Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 232163, 15 pages doi:10.1155/2011/232163 Research Article A Hybrid-Extragradient Scheme for System of Equilibrium Problems, Nonexpansive Mappings, and Monotone Mappings Jian-Wen Peng, 1 Soon-Yi Wu, 2 and Gang-Lun Fan 2 1 School of Mathematics, Chongqing Normal University, Chongqing 400047, China 2 Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan Correspondence should be addressed to Jian-Wen Peng, jwpeng6@yahoo.com.cn Received 21 October 2010; Accepted 24 November 2010 Academic Editor: Jen Chih Yao Copyright q 2011 Jian-Wen Peng 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. We introduce a new iterative scheme based on both hybrid method and extragradient method for finding a common element of the solutions set of a system of equilibrium problems, the fixed points set of a nonexpansive mapping, and the solutions set of a variational inequality problems for a monotone and k-Lipschitz continuous mapping in a Hilbert space. Some convergence results for the iterative sequences generated by these processes are obtained. The results in this paper extend and improve some known results in the literature. 1. Introduction In this paper, we always assume that H is a real Hilbert space with inner product ·, · and induced norm ·and C is a nonempty closed convex subset of H, S : C → C is a nonexpansive mapping; that is, Sx − Sy≤x − y for all x,y ∈ C, P C denotes the metric projection of H onto C,andFixS denotes the fixed points set of S. Let {F k } k∈Γ be a countable family of bifunctions from C × C to R, where R is the set of real numbers. Combettes and Hirstoaga 1 introduced the following system of equilibrium problems: finding x ∈ C, such that ∀k ∈ Γ, ∀y ∈ C, F k  x, y  ≥ 0, 1.1 where Γ is an arbitrary index set. If Γ is a singleton, the problem 1.1 becomes the following equilibrium problem: finding x ∈ C, such that F  x, y  ≥ 0, ∀y ∈ C. 1.2 2 Fixed Point Theory and Applications The set of solutions of 1.2 is denoted by EPF. And it is easy to see that the set of solutions of 1.1 can be written as  k∈Γ EPF k . Given a mapping T : C → H,letFx, yTx,y − x for all x, y ∈ C. Then, the problem 1.2 becomes the following variational inequality: finding x ∈ C, such that  Tx,y − x  ≥ 0, ∀y ∈ C. 1.3 The set of solutions of 1.3 is denoted by VIC, A. The problem 1.1 is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance, 1–4. In 1953, Mann 5 introduced t he following iteration algorithm: let x 0 ∈ C be an arbitrary point, let {α n } be a real sequence in 0, 1, and let the sequence {x n } be defined by x n1  α n x n   1 − α n  Sx n . 1.4 Mann iteration algorithm has been extensively investigated for nonexpansive mappings, for example, please see 6, 7. Takahashi et al. 8 modified the Mann iteration method 1.4 and introduced the following hybrid projection algorithm: x 0 ∈ H, C 1  C, x 1  P C 1 x 0 , y n  α n x n   1 − α n  Sx n , C n1   z ∈ C n :   y n − z   ≤  x n − z   , x n1  P C n1 x 0 , ∀n ∈ N, 1.5 where 0 ≤ α n <a<1. They proved that the sequence {x n } generated by 1.5 converges strongly to P FixS x 0 . In 1976, Korpelevi ˇ c 9 introduced the following so-called extragradient algorithm: x 0  x ∈ C, y n  P C  x n − λAx n  , x n1  P C  x n − λAy n  1.6 for all n ≥ 0, where λ ∈ 0, 1/k, A is monotone and k-Lipschitz continuous mapping of C into R n . She proved that, if VIC, A is nonempty, the sequences {x n } and {y n }, generated by 1.6, converge to the same point z ∈ VIC, A. Some methods have been proposed to solve the problem 1.2; see, for instance, 10, 11 and the references therein. S. Takahashi and W. Takahashi 10 introduced the following iterative scheme by the viscosity approximation method for finding a common element of the Fixed Point Theory and Applications 3 set of the solution 1.2 and the set of fixed points of a nonexpansive mapping in a real Hilbert space: starting with an arbitrary initial x 1 ∈ C, 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, x n1  α n f  x n    1 − α n  Su n ,n≥ 1. 1.7 They proved that under certain appropriate conditions imposed on {α n } and {r n },the sequences {x n } and {u n } converge strongly to z ∈ FixS ∩ EPF, where z  P FixS∩EPF fz. Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E.Letf be a bifunction from C × C to R,andletS be a relatively nonexpansive mapping from C into itself such that FixS ∩ EPf /  ∅. Takahashi and Zembayashi 11 introduced the following hybrid method in Banach space: let {x n } be a sequence generated by x 0  x ∈ C, C 0  C,and y n  J −1  α n Jx n   1 − α n  JSx n  , u n ∈ C, such that f  u n ,y   1 r n  y − u n ,Ju n − Jy n  ≥ 0, ∀y ∈ C, C n1   z ∈ C n : φ  z, u n  ≤ φ  z, x n   , x n1   C n1 x 1.8 for every n ∈ N ∪{0}, where J is the duality napping on E, φx, yy 2 − 2y, Jx  x 2 for all x, y ∈ E,and  C x  arg min y∈C φy, x for all x ∈ E. They proved that the sequence {x n } generated by 1.8 converges strongly to  FixS∩EPf x if {α n }⊂0, 1 satisfies lim inf n →∞ α n 1 − α n  > 0and{r n }⊂a, ∞ for some a>0. On the other hand, Combettes and Hirstoaga 1 introduced an iterative scheme for finding a common element of the set of solutions of problem 1.1 in a Hilbert space and obtained a weak convergence theorem. Peng and Yao 4 introduced a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions of problem 1.1, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping in a Hilbert space and obtained two strong convergence theorems. Colao et al. 3 introduced an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. Peng et al. 12 introduced a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed points set of a family of infinitely nonexpansive mappings, and the solution set of a variational inequality for a relaxed coercive mapping in a Hilbert space and obtained a strong convergence theorem. In this paper, motivated by the above results, we introduce a new hybrid extragradient method to find a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of a nonexpansive mapping, and the set of solutions of the variational inequality for monotone and k-Lipschitz continuous mappings in a Hilbert space 4 Fixed Point Theory and Applications and obtain some strong convergence theorems. Our results unify, extend, and improve those corresponding results in 8, 11 and the references therein. 2. Preliminaries Let symbols → and  denote strong and weak convergence, respectively. It is well known that   λx   1 − λ  y   2  λ  x  2   1 − λ    y   2 − λ  1 − λ    x − y   2 2.1 for all x, y ∈ H and λ ∈ R. For any x ∈ H, there exists a unique nearest point in C denoted by P C x such that x − P C x≤x − y for all y ∈ C. The mapping P C is called the metric projection of H onto C. We know that P C is a nonexpansive mapping from H onto C. It is also known that P C x ∈ C and  x − P C  x  ,P C  x  − y  ≥ 0 2.2 for all x ∈ H and y ∈ C. It is easy to see that 2.2 is equivalent to   x − y   2 ≥  x − P C  x   2    y − P C  x    2 2.3 for all x ∈ H and y ∈ C. A mapping A of C into H is called monotone if Ax − Ay, x − y≥0 for all x, y ∈ C.A mapping A : C → H is called L-Lipschitz continuous if there exists a positive real number L such that Ax − Ay≤Lx − y for all x, y ∈ C. Let A be a monotone mapping of C into H. In the context of the variational inequality problem, the characterization of projection 2.2 implies the following: u ∈ VI  C, A  ⇒ u  P C  u − λAu  , ∀λ>0, u  P C  u − λAu  , for some λ>0 ⇒ u ∈ VI  C, A  . 2.4 For solving the equilibrium problem, let us assume that the bifunction F satisfies the following conditions which were imposed in 2: A1 Fx, x0 for all x ∈ C; A2 F is monotone; that is, Fx, yFy, x ≤ 0 for any x, y ∈ C; A3 for each x, y, z ∈ C, lim t↓0 F  tz   1 − t  x, y  ≤ F  x, y  ; 2.5 A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous. We recall some lemmas which will be needed in the rest of this paper. Fixed Point Theory and Applications 5 Lemma 2.1 See 2. Let C be a nonempty closed convex subset of H, and let F be a bifunction from C × C to R satisfying (A1)–(A4). Let r>0 and x ∈ H. Then, there exists z ∈ C such that F  z, y   1 r  y − z, z − x  ≥ 0, ∀y ∈ C. 2.6 Lemma 2.2 See 1. Let C be a nonempty closed convex subset of H, and let F be a bifunction from C × C to R satisfying (A1)–(A4). For r>0 and x ∈ H, define a mapping T F r : H → 2 C as follows: T F r  x    z ∈ C : F  z, y   1 r  y − z, z − x  ≥ 0, ∀y ∈ C  2.7 for all x ∈ H. Then, the following statements hold: 1 T F r is single-valued; 2 T F r is firmly nonexpansive; that is, for any x, y ∈ H,    T F r  x  − T F r  y     2 ≤  T F r  x  − T F r  y  ,x− y  ; 2.8 3 FixT F r EPF; 4 EP F is closed and convex. 3. Main Results In this section, we will introduce a new algorithm based on hybrid and extragradient method to find a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of a nonexpansive mapping, and the set of solutions of the variational inequality for monotone and k-Lipschitz continuous mappings in a Hilbert space and show that the sequences generated by the processes converge strongly to a same point. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H.LetF k , k ∈ {1, 2, ,M} be a family of bifunctions from C × C to R satisfying (A1)–(A4), let A be a monotone and k-Lipschitz continuous mapping of C into H, and let S be a nonexpansive mapping from C into itself such that ΩFixS ∩ VIC, A ∩   M k1 EPF k  /  ∅. Pick any x 0 ∈ H, and set C 1  C.Let {x n }, {y n }, {w n }, and {u n } be sequences generated by x 1  P C 1 x 0 and u n  T F M r M·n T F M−1 r M−1,n ···T F 2 r 2,n T F 1 r 1,n x n , y n  P C  u n − λ n Au n  , w n  α n x n   1 − α n  SP C  u n − λ n Ay n  , C n1  { z ∈ C n :  w n − z  ≤  x n − z  } , x n1  P C n1 x 0 3.1 6 Fixed Point Theory and Applications for each n ∈ N.If{λ n }⊂a, b for some a, b ∈ 0, 1/k, {α n }⊂c, d for some c, d ∈ 0, 1, and {r k,n }⊂0, ∞ satisfies lim inf n →∞ r k,n > 0 for each k ∈{1, 2, ,M},then{x n }, {u n }, {y n }, and {w n } generated by 3.1 converge strongly to P Ω x 0 . Proof. It is obvious that C n is closed for each n ∈ N. Since C n1   z ∈ C n :  w n − x n  2  2  w n − x n ,x n − z  ≤ 0  , 3.2 we also have that C n is convex for each n ∈ N.Thus,{x n }, {u n }, {y n },and{w n } are welldefined. By taking Θ k n  T F k r k·n T F k−1 r k−1,n ···T F 2 r 2,n T F 1 r 1,n for k ∈{1, 2, ,M} and n ∈ N, Θ 0 n  I for each n ∈ N, where I is the identity mapping on H. Then, it is easy to see that u n Θ M n x n . We divide the proof into several steps. Step 1. We show by induction that Ω ⊂ C n for n ∈ N. It is obvious that Ω ⊂ C  C 1 .Suppose that Ω ⊂ C n for some n ∈ N.Letv ∈ Ω. Then, by Lemma 2.2 and v  P C v − λ n AvΘ M n v, we have  u n − v      Θ M n x n − Θ M n v    ≤  x n − v  , ∀n ∈ N. 3.3 Putting v n  P C u n − λ n Ay n  for each n ∈ N,from2.3 and the monotonicity of A, we have  v n − v  2 ≤   u n − λ n Ay n − v   2 −   u n − λ n Ay n − v n   2   u n − v  2 −  u n − v n  2  2λ n  Ay n ,v− v n    u n − v  2 −  u n − v n  2  2λ n  Ay n − Av, v − y n    Av, v − y n    Ay n ,y n − v n  ≤  u n − v  2 −  u n − v n  2  2λ n  Ay n ,y n − v n    u n − v  2 −   u n − y n   2 − 2  u n − y n ,y n − v n  −   y n − v n   2  2λ n  Ay n ,y n − v n    u n − v  2 −   u n − y n   2 −   y n − v n   2  2  u n − λ n Ay n − y n ,v n − y n  . 3.4 Moreover, from y n  P C u n − λ n Au n  and 2.2, we have  u n − λ n Au n − y n ,v n − y n  ≤ 0. 3.5 Fixed Point Theory and Applications 7 Since A is k-Lipschitz continuous, it follows that  u n − λ n Ay n − y n ,v n − y n    u n − λ n Au n − y n ,v n − y n    λ n Au n − λ n Ay n ,v n − y n  ≤  λ n Au n − λ n Ay n ,v n − y n  ≤ λ n k   u n − y n     v n − y n   . 3.6 So, we have  v n − v  2 ≤  u n − v  2 −   u n − y n   2 −   y n − v n   2  2λ n k   u n − y n     v n − y n   ≤  u n − v  2 −   u n − y n   2 −   y n − v n   2  λ 2 n k 2   u n − y n   2    v n − y n   2   u n − v  2   λ 2 n k 2 − 1    u n − y n   2 ≤  u n − v  2 . 3.7 From 3.7 and the definition of w n , we have  w n − v  2 ≤ α n  x n − v  2   1 − α n   Sv n − v  2 ≤ α n  x n − v  2   1 − α n   v n − v  2 ≤ α n  x n − v  2   1 − α n    u n − v  2   λ 2 n k 2 − 1    u n − y n   2  3.8 ≤ α n  x n − v  2   1 − α n   x n − v  2   1 − α n   λ 2 n k 2 − 1    u n − y n   2   x n − v  2   1 − α n   λ 2 n k 2 − 1    u n − y n   2 ≤  x n − v  2 , 3.9 and hence v ∈ C n1 . This implies that Ω ⊂ C n for all n ∈ N. Step 2. We show that lim n →∞ x n − w n →0 and lim n →∞ u n − y n   0. Let l 0  P Ω x 0 .Fromx n  P C n x 0 and l 0 ∈ Ω ⊂ C n , we have  x n − x 0  ≤  l 0 − x 0  , ∀n ∈ N. 3.10 Therefore, {x n } is bounded. From 3.3–3.9, we also obtain that {w n }, {v n },and{u n } are bounded. Since x n1 ∈ C n1 ⊆ C n and x n  P C n x 0 , we have  x n − x 0  ≤  x n1 − x 0  , ∀n ∈ N. 3.11 Therefore, lim n →∞ x n − x 0  exists. 8 Fixed Point Theory and Applications 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, ∀n ∈ N. 3.12 So  x n − x n1  2    x n − x 0    x 0 − x n1   2   x n − x 0  2  2  x n − x 0 ,x 0 − x n1    x 0 − x n1  2   x n − x 0  2  2  x n − x 0 ,x 0 − x n  x n − x n1    x 0 − x n1  2   x n − x 0  2 − 2  x 0 − x n ,x 0 − x n  − 2  x 0 − x n ,x n − x n1    x 0 − x n1  2 ≤x n − x 0  2 − 2x n − x 0  2  x 0 − x n1  2  −  x n − x 0  2   x 0 − x n1  2 , 3.13 which implies that lim n →∞  x n1 − x n   0. 3.14 Since x n1 ∈ C n1 , we have w n − x n1 ≤x n − x n1 , and hence  x n − w n  ≤  x n − x n1    x n1 − w n  ≤ 2  x n − x n1  , ∀n ∈ N. 3.15 It follows from 3.14 that x n − w n →0. For v ∈ Ω, it follows from 3.9 that   u n − y n   2 ≤ 1  1 − α n   1 − λ 2 n k 2    x n − v  2 −  w n − v  2   1  1 − α n   1 − λ 2 n k 2    x n − v  −  w n − v    x n − v    w n − v   ≤ 1  1 − α n   1 − λ 2 n k 2   x n − w n    x n − v    w n − v   , 3.16 which implies that lim n →∞ u n − y n   0. Step 3. We now show that lim n →∞    Θ k n x n − Θ k−1 n x n     0,k 1, 2, ,M. 3.17 Fixed Point Theory and Applications 9 Indeed, let v ∈ Ω. It follows form the firmly nonexpansiveness of T F k r k,n that we have, for each k ∈{1, 2, ,M},    Θ k n x n − v    2     T F k r k,n Θ k−1 n x n − T F k r k,n v    2 ≤  Θ k n x n − v, Θ k−1 n x n − v   1 2     Θ k n x n − v    2     Θ k−1 n x n − v    2 −    Θ k n x n − Θ k−1 n x n    2  . 3.18 Thus, we get    Θ k n x n − v    2 ≤    Θ k−1 n x n − v    2 −    Θ k n x n − Θ k−1 n x n    2 ,k 1, 2, ,M, 3.19 which implies that, for each k ∈{1, 2, ,M},    Θ k n x n − v    2 ≤    Θ 0 n x n − v    2 −    Θ k n x n − Θ k−1 n x n    2 −    Θ k−1 n x n − Θ k−2 n x n    2 −···−    Θ 2 n x n − Θ 1 n x n    2 −    Θ 1 n x n − Θ 0 n x n    2 ≤  x n − v  2 −    Θ k n x n − Θ k−1 n x n    2 . 3.20 By 3.8, u n Θ M n x n ,and3.20, we have, for each k ∈{1, 2, ,M},  w n − v  2 ≤ α n  x n − v  2   1 − α n   u n − v  2 ≤ α n  x n − v  2   1 − α n     Θ k n x n − v    2 , ∀k ∈ { 1, 2, ,M } ≤ α n  x n − v  2   1 − α n    x n − v  2 −    Θ k n x n − Θ k−1 n x n    2  ≤  x n − v  2 −  1 − α n     Θ k n x n − Θ k−1 n x n    2 , 3.21 which implies that  1 − α n     Θ k n x n − Θ k−1 x n    ≤  x n − v  2 −  w n − v  2    x n − v    w n − v    x n − v  −  w n − v   ≤   x n − v    w n − v    x n − w n  . 3.22 It follows from x n − w n →0and0<c≤ α n ≤ d<1that3.17 holds. Step 4. We now show that lim n →∞ Sv n − v n   0. 10 Fixed Point Theory and Applications It follows from 3.17 that x n − u n →0. Since x n − y n ≤x n − u n   u n − y n ,we get lim n →∞   x n − y n    0. 3.23 We observe that   v n − y n      P C  u n − λ n Ay n  − P C  u n − λ n Au n    ≤   λ n Au n − λ n Ay n   ≤ λ n k   u n − y n   , 3.24 which implies that lim n →∞   v n − y n    0. 3.25 Since x n − w n   x n − α n x n − 1 − α n Sv n   1 − α n x n − Sv n ,weobtain lim n →∞  x n − Sv n   0. 3.26 Since Sv n − v n ≤Sv n − x n   x n − y n   y n − v n ,weget lim n →∞  Sv n − v n   0. 3.27 Step 5. We show that x n → w, where w  P Ω x 0 . As {x n } is bounded, there exists a subsequence {x n i } which converges weakly to w. From Θ k n x n − Θ k−1 n x n →0 for each k  1, 2, ,M,weobtainthatΘ k n i x n i wfor k  1, 2, ,M. It follows from x n − w n →0, v n − y n →0, and u n − y n →0thatw n i w, y n i w,andv n i w. In order to show that w ∈ Ω, we first show that w ∈  M k1 EPF k . Indeed, by definition of T F k r k,n , we have that, for each k ∈{1, 2, ,M}, F k  Θ k n x n ,y   1 r k,n  y − Θ k n x n , Θ k n x n − Θ k−1 n x n  ≥ 0, ∀y ∈ C. 3.28 From A2, we also have 1 r k,n  y − Θ k n x n , Θ k n x n − Θ k−1 n x n  ≥ F k  y, Θ k n x n  , ∀y ∈ C. 3.29 [...]... 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The proof is now complete By Theorem 3.1, we can easily obtain some new results as follows Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H Let F be a bifunction from C × C to R satisfying (A1 )– (A4 ), let A be a monotone and k-Lipschitz continuous mapping of C into H, and let S be a nonexpansive mapping from C into itself such that Ω Fixed Point Theory and Applications... mappings and monotone mappings, Nonlinear Analysis: Theory, Methods & Applications, vol 71, no 12, pp 6001–6010, 2009 5 W R Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol 4, pp 506–510, 1953 6 A Genel and J Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol 22, no 1, pp 81–86, 1975 7 S Reich, “Weak convergence theorems for. .. the proof Corollary 3.4 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 C Let from C into itself such that Ω Fix S ∩ VI C, A / ∅ Pick any x0 ∈ H, and set C1 {xn }, {yn }, and {wn } be sequences generated by x1 PC1 x0 and yn αn xn wn Cn 1 PC xn − λn Axn , 1 − αn SPC un − λn Ayn , . element of the solutions set of a system of equilibrium problems, the fixed points set of a nonexpansive mapping, and the solutions set of a variational inequality problems for a monotone and k-Lipschitz. W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications,. Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975. 7 S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications,