 ≤  x n − u  2 , (3:7) which implies that u ∈ C n +1 . Therefore, ∞  n =1 Fix(S n ) ∩  ⊂ C n+1 , ∀n ≥ 1 . This implies that {x n+1 } is well defined. □ Conclusion 3.3. The sequences {x n }, {z n } and {t n } are all bounded and lim n®∞ || x n - x 0 || exists. Proof. From x n+1 = P C n +1 [x 0 ] , we have x 0 − x n+1 , x n+1 − y ≥0, ∀ y ∈ C n+1 . Since  ∞ n =1 Fix(S n ) ∩  ⊂ C n+ 1 , we also have x 0 − x n+1 , x n+1 − u≥0, ∀u ∈ ∞  n =1 Fix(S n ) ∩  . So, for u ∈  ∞ n =1 Fix(S n ) ∩  , we have 0 ≤  x 0 − x n+1 , x n+1 − u  = x 0 − x n+1 , x n+1 − x 0 + x 0 − u = −x 0 − x n+1  2 + x 0 − x n+1 , x 0 − u ≤−  x 0 − x n+1  2 +  x 0 − x n+1  x 0 − u  . Hence,  x 0 − x n+1 ≤x 0 − u , ∀u ∈ ∞  n =1 Fix(S n ) ∩  , (3:8) which implies that {x n } is bounded. From (3.6) and (3.7), we can deduce that {z n } and {t n } are also bounded. From x n = P C n [x 0 ] and x n+1 = P C n +1 [x 0 ] ∈ C n+1 ⊂ C n , we have  x 0 − x n , x n − x n+1  ≥ 0 . (3:9) Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 6 of 10 As above one can obtain that 0 ≤−  x 0 − x n  2 +  x 0 − x n  x 0 − x n+1 , and therefore  x 0 − x n  ≤  x 0 − x n+1  . This together with the boundedness of the sequence {x n } imply that lim n®∞ || x n - x 0 || exists. Conclusion 3.4.lim n®∞ ||x n+1 - x n || = lim n®∞ ||x n - y n || = lim n®∞ ||x n - z n || = lim n®∞ ||x n - t n || = 0 and lim n®∞ ||x n - W n x n || = lim n®∞ ||x n - Wx n || = 0. Proof. It is well known that in Hilbert spaces H, the following identity holds:  x − y  2 =  x 2 − y  2 − 2x − y , y , ∀x, y ∈ H . Therefore,  x n+1 − x n  2 =  (x n+1 − x 0 ) − (x n − x 0 ) 2 =  x n+1 − x 0  2 −  x n − x 0  2 − 2  x n+1 − x n , x n − x 0 , and by (3.9)  x n+1 − x n  2 ≤  x n+1 − x 0  2 −  x n − x 0  2 . Since lim n®∞ ||x n - x 0 || exists, we get ||x n+1 - x 0 || 2 -||x n - x 0 || 2 ® 0. Therefore, lim n → ∞  x n+1 − x n  =0 . Since x n+1 Î C n , we have  z n − x n+1  ≤  x n − x n+1 , and hence  x n − z n  ≤  x n − x n+1  +  x n+1 − z n  ≤ 2  x n+1 − x n  → 0. For each u ∈  ∞ n =1 Fix(S n ) ∩  , from (3.7), we have  x n − y n  2 ≤ 1 (1 − α n )(1 − λ 2 n k 2 ) ( x n − u 2 −z n − u 2 ) ≤ 1 (1 − α n )(1 − λ 2 n k 2 ) ( x n − u  +  z n − u )  x n − z n  . Since ||x n -z n || ® 0 and the sequences { x n }and{z n } are bounded, we obtain ||x n - y n || ® 0. We note that following the same idea as in (3.6) one obtains that  t n − u 2 ≤x n − u 2 +(λ 2 n k 2 − 1)  y n − t n  2 . Hence,  z n − u 2 ≤ α n  x n − u 2 +(1− α n )  t n − u 2 ≤ α n  x n − u 2 +(1− α n )( x n − u 2 +(λ 2 n k 2 − 1)  y n − t n  2 ) =  x n − u 2 +(1− α n )(λ 2 n k 2 − 1)  y n − t n  2 . Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 7 of 10 It follows that  t n − y n  2 ≤ 1 (1 − α n )(1 − λ 2 n k 2 ) ( x n − u 2 −z n − u 2 ) ≤ 1 (1 − α n )(1 − λ 2 n k 2 ) ( x n − u  +  z n − u )  x n − z n  → 0. Since A is k-Lipschitz-continuous, we have ||Ay n -At n || ® 0. From  x n − t n  ≤  x n − y n  +  y n − t n  , we also have  x n − t n  → 0 . Since z n = a n x n +(1-a n )W n t n , we have ( 1 − α n )( W n t n − t n ) = α n ( t n − x n ) + ( z n − t n ). Then, (1 − c)  W n t n − t n ≤(1 − α n )  W n t n − t n  ≤ α n  t n − x n  +  z n − t n  ≤ ( 1+α n )  t n − x n  +  z n − x n  and hence || t n -W n t n || ® 0. To conclude,  x n − W n x n ≤x n − t n  +  t n − W n t n  +  W n t n − W n x n  ≤x n − t n  +  t n − W n t n  +  t n − x n  ≤ 2  x n − t n  +  t n − W n t n  . So, || x n -W n x n || ® 0 too. On the other hand, since {x n }isbounded,fromLemma 2.3, we have lim n®∞ ||W n x n - Wx n || = 0. Therefore, we have lim n → ∞  x n − Wx n  =0 . □ Finally, according to Conclusions 3.3-3.5, we prove the remainder of Theorem 3.1. Proof. By Conclusions 3.3-3.5, we have proved that lim n → ∞  x n − Wx n  =0 . Furthermore, since {x n } is bounded, it has a subsequence {x n j } which converges weakly to some ˜ u ∈ C ; hence, we have lim j→∞  x n j − Wx n j = 0 .Notethat,from Lemma 2.4, it follows that I-Wis demiclosed at zero. Thus ˜ u ∈ Fix ( W ) .Since t n = P C n [x n − λ n Ay n ] , for every x Î C n we have x n − λ n A y n − t n , t n − x≥ 0 hence, x − t n , Ay n ≥x − t n , x n − t n λ n  . Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 8 of 10 Combining with monotonicity of A we obtain x − t n , Ax≥x − t n , At n  = x − t n , At n − Ay n  + x − t n , Ay n  ≥x − t n , At n − Ay n  + x − t n , x n − t n λ n  . Since lim n®∞ (x n - t n )=lim n®∞ (y n - t n )=0,A is Li pschitz continuous and l n ≥ a >0, we deduce that x − ˜ u, Ax = lim n j →∞ x − t n j , Ax≥0 . This implies that ˜ u ∈  . Consequently, ˜ u ∈  ∞ n =1 Fix(S n ) ∩  That is, ω w (x n ) ⊂  ∞ n =1 Fix(S n ) ∩  . In (3.8), if we take u = P  ∞ n =1 Fix(S n )∩ [x 0 ] , we get  x 0 − x n+1 ≤x 0 − P  ∞ n =1 Fix(S n )∩ [x 0 ]  . (3:10) Notice that ω w (x n ) ⊂  ∞ n =1 Fix(S n ) ∩  . Then, (3.10) and Lemma 2.5 ensure the strong convergence of {x n+1 }to P  ∞ n =1 Fix(S n )∩ [x 0 ] . Consequently, {y n }and{z n }alsocon- verge strongly to P  ∞ n =1 Fix(S n )∩ [x 0 ] . This completes the proof. Remark 3.5. Our algorithm (3.1) is simpler than the one in [23] and we extend the single mapping in [23] to an infinite family mappings. At the same time, the proofs are also simple. Acknowledgements The authors are extremely grateful to the referees for their useful comments and suggestions which helped to improve this paper. Yonghong Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 100-2221-E-230-012. Jen-Chih Yao was partially supported by the Grant NSC 99-2115-M-037- 002-MY3. Author details 1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China 2 Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan 3 Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 32023, Taiwan 4 Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan Authors’ contributions All authors participated in the design of the study and performed the converegnce analysis. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 14 March 2011 Accepted: 17 September 2011 Published: 17 September 2011 References 1. Stampacchia, G: Formes bilineaires coercitives sur les ensembles convexes. CR Acad Sci Paris. 258, 4413–4416 (1964) 2. Lions, JL, Stampacchia, G: Variational inequalities. 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Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Yao et al. Fixed Point Theory and Applications 2011, 2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53 Page 10 of 10 . method for variational inequalities. Carpathian J Math. 24, 139–148 (2008) 14. He, BS, Yang, ZH, Yuan, XM: An approximate proximal-extragradient type method for monotone variational inequalities. J. variational inequality is denoted by Ω. Variational inequality theory has emergedasanimportanttoolinstudyingawide class of obstacle, unilateral and equilibrium problems, which arise in several. several branches of pure and applied sciences in a unified and general framework. Several numerical methods have be en developed fo r solving variational inequalities and related optimiza- tion