Strong convergence theorems for fixed point problems, variational inequality problems, and equilibrium problems

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1 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 DOI /s R E S E A R C H Open Access Strong convergence theorems for fixed point problems, variational inequality problems, and equilibrium problems Zhangsong Yao 1, Yeong-Cheng Liou 2,3*,Li-JunZhu 4, Ching-Hua Lo 2 and Chen-Chang Wu 2 * Correspondence: simplex_liou@hotmail.com 2 Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan 3 Center for General Education, Kaohsiung Medical University, Kaohsiung, 807, Taiwan Full list of author information is available at the end of the article Abstract In this paper, we first introduce an iterative algorithm for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the solution set of the variational inequality problem for a monotone mapping in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets under some mild conditions. MSC: 49J30; 47H09; 47H20 Keywords: nonexpansive mapping; equilibrium problem; fixed point; variational inequality 1 Introduction Equilibrium problems theory provides us a natural, novel and unified framework to study a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimization. The ideas and techniques of this theory are being used in a variety of diverse areas and proved to be productive and innovative. It has been shown by Blum and Oettli [1] and Noor and Oettli [2] that variational inequalities and mathematical programming problems can be viewed as special realization of the abstract equilibrium problems. Equilibrium problems have numerous applications, including but not limited to problems in economics, game theory, finance, traffic analysis, circuit network analysis, and mechanics. There are a substantial number of numerical methods including projection technique and its variant forms, Wiener-Hopf equations, the auxiliary principle technique, and resolvent equations methods for solving variational inequalities. However, it is well known that projection, Wiener-Hopf equations, and proximal and resolvent equations techniques cannot be extended and generalized to suggest and analyze similar iterative methods for solving equilibrium problems. This fact has motivated us to use the auxiliary principle technique for solving the equilibrium problems. MA Noor and KI Noor [3]andMANoor [4] used this technique to suggest implicit and explicit iterative methods for solving equilibrium problems. Related to this problem, much attention has been given to finding the fixed point of nonexpansive and contraction mappings. It is natural to combine these two 2015 Yao et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 2 of 15 different problems and study the problem of finding the equilibrium problem coupled with fixed point problem. In this direction, Combettes and Hirstoaga [5]introducedaniterative scheme for finding the best approximation to the initial data of the equilibrium problem and proved a strong convergence result. See also [6 16] and the references therein for more details. It is our purpose in this paper that we introduce an iterative algorithm for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the solution set of the variational inequality problem. We show the proposed algorithm has strong convergence. Let H be a real Hilbert space with inner product, and norm,andletc be a closed convex subset of H. A mapping A of C into H is called monotone if Au Av, u v 0, u, v C. A : C H is called α-inverse-strongly-monotone if there exists a positive real number α such that Au Av, u v α Au Av 2, u, v C (see Browder and Petryshyn [17]; Liu and Nashed [18]). It is obvious that any α-inversestrongly monotone mapping A is monotone and 1 -Lipschitz continuous. A mapping S : α C H is said to be nonexpansive if Sx Sy x y, x, y C. We denote by Fix(S)thesetoffixedpointsofS. Recall that the classical variational inequality, denoted by VI(A, C), is to find x C such that Ax, v x 0 for all v C. The variational inequalities have been widely studied in the literature; see, e.g., [19 25] and the references therein. For finding an element of Fix(S) VI(A, C) under the assumption that a mapping A of C into H is α-inverse-strongly-monotone, Takahashi and Toyoda [26] introducedthe following iterative scheme: x n+1 = α n x n +(1 α n )SP C (x n λ n Ax n ) for every n =0,1,2,...,whereP C is the metric projection of H onto C, x 0 = x C, {α n } is a sequence in (0, 1), and {λ n } is a sequence in (0, 2α). On the other hand, for solving the variational inequality problem in the finite-dimensional Euclidean space R n,korpelevich [27] introduced the following so-called extragradient method: x 0 = x C, y n = P C (x n λax n ), x n+1 = P C (x n λay n )

3 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 3 of 15 for every n = 0,1,2,..., where λ (0, 1/k). Recently, Nadezhkina and Takahashi [6] introduced another extragradient method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem. They obtained the following result. Theorem 1.1 ([6]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be a monotone, k-lipschitz continuous mapping, and let S : C C be a nonexpansive mapping such that Fix(S) VI(A, C). Let the sequences {x n }, {y n } be generated by x 0 = x H, y n = P C (x n λ n Ax n ), x n+1 = α n x n +(1 α n )SP C (x n λ n Ay n ), n 0, (1.1) where {λ n } [a, b] for some a, b (0, 1/k) and {α n } [c, d] for some c, d (0, 1). Then the sequences {x n }, {y n } generated by (1.1) converge weakly to the same point P Fix(S) VI(A,C) (x 0 ). The iterative scheme (1.1) in Theorem 1.1 has only weak convergence. In order to obtain strong convergence theorem, very recently, Zeng and Yao [11] further introduced a new extragradient method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem. They proved the following interesting result. Theorem 1.2 ([11]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be a monotone, k-lipschitz continuous mapping, and let S : C C be a nonexpansive mapping such that Fix(S) VI(A, C). Let the sequences {x n }, {y n } be generated by x 0 = x H, y n = P C (x n λ n Ax n ), x n+1 = α n x 0 +(1 α n )SP C (x n λ n Ay n ), n 0, (1.2) where {λ n } and {α n } satisfy the conditions: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) {α n } (0, 1), n=0 α n =, lim n α n =0. Then the sequences {x n } and {y n } converge strongly to the same point P Fix(S) VI(A,C) (x 0 ) provided lim n x n+1 x n =0. Now we concern ourselves with the following equilibrium problem: Find x C such that EP : F(x, y) 0 for all y C, (1.3) where F is an equilibrium bifunction of C C into R. The set of solutions of (1.3)isdenoted by EP(F). Given a mapping T : C H, letf(x, y) =Tx, y x for all x, y C. Thenz EP(F)ifandonlyifTz, y z 0 for all y C.

4 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 4 of 15 For solving equilibrium problem (1.1), Takahashi and Takahashi [9] introducedanit- erative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let S : C H be a nonexpansive mapping. Starting with arbitrary initial x 1 H, define sequences {x n } and {u n } recursively by { F(u n, y)+ 1 r n y u n, u n x n 0, y C, (1.4) x n+1 = α n f (x n )+(1 α n )Su n, n 0. They proved that the sequences {x n } and {u n } converge strongly to z Fix(S) EP(F)with the following restrictions on the algorithm parameters {α n } and {r n }: (i) lim n α n =0and n=0 α n = ; (ii) lim inf n r n >0; (iii) (A1): n=0 α n+1 α n < ;and(r1): n=0 r n+1 r n <. Motivated and inspired by the work of Zeng and Yao [11], Takahashi and Takahashi [9], in this paper, we first introduce an iterative algorithm for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the solution set of the variational inequality problem for a monotone mapping in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets under some mild conditions. Our result includes the result of Zeng and Yao [11]asaspecialcase. 2 Preliminaries Let C be a nonempty closed convex subset of a real Hilbert space H.Itiswellknownthat, for any u H,thereexistsauniquey 0 C such that u y 0 = inf { u y : y C }. We denote y 0 by P C u,wherep C is called the metric projection of H onto C. Themetric projection P C of H onto C is characterized by the following properties: (i) P C x P C y x y for all x, y H, (ii) x y, P C x P C y P C x P C y 2 for every x, y H, (iii) x P C x, y P C x 0 for all x H, y C, (iv) x y 2 x P C x 2 + y P C x 2 for all x H, y C. Let A be a monotone mapping of C into H. In the context of the variational inequality problem, it is easy to see from (iv) that u VI(A, C) u = P C (u λau), λ >0. A set-valued mapping B : H 2 H is called monotone if, for all x, y H, f Bx and g By imply x y, f g 0. A monotone mapping B : H 2 H is maximal if its graph G(B) is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping B is maximal if and only if, for (x, f ) H H, x y, f g 0for every (y, g) G(B)impliesf Bx.LetA be a monotone mapping of C into H,andletN C v be the normal cone to C at v C; i.e., N C v = { w H : v u, w 0, u C }.

5 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 5 of 15 Define Bv = { Av + N C v, ifv C,, ifv / C. Then B is maximal monotone and 0 Bv if and only if v VI(A, C). In this paper, for solving the equilibrium problems for a bifunction F : C C R, we assume that F satisfies the following conditions: (H1) F(x, x)=0for all x C; (H2) F is monotone, i.e., F(x, y)+f(y, x) 0 for all x, y C; (H3) for each x, y, z C, lim t 0 F(tz +(1 t)x, y) F(x, y); (H4) for each x C, y F(x, y) is convex and lower semi-continuous. In the sequel, we shall need the following lemmas for proving our main result. Lemma 2.1 ([5]) Let C be a nonempty closed convex subset of H, and let F be a bifunction of C CintoRsatisfying conditions (H1)-(H4). Let r >0and x C. Then there exists y C such that F(y, z)+ 1 z y, y x 0 for all z C. r Lemma 2.2 ([5]) Assume that F satisfies the same assumptions as in Lemma 2.1. For r >0 and x C, define a mapping T r : H Casfollows: T r (x)= {y C : F(y, z)+ 1r } z y, y x 0, z C for all y H. Then the following hold: (1) T r is single-valued; (2) T r is firmly nonexpansive, i.e., for any x, y H, T r x T r y 2 T r x T r y, x y ; (3) Fix(T r )=EP(F); (4) EP(F) is closed and convex. Lemma 2.3 ([28]) Assume {a n } is a sequence of nonnegative real numbers such that a n+1 (1 γ n )a n + δ n, where {γ n } is a sequence in (0, 1) and {δ n } is a sequence such that (1) n=1 γ n = ; (2) lim sup n δ n /γ n 0 or n=1 δ n <. Then lim n a n =0. 3 Main results In this section, we first introduce the following iterative algorithm.

6 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 6 of 15 Algorithm 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H.LetF be a bifunction from C C R.LetA : C H be a nonlinear mapping, and let S : C C a mapping. For fixed u C and given x 0 C arbitrarily, suppose the sequences {x n }, {y n }, and {u n } are generated iteratively by F(u n, y)+ 1 r n y u n, u n x n 0, y C, y n = P C (u n λ n Au n ), (3.1) x n+1 = α n u + β n x n +(1 α n β n )SP C (u n λ n Ay n ), where {α n } and {β n } are two sequences in (0, 1) and {λ n } and {r n } are two sequence in (0, ). Next we prove the strong convergence of Algorithm 3.1. Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C C R satisfying (H1)-(H4). Let A be a monotone and k-lipschitz continuous mapping of C into H, and let S be a nonexpansive mapping of C into itself such that Fix(S) VI(A, C) EP(F). Assume that: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) lim n α n =0and n=0 α n = ; (c) lim sup n β n <1. Then the sequences {x n }, {y n }, and {u n } generated by (3.1) converge strongly to P Fix(S) VI(A,C) EP(F) (u) if and only if lim n x n+1 x n =0. Proof The necessity is obvious. Next we prove the sufficiency. Let x Fix(S) VI(A, C) EP(F), and let {T rn } be a sequence of mappings defined as in Lemma 2.2.Thenwehavex = P C (x λ n Ax )=T rn x. Set z n = P C (u n λ n Ay n ) for all n 0. From the property (iv) of P C,wehave z n x 2 u n λ n Ay n x 2 u n λ n Ay n z n 2 = un x 2 2λ n Ayn, u n x + λ 2 n Ay n 2 u n z n 2 +2λ n Ay n, u n z n λ 2 n Ay n 2 = u n x 2 u n z n 2 +2λ n Ayn, x z n = un x 2 u n z n 2 +2λ n Ayn Ax, x y n +2λ n Ax, x y n +2λn Ay n, y n z n. (3.2) Using the fact that A is monotonic and x is a solution of the variational inequality problem VI(A, C), we have Ayn Ax, x y n 0 and Ax, x y n 0. (3.3) It follows from (3.2)and(3.3)that z n x 2 u n x 2 u n z n 2 +2λ n Ay n, y n z n = un x 2 u n y n 2 2u n y n, y n z n

7 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 7 of 15 y n z n 2 +2λ n Ay n, y n z n = u n x 2 u n y n 2 +2u n λ n Ay n y n, z n y n y n z n 2. (3.4) By using the property (iii) of P C,wehaveu n λ n Au n y n, z n y n 0. Therefore, we get u n λ n Ay n y n, z n y n = u n λ n Au n y n, z n y n + λ n Au n Ay n, z n y n λ n Au n Ay n, z n y n λ n Au n Ay n z n y n λ n k u n y n z n y n. (3.5) Combining (3.4)and(3.5), we obtain z n x 2 u n x 2 u n y n 2 y n z n 2 +2λ n k u n y n z n y n un x 2 u n y n 2 y n z n 2 + λ 2 n k2 u n y n 2 + z n y n 2 = un x 2 + ( λ 2 n k2 1 ) u n y n 2 u n x 2 = T rn x n T rn x 2 xn x 2. (3.6) From (3.1), we deduce that x n+1 x = α n ( u x ) + β n ( xn x ) +(1 α n β n ) ( Sz n x ) u x + β n x n x +(1 α n β n ) z n x u x +(1 αn ) xn x. (3.7) It followsfrom (3.7) and induction that x n p max { u x, x 0 x }, n 0. Hence {x n } is bounded. It is clear that {y n }, {u n },and{z n } are all bounded. Next, we show y n Sy n 0. From x n+1 = α n u + β n x n +(1 α n β n )Sz n,wehave x n Sz n x n x n+1 + x n+1 Sz n x n x n+1 + α n u Sz n + β n x n Sz n,

8 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 8 of 15 that is, x n Sz n 1 1 β n x n x n+1 + α n 1 β n u Sz n. This together with x n+1 x n 0andα n 0impliesthat lim n x n Sz n =0. (3.8) Since T rn is firmly nonexpansive, we have u n x 2 = T rn x n T rn x 2 T rn x n T rn x, x n x = u n x, x n x = 1 ( un x 2 + xn x 2 x n u n 2) 2 and hence u n x 2 x n x 2 x n u n 2. (3.9) By (3.1), we have xn+1 x 2 = αn ( u x ) + β n ( xn x ) +(1 α n β n ) ( Sz n x ) 2 u x 2 + β n xn x 2 +(1 α n β n ) Szn x 2 u x 2 + β n xn x 2 +(1 α n β n ) zn x 2. (3.10) From (3.6)and(3.10), we have x n+1 x 2 u x 2 + β n x n x 2 +(1 α n β n ) [ u n x 2 + ( λ 2 n k2 1 ) u n y n 2] u x 2 + β n xn x 2 +(1 α n β n ) [ xn x 2 + ( λ 2 n k2 1 ) u n y n 2] u x 2 + xn x 2 +(1 α n β n ) ( λ 2 n k2 1 ) u n y n 2. Then we derive (1 α n β n ) ( 1 λ 2 n k2) u n y n 2 x n x 2 x n+1 p 2 + α n u x 2 ( x n x + x n+1 x ) x n+1 x n + α n u x 2. (3.11)

9 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 9 of 15 It is clear that lim inf n (1 α n β n )(1 λ 2 n k2 )>0.So,from(3.11), we have lim n u n y n =0. (3.12) From (3.6), (3.9), and (3.10), we have x n+1 x 2 u x 2 + β n x n x 2 +(1 α n β n ) z n x 2 u x 2 + β n x n x 2 +(1 α n β n ) u n x 2 u x 2 + β n x n x 2 +(1 α n β n ) [ xn x 2 x n u n 2] u x 2 + xn x 2 (1 α n β n ) x n u n 2, that is, (1 α n β n ) x n u n 2 u x 2 + x n x 2 x n+1 x 2 u x 2 + ( x n x + x n+1 x ) x n+1 x n, which implies that We have lim n x n u n = 0. (3.13) Sy n y n Sy n Sz n + Sz n x n + x n u n + u n y n y n z n + Sz n x n + x n u n + u n y n = P C (u n λ n Au n ) P C (u n λ n Ay n ) + Sz n x n + x n u n + u n y n λ n Au n Ay n + Sz n x n + x n u n + u n y n (λ n k +1) u n y n + Sz n x n + x n u n. This together with (3.8), (3.12), and (3.13)impliesthat lim n Sy n y n =0. Next we prove lim supu z 0, x n z 0 0, n where z 0 = P Fix(S) VI(A,C) EP(F) (u). First, we show that lim supu z 0, Sy n z 0 0. n

10 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 10 of 15 To show this inequality, we can choose a subsequence {y nj } of {y n } such that lim j u x, Sy nj x = lim sup u x, Sy n x. n Since {y nj } is bounded, there exists a subsequence {y nji } of {y nj } which converges weakly to w. Without loss of generality, we can assume that y nj w weakly. From Sy n y n 0, we obtain Sy nj w weakly. First we show w EP(F). By u n = T rn x n,wehave F(u n, y)+ 1 r n y u n, u n x n 0, y C. From the monotonicity of F,we have 1 r n y u n, u n x n F(u n, y) F(y, u n ) and hence y u nj, u n j x nj F(y, u nj ). r nj Since u n j x nj r nj 0andu nj w weakly, from the lower semi-continuity of F(x, y) onthe second variable y,wehave F(y, w) 0 for all y C.Fort with 0 < t 1andy C,lety t = ty +(1 t)w.sincey C and w C,we have y t C and hence F(y t, w) 0. So, from the convexity of the equilibrium bifunction F(x, y) onthesecondvariabley, wehave 0=F(y t, y t ) tf(y t, y)+(1 t)f(y t, w) tf(y t, y) and hence F(y t, y) 0. Then we have F(w, y) 0 for all y C and hence w EP(F). Second, we show that w VI(A, C). Set Bv = { Av + N C v, ifv C,, ifv / C. Then B is maximal monotone. Let (v, u) G(B). Since u Av N C v and y n C,wehave v y n, u Av 0.

11 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 11 of 15 On the other hand, from y n = P C (u n λ n Au n ), we have v yn, y n (u n λ n Au n ) 0 and hence v y n, y n u n + Au n 0. λ n It follows that v y ni, u v y ni, Av v y ni, y n i u ni + Au ni λ ni = v y ni, Av y n i u ni Au ni λ ni = v y ni, Av Ay ni + v y ni, Ay ni Au ni v y ni, y n i u ni λ ni v y ni, Ay ni Au ni v y ni, y n i u ni, λ ni which implies that v w, u 0. We have w B 1 (0) and hence w VI(A, C). Thirdly, we prove that w Fix(S). Assume that w / Fix(S). Since y nj w and w Sw,by Opial s condition we have lim inf j y n j w < lim inf y n j Sw j ( lim inf ynj Sy nj + Sy nj Sw ) j lim inf y n j w, j which is a contradiction. Then we get w Fix(S). Hence, we deduce that w Fix(S) VI(A, C) EP(F). Therefore, from the property (iii) of P C,wehave lim supu z 0, x n z 0 = lim supu z 0, Sy n z 0 n n = lim u z 0, Sy nj z 0 j = u z 0, w z 0 0. (3.14) Finally, we show x n z 0,wherez 0 = P Fix(S) VI(A,C) EP(F) (u). From (3.1), we have x n+1 z 0 2 = αn (u z 0 )+β n (x n z 0 )+(1 α n β n )(Sz n z 0 ) 2 βn (x n z 0 )+(1 α n β n )(Sz n z 0 ) 2

12 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 12 of 15 +2α n u z 0, x n+1 z 0 [ (1 α n β n ) Sz n z 0 + β n x n z 0 ] 2 +2α n u z 0, x n+1 z 0 (1 α n ) 2 x n z 0 +2α n u z 0, x n+1 z 0 (1 α n ) x n z 0 +2α n u z 0, x n+1 z 0. (3.15) Hence, by Lemma 2.3, (3.14), and (3.15), we conclude that the sequence x n converges strongly to z 0.Since y n x n 0and u n x n,wehavey n z 0 and u n z 0.This completes the proof. 4 Applications Taking β n 0 for all n 0in(3.1), we immediately obtain the following result. Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C C R satisfying (H1)-(H4). Let A be a monotone and k-lipschitz continuous mapping of C into H, and let S be a nonexpansive mapping of C into itself such that Fix(S) VI(A, C) EP(F). Let {α n } be a sequence in (0, 1), and let {λ n } and {r n } be two sequences in (0, ). For fixed u Candgivenx 0 Carbitrarily, let the sequences {x n }, {y n }, and {u n } be generated iteratively by F(u n, y)+ 1 r n y u n, u n x n 0, y C, y n = P C (u n λ n Au n ), (4.1) x n+1 = α n u +(1 α n β n )SP C (u n λ n Ay n ). Suppose the following conditions are satisfied: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) lim n α n =0and n=0 α n =. Then the sequences {x n }, {y n }, and {u n } generated by (4.1) converge strongly to P Fix(S) VI(A,C) EP(F) (u) if and only if lim n x n+1 x n =0. In (3.1), we put F(x, y) = 0 for all x, y C and r n =1foralln N. Thenwehaveu n = P C x n = x n. Then we obtain the following theorem. Theorem 4.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-lipschitz continuous mapping of C into H, and let S be a nonexpansive mapping of C into itself such that Fix(S) VI(A, C). Let {α n } and {β n } be two sequences in (0, 1), and let {λ n } be a sequence in (0, ). For fixed u Candgivenx 0 Carbitrarily, let the sequences {x n } and {y n } be generated iteratively by { y n = P C (x n λ n Ax n ), (4.2) x n+1 = α n u + β n x n +(1 α n β n )SP C (x n λ n Ay n ). Suppose the following conditions are satisfied: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) lim n α n =0and n=0 α n = ; (c) lim sup n β n <1.

13 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 13 of 15 Then the sequences {x n } and {y n } generated by (4.2) converge strongly to P Fix(S) VI(A,C) (u) if and only if lim n x n+1 x n =0. In (4.2), we put β n 0 for all n 0. Then we obtain the following result. Theorem 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-lipschitz continuous mapping of C into H, and let S be a nonexpansive mapping of C into itself such that Fix(S) VI(A, C). Let {α n } be a sequence in (0, 1), and let {λ n } be a sequence in (0, ). For fixed u Candgivenx 0 Carbitrarily, let the sequences {x n } and {y n } be generated iteratively by { y n = P C (x n λ n Ax n ), x n+1 = α n u +(1 α n β n )SP C (x n λ n Ay n ). (4.3) Suppose the following conditions are satisfied: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) lim n α n =0and n=0 α n =. Then the sequences {x n } and {y n } generated by (4.3) converge strongly to P Fix(S) VI(A,C) (u) if and only if lim n x n+1 x n =0. Remark 4.4 It is clear that Theorem 4.3 indicates the result in Zeng and Yao [11]. A mapping T : C C is called strictly pseudocontractive if there exists k with 0 k <1 such that Tx Ty 2 x y 2 + k (I T)x (I T)y 2 for all x, y C.PutA = I T,thenwehave (I A)x (I A)y 2 x y 2 + k Ax Ay 2. On the other hand, (I A)x (I A)y 2 = x y 2 + Ax Ay 2 2x y, Ax Ay. Hence we have x y, Ax Ay 1 k 2 Ax Ay 2 0. This shows that if T is a strictly pseudocontractive mapping; then I T is a monotone and 2 -Lipschitz continuous mapping. Note that Fix(T)=VI(I T, C). Hence it is easy 1 k to obtain the following theorems. Theorem 4.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C C R satisfying (H1)-(H4). Let T be a k-strictly pseudocontractive mapping of C into H, and let S be a nonexpansive mapping of C into itself such that Fix(S) Fix(T) EP(F). Let {α n } and {β n } be two sequences in (0, 1), and let {λ n } and {r n } be

14 Yao et al. Journal of Inequalities and Applications (2015) 2015:198 Page 14 of 15 two sequences in (0, ). For fixed u Candgivenx 0 Carbitrarily, let the sequences {x n }, {y n }, and {u n } be generated iteratively by F(u n, y)+ 1 r n y u n, u n x n 0, y C, y n = P C (u n λ n (I T)u n ), x n+1 = α n u + β n x n +(1 α n β n )SP C (u n λ n (I T)y n ). (4.4) Suppose the following conditions are satisfied: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) lim n α n =0and n=0 α n = ; (c) lim sup n β n <1. Then the sequences {x n }, {y n }, and {u n } generated by (4.4) converge strongly to P Fix(S) Fix(T) EP(F) (u) if and only if lim n x n+1 x n =0. Theorem 4.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a k-strictly pseudocontractive mapping of C into H, and let S be a nonexpansive mapping of CintoitselfsuchthatFix(S) Fix(T). Let {α n } and {β n } be two sequences in (0, 1), and let {λ n } and {r n } be two sequences in (0, ). For fixed u Candgivenx 0 Carbitrarily, let the sequences {x n } and {y n } be generated iteratively by { y n = P C (x n λ n (I T)x n ), x n+1 = α n u + β n x n +(1 α n β n )SP C (x n λ n (I T)y n ). (4.5) Suppose the following conditions are satisfied: (a) {λ n k} (0, 1 δ) for some δ (0, 1); (b) lim n α n =0and n=0 α n = ; (c) lim sup n β n <1. Then the sequences {x n } and {y n } generated by (4.5) converge strongly to P Fix(S) Fix(T) (u) if and only if lim n x n+1 x n =0. Competing interests The authors declare that they have no competing interests. Authors contributions All authors read and approved the final manuscript. Author details 1 School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, , China. 2 Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan. 3 Center for General Education, Kaohsiung Medical University, Kaohsiung, 807, Taiwan. 4 School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, , China. Acknowledgements Yeong-Cheng Liou was supported in part by NSC E MY3 and NSC E MY3. Li-Jun Zhu was supported in part by NNSF of China ( ). This research is supported partially by Kaohsiung Medical University Aim for the Top Universities Grant, Grant No. KMU-TP103F00. Received: 13 February 2015 Accepted: 29 May 2015 References 1. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, (1994) 2. Noor, MA, Oettli, W: On general nonlinear complementarity problems and quasi equilibria. Matematiche 49, (1994) 3. Noor, MA, Noor, KI: On equilibrium problems. Appl. Math. E-Notes 4, (2004)

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