Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 134148, 17 pages doi:10.1155/2008/134148 Research Article An Extragradient Approximation Method for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings Rabian Wangkeeree Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Rabian Wangkeeree, rabianw@nu.ac.th Received 28 February 2008; Accepted 13 July 2008 Recommended by Huang Nanjing We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al. 2007 and many others. Copyright q 2008 Rabian Wangkeeree. 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. 1. Introduction Let H be a real Hilbert space with inner product ·, · and norm ·,andletC be a closed convex subset of H.LetF be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for φ : C × C → R is to find x ∈ C such that φx, y ≥ 0 ∀y ∈ C. 1.1 The set of solutions of 1.1 is denoted by EPφ. Given a mapping T : C → H,letφx, y Tx,y − x for all x, y ∈ C. Then z ∈ EPφ if and only if Tz,y − z≥0 for all y ∈ C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of 1.1. In 1997, Fl ˚ am and Antipin 1 introduced an iterative scheme of finding the best approximation to initial data when EPφ is nonempty and proved a strong convergence theorem. Let A : C → H be a mapping. The classical variational inequality, denoted by VIA, C, is to find x ∗ ∈ C such that  Ax ∗ ,v− x ∗  ≥ 0 1.2 2 Fixed Point Theory and Applications for all v ∈ C. The variational inequality has been extensively studied in the literature. See, for example, 2, 3 and the references therein. A mapping A of C into H is called α-inverse- strongly monotone 4, 5 if there exists a positive real number α such that Au − Av, u − v≥αAu − Av 2 1.3 for all u, v ∈ C. It is obvious that any α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. A mapping S of C into itself is called nonexpansive if Su − Sv≤u − v 1.4 for all u, v ∈ C. We denote by FS the set of fixed points of S. For finding an element of FS ∩ VIA, C, under the assumption that a set C ⊆ H is nonempty, closed, and convex, a mapping S : C → C is nonexpansive and a mapping A : C → H is α-inverse-strongly monotone, Takahashi and Toyoda 6 introduced the following iterative scheme: x n1  α n x n   1 − α n  SP C  x n − λ n Ax n  1.5 for every n  0, 1, 2, ,where x 0  x ∈ C, {α n } is a sequence in 0, 1,and{λ n } is a sequence in 0, 2α. They proved that if FS ∩ VIA, C /  ∅, then the sequence {x n } generated by 1.5 converges weakly to some z ∈ FS∩VIA, C. Recently, motivated by the idea of Korpelevi ˇ c’s extragradient method 7, Nadezhkina and Takahashi 8 introduced an iterative scheme for finding an element of FS ∩ VIA, C and the weak convergence theorem is presented. Moreover, Zeng and Yao 9 proposed some new iterative schemes for finding elements in FS ∩ VIA, C and obtained the weak convergence theorem for such schemes. Very recently, Yao et al. 10 introduced the following iterative scheme for finding an element of FS ∩VIA, C under some mild conditions. Let C be a closed convex subset of a real Hilbert space H, A : C → H a monotone, L-Lipschitz continuous mapping, and S a nonexpansive mapping of C into itself such that FS∩VIA, C /  ∅. Suppose that x 1  u ∈ C and {x n }, {y n } are given by y n  P C  x n − λ n Ax n  , x n1  α n u  β n x n  γ n SP C  x n − λ n Ay n  ∀n ∈ N, 1.6 where {α n }, {β n }, {γ n }⊆0, 1 and {λ n }⊆0, 1 satisfy some parameters controlling conditions. They proved that the sequence {x n } defined by 1.6 converges strongly to a common element of FS ∩ VIA, C. On the other hand, S. Takahashi and W. Takahashi 11 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution 1.1 and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let S : C → C be a nonexpansive mapping. Starting with arbitrary initial x 1 ∈ C, define sequences {x n } and {u n } recursively by φ  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 ∈ N. 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 ∈ FS ∩ EPφ, where z  P FS∩EPφ fz. Rabian Wangkeeree 3 Moreover, Aoyama et al. 12 introduced an iterative scheme for finding a common fixed point of a countable family of nonexpansive mappings in Banach spaces and obtained the strong convergence theorem for such scheme. In this paper, motivated by Yao et al. 10, S. Takahashi and W. Takahashi 11 and Aoyama et al. 12, we introduce a new extragradient method 4.2 which is mixed the iterative schemes considered in 10–12 for finding a common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the solution set of the classical variational inequality problem for a monotone L-Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Further, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. The results obtained in this paper improve and extend the recent ones announced by Yao et al. results 10 and many others. 2. Preliminaries Let H be a real Hilbert space with norm · and inner product ·, · and let C be a closed convex subset of H. For every point x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that   x − P C x   ≤x − y∀y ∈ C. 2.1 P C is called the metric projection of H onto C. It is well known that P C is a nonexpansive mapping of H onto C and satisfies  x − y, P C x − P C y  ≥   P C x − P C y   2 2.2 for every x, y ∈ H. Moreover, P C x is characterized by the following properties: P C x ∈ C and  x − P C x, y − P C x  ≤ 0, 2.3 x − y 2 ≥   x − P C x   2    y − P C x   2 2.4 for all x ∈ H, y ∈ C. For more details, see 13. It is easy to see that the following is true: u ∈ VIA, C ⇐⇒ u  P C u − λAu,λ>0. 2.5 A set-valued mapping T : H → 2 H is called monotone if for all x, y ∈ H, f ∈ Tx, and g ∈ Ty imply x − y, f − g≥0. A monotone mapping T : H → 2 H is maximal if the graph of GT of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for x, f ∈ H × H, x − y, f − g≥0 for every y, g ∈ GT implies f ∈ Tx.LetB be a monotone map of C into H, L-Lipschitz continuous mapping and let N C v be the normal cone to C at v ∈ C,thatis,N C v  {w ∈ H : u − v, w≥0 for all u ∈ C}. Define Tv  ⎧ ⎨ ⎩ Bv  N C v, v ∈ C; ∅,v / ∈ C. 2.6 Then T is the maximal monotone and 0 ∈ Tv if and only if v ∈ VIC, B;see14. 4 Fixed Point Theory and Applications The following lemmas will be useful for proving the convergence result of this paper. Lemma 2.1 see 15. Let E, ·, · be an inner product space. Then for all x, y, z ∈ E and α, β, γ ∈ 0, 1 with α  β  γ  1,one has αx  βy  γz 2  αx 2  βy 2  γz 2 − αβx − y 2 − αγx − z 2 − βγy − z 2 . 2.7 Lemma 2.2 see 16. Let {x n } and {z n } be bounded sequences in a Banach space E and let {β n } be a sequence in 0, 1 with 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1. Suppose x n1 1 − β n z n  β n x n for all integers n ≥ 1 and lim sup n→∞ z n1 − z n −x n1 − x n  ≤ 0. Then, lim n→∞ z n − x n   0. Lemma 2.3 see 17. Assume {a n } is a sequence of nonnegative real numbers such that a n1 ≤  1 − α n  a n  δ n ,n≥ 1, 2.8 where {α n } is a sequence in 0, 1 and {δ n } is a sequence in R suchthat i  ∞ n1 α n  ∞ and ii lim sup n→∞ δ n /α n  ≤ 0 or  ∞ n1 |δ n | < ∞. Then lim n→∞ a n  0. Lemma 2.4 see 12, Lemma 3.2. Let C be a nonempty closed subset of a Banach space and let {S n } be a sequence of mappings of C into itself. Suppose that  ∞ n1 sup{S n1 z− S n z : z ∈ C} < ∞. Then, for each y ∈ C, {S n y} converges strongly to some point of C. Moreover, let S be a mapping of C into itself defined by Sy  lim n→∞ S n y ∀y ∈ C. 2.9 Then lim n→∞ sup{Sz − S n z : z ∈ C}  0. For solving the equilibrium problem for a bifunction φ : C × C → R, let us assume that φ satisfies the following conditions: A1 φx, x0 for all x ∈ C; A2 φ is monotone, that is, φx, yφy, x ≤ 0 for all x, y ∈ C; A3 for each x, y, z ∈ C, lim t→0 φtz 1 − tx, y ≤ φx, y; A4 for each x ∈ C, y → φx, y is convex and lower semicontinuous. The following lemma appears implicitly in 18. Lemma 2.5 see 18. Let C be a nonempty closed convex subset of H and let φ be a bifunction of C × C into R satisfying (A1)–(A4). Let r>0 and x ∈ H. Then, there exists z ∈ C such that φz, y 1 r y − z, z − x≥0 ∀y ∈ C. 2.10 The following lemma was also given in 1. Rabian Wangkeeree 5 Lemma 2.6 see 1. Assume that φ : C × C → R satisfies (A1)–(A4). For r>0 and x ∈ H, define a mapping T r : H → C as follows: T r x  z ∈ C : φz, y 1 r y − z, z − x≥0 ∀y ∈ C  2.11 for all z ∈ H. Then, the following hold: i T r is single-valued; ii T r is firmly nonexpansive, that is, for any x, y ∈ H, T r x − T r y 2 ≤T r x − T r y, x − y; iii FT r EPφ; iv EPφ is closed and convex. 3. Main results In this section, we prove a strong convergence theorem. Theorem 3.1. Let C be a closed convex subset of a real Hilbert space H.Letφ be a bifunction from C× C to R satisfying (A1)–(A4), A : C → H a monotone L-Lipschitz continuous mapping and let {S n } be a sequence of nonexpansive mappings of C into itself such that ∩ ∞ n1 FS n  ∩ VIA, C ∩ EPφ /  ∅. Let the sequences {x n }, {u n }, and {y n } be generated by x 1  x ∈ C chosen arbitrarily, φ  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  , x n1  α n f  x n   β n x n  γ n S n P C  u n − λ n Ay n  ∀n ≥ 1, 3.1 where {α n }, {β n }, {γ n }⊆0, 1, {λ n }⊆0, 1, and {r n }⊆0, ∞ satisfy the following conditions: C1 α n  β n  γ n  1, C2 lim n→∞ α n  0,  ∞ n1 α n  ∞, C3 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1, C4 lim n→∞ λ n  0, C5 lim inf n→∞ r n > 0,  ∞ n1 |r n1 − r n | < ∞. Suppose that  ∞ n1 sup{S n1 z − S n z : z ∈ B} < ∞ for any bounded subset B of C.Let S be a mapping of C into itself defined by Sy  lim n→∞ S n y for all y ∈ C and suppose that FS∩ ∞ n1 FS n . Then the sequences {x n }, {u n }, and {y n } converge strongly to the same point q ∈∩ ∞ n1 FS n  ∩ VIA, C ∩ EPφ,whereq  P ∩ ∞ n1 FS n ∩VIA,C∩EPφ fq. Proof. Let Q  P ∩ ∞ n1 FS n ∩VIA,C∩EPφ . Since f is a contraction with α ∈ 0, 1,weobtain   Qfx − Qfy   ≤   fx − fy   ≤ αx − y∀x, y ∈ C. 3.2 Therefore, Qf is a contraction of C into itself, which implies that there exists a unique element q ∈ C such that q  Qfq. Then we divide the proof into several steps. 6 Fixed Point Theory and Applications Step 1 {x n } is bounded. Indeed, put t n  P C u n − λ n Ay n  for all n ≥ 1. Let x ∗ ∈∩ ∞ n1 FS n  ∩ VIA, C ∩ EPφ.From2.5 we have x ∗  P C x ∗ − λ n Ax ∗ . Also it follows from 2.4 that   t n − x ∗   2 ≤   u n − λ n Ay n − x ∗   2 −   u n − λ n Ay n − t n   2    u n − x ∗   2 − 2λ n  Ay n ,u n − x ∗   λ 2 n   Ay n   2 −   u n − t n   2  2λ n  Ay n ,u n − t n  − λ 2 n   Ay n   2    u n − x ∗   2  2λ n  Ay n ,x ∗ − t n  −   u n − t n   2    u n − x ∗   2 −   u n − t n   2  2λ n  Ay n − Ax ∗ ,x ∗ − y n   2λ n  Ax ∗ ,x ∗ − y n   2λ n  Ay n ,y n − t n  . 3.3 Since A is monotone and x ∗ is a solution of the variational inequality problem VIA, C,we have  Ay n − Ax ∗ ,x ∗ − y n  ≤ 0,  Ax ∗ ,x ∗ − y n  ≤ 0. 3.4 This together with 3.3 implies that   t n − x ∗   2 ≤   u n − x ∗   2 −   u n − t n   2  2λ n  Ay n ,y n − t n     u n − x ∗   2 −   u n − y n y n − t n    2  2λ n  Ay n ,y n − t n     u n − x ∗   2 −   u n − y n   2 − 2  u n − y n ,y n − t n  −   y n − t n   2  2λ n  Ay n ,y n − t n     u n − x ∗   2 −   u n − y n   2 −   y n − t n   2  2  u n − λ n Ay n − y n ,t n − y n  . 3.5 From 2.3, we have  u n − λ n Au n − y n ,t n − y n  ≤ 0, 3.6 so that  u n − λ n Ay n − y n ,t n − y n    u n − λ n Au n − y n ,t n − y n    λ n Au n − λ n Ay n ,t n − y n  ≤  λ n Au n − λ n Ay n ,t n − y n  ≤ λ n   Au n − Ay n     t n − y n   ≤ λ n L   u n − y n     t n − y n   . 3.7 Hence it follows from 3.5 and 3.7 that   t n − x ∗   2 ≤   u n − x ∗   2 −   u n − y n   2 −   y n − t n   2  2λ n L   u n − y n     t n − y n   ≤   u n − x ∗   2 −   u n − y n   2 −   y n − t n   2  λ n L    u n − y n   2    y n − t n   2   u n − x ∗  2   λ n L − 1    u n − y n   2   λ n L − 1    y n − t n   2 . 3.8 Rabian Wangkeeree 7 Since λ n → 0asn →∞, there exists a positive integer N 0 such that λ n L − 1 ≤−1/3, when n ≥ N 0 . Hence it follows from 3.8 that   t n − x ∗   ≤   u n − x ∗   . 3.9 Observe that   u n − x ∗      T r n x n − T r n x ∗   ≤   x n − x ∗   , 3.10 and hence   t n − x ∗   ≤   x n − x ∗   . 3.11 Thus, we can calculate   x n1 − x ∗      α n f  x n   β n x n  γ n S n t n − x ∗   ≤ α n   f  x n  − x ∗    β n   x n − x ∗    γ n   t n − x ∗   ≤ α n   f  x n  − f  x ∗     α n   f  x ∗  − x ∗    β n   x n − x ∗    γ n   x n − x ∗   ≤  1 − α n 1 − α    x n − x ∗    α n   f  x ∗  − x ∗     1 − α n 1 − α    x n − x ∗    α n 1 − α   f  x ∗  − x ∗   1 − α . 3.12 It follows from induction that   x n − x ∗   ≤ max    x 1 − x ∗   ,   f  x ∗  − x ∗   1 − α  ,n≥ N 0 . 3.13 Therefore, {x n } is bounded. Hence, so are {t n }, {S n t n }, {Au n }, {Ay n },and{fx n }. Step 2 lim n→∞ x n1 − x n   0. Indeed, we observe that for any x, y ∈ C,    I − λ n A  x −  I − λ n A  y   2    x − y − λ n Ax − Ay   2  x − y 2 − 2λ n x − y, Ax − Ay  λ 2 n Ax − Ay 2 ≤x − y 2  λ 2 n L 2 x − y 2   1  λ 2 n L 2  x − y 2 , 3.14 which implies that    I − λ n A  x −  I − λ n A  y   ≤  1  λ n L  x − y. 3.15 Thus   t n1 − t n   ≤   P C  u n1 − λ n1 Ay n1  − P C  u n − λ n Ay n    ≤   u n1 − λ n1 Ay n1 −  u n − λ n Ay n        u n1 − λ n1 Au n1  −  u n − λ n1 Au n   λ n1  Au n1 − Ay n1 − Au n   λ n Ay n   ≤    u n1 − λ n1 Au n1  −  u n − λ n1 Au n     λ n1    Au n1      Ay n1      Au n     λ n   Ay n   ≤  1  λ n1 L    u n1 − u n    λ n1    Au n1      Ay n1      Au n     λ n   Ay n   . 3.16 8 Fixed Point Theory and Applications On the other hand, from u n  T r n x n and u n1  T r n1 x n1 , we note that φ  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0 ∀y ∈ C , 3.17 φ  u n1 ,y   1 r n1  y − u n1 ,u n1 − x n1  ≥ 0 ∀y ∈ C. 3.18 Putting y  u n1 in 3.17 and y  u n in 3.18, we have φ  u n ,u n1   1 r n  u n1 − u n ,u n − x n  ≥ 0, φ  u n1 ,u n   1 r n1  u n − u n1 ,u n1 − x n1  ≥ 0. 3.19 So, from A2, we have  u n1 − u n , u n − x n r n − u n1 − x n1 r n1  ≥ 0 3.20 and hence  u n1 − u n ,u n − u n1  u n1 − x n − r n r n1  u n1 − x n1   ≥ 0. 3.21 Without loss of generality, let us assume that there exists a real number c such that r n >c>0 for all n ∈ N. Then, we have   u n1 − u n   2 ≤  u n1 − u n ,x n1 − x n   1 − r n r n1   u n1 − x n1   ≤   u n1 − u n      x n1 − x n        1 − r n r n1       u n1 − x n1    3.22 and hence   u n1 − u n   ≤   x n1 − x n    1 r n1   r n1 − r n     u n1 − x n1   ≤   x n1 − x n    1 c   r n1 − r n   M, 3.23 where M  sup{u n − x n  : n ∈ N}. It follows from 3.16 and the last inequality that   t n1 − t n   ≤  1  λ n1 L    x n1 − x n     1  λ n1 L  1 c   r n1 − r n   M  λ n1    Au n1      Ay n1      Au n     λ n Ay n . 3.24 Rabian Wangkeeree 9 Setting z n α n fx n γ n S n t n /1 − β n ,weobtainx n1 1− β n z n  β n x n for all n ∈ N.Thus, we have   z n1 − z n        α n1 f  x n1   γ n1 S n1 t n1 1 − β n1 − α n f  x n   γ n S n t n 1 − β n          α n1 1 − β n1 f  x n1   γ n1 1 − β n1  S n1 t n1 − S n t n  − α n 1 − β n f  x n  −  1 − α n 1 − β n  S n t n   1 − α n1 1 − β n1 S n t n      ≤ α n1 1 − β n1   f  x n1  − S n t n    α n 1 − β n   S n t n − f  x n     γ n1 1 − β n1   S n1 t n1 − S n t n   . 3.25 It follows from 3.24 that   S n1 t n1 − S n t n   ≤   S n1 t n1 − S n1 t n      S n1 t n − S n t n   ≤   t n1 − t n      S n1 t n − S n t n   ≤  1  λ n1 L    x n1 − x n     1  λ n1 L  1 c   r n1 − r n   M  λ n1    Au n1      Ay n1      Au n     λ n   Ay n      S n1 t n − S n t n   . 3.26 Combining 3.25 and 3.26, we have   z n1 −z n   −   x n1 −x n   ≤ α n1 1 − β n1   f  x n1  − S n t n    α n 1 − β n   S n t n − f  x n     γ n1 1 − β n1  1λ n1 L    x n1 −x n    γ n1 1 − β n1  1λ n1 L  1 c |r n1 −r n |M  γ n1 1 − β n1 λ n1    Au n1      Ay n1      Au n     γ n1 1 − β n1 λ n   Ay n    γ n1 1 − β n1   S n1 t n − S n t n   −   x n1 − x n   ≤ α n1 1 − β n1   f  x n1  − S n t n    α n 1 − β n   S n t n − f  x n     γ n1 1 − β n1 λ n1 Lx n1 − x n   γ n1 1 − β n1  1  λ n1 L  1 c |r n1 − r n |M  γ n1 1 − β n1 λ n1    Au n1      Ay n1      Au n     γ n1 1 − β n1 λ n   Ay n    γ n1 1 − β n1 sup    S n1 t − S n t   : t ∈  t n  . 3.27 10 Fixed Point Theory and Applications This together with C1–C5 and lim n→∞ sup{S n1 t − S n t : t ∈{t n }}  0 implies that lim sup n→∞    z n1 − z n   −   x n1 − x n    ≤ 0. 3.28 Hence, by Lemma 2.2,weobtainz n − x n →0asn →∞. It then follows that lim n→∞   x n1 − x n    lim n→∞  1 − β n    z n − x n    0. 3.29 By 3.23 and 3.24, we also have lim n→∞   t n1 − t n    lim n→∞   u n1 − u n    0. 3.30 Step 3 lim n→∞ St n − t n   0. Indeed, pick any x ∗ ∈∩ ∞ n1 FS n  ∩ VIA, C ∩ EPφ,toobtain   u n − x ∗   2    T r n x n − T r n x ∗   2 ≤  T r n x n − T r n x ∗ ,x n − x ∗    u n − x ∗ ,x n − x ∗   1 2    u n − x ∗   2    x n − x ∗   2 −   x n − u n   2  . 3.31 Therefore, u n − x ∗  2 ≤x n − x ∗  2 −x n − u n  2 .FromLemma 2.1 and 3.9, we obtain, when n ≥ N 0 ,that   x n1 − x ∗   2    α n f  x n   β n x n  γ n S n t n − x ∗   2 ≤ α n   f  x n  − x ∗   2  β n   x n − x ∗   2  γ n   S n t n − x ∗   2 ≤ α n   f  x n  − x ∗   2  β n   x n − x ∗   2  γ n   t n − x ∗   2 ≤ α n   f  x n  − x ∗   2  β n   x n − x ∗   2  γ n   u n − x ∗   2 ≤ α n   f  x n  − x ∗   2  β n   x n − x ∗   2  γ n    x n − x ∗   2 −   x n − u n   2  ≤ α n   f  x n  − x ∗   2   1 − α n    x n − x ∗   2 − γ n   x n − u n   2 3.32 and hence γ n   x n − u n   2 ≤ α n   f  x n  − x ∗   2    x n − x ∗   2 −   x n1 − x ∗   2 ≤ α n   f  x n  − x ∗   2    x n − x n1      x n − x ∗      x n1 − x ∗    . 3.33 It now follows from the last inequality, C1, C2, C3 and 3.29,that lim n→∞   x n − u n    0. 3.34 Noting that   y n − x n      P C  u n − λ n Au n  − x n   ≤   u n − x n    λ n   Au n   −→ 0asn −→ ∞ ,   y n − t n      P C  u n − λ n Au n  − P C  u n − λ n Ay n    ≤ λ n   Au n − Ay n   −→ 0asn −→ ∞ . 3.35 [...]... Yao, An extragradient method for fixed point problems and variational inequality problems, ” Journal of Inequalities and Applications, vol 2007, Article ID 38752, 12 pages, 2007 11 S Takahashi and W Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 331, no 1, pp 506–515, 2007 Rabian Wangkeeree... 17 12 K Aoyama, Y Kimura, W Takahashi, and M Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol 67, no 8, pp 2350–2360, 2007 13 W Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000 14 R T Rockafellar, “On the maximality of sums of nonlinear monotone... by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol 128, no 1, pp 191–201, 2006 9 L.-C Zeng and J.-C Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, ” Taiwanese Journal of Mathematics, vol 10, no 5, pp 1293–1303, 2006 10 Y Yao, Y.-C Liou, and J.-C Yao,... sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol 305, no 1, pp 227–239, 2005 17 H.-K Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol 298, no 1, pp 279–291, 2004 18 E Blum and W Oettli, “From optimization and variational inequalities to equilibrium problems, ”... 1998 6 W Takahashi and M Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol 118, no 2, pp 417–428, 2003 7 G M Korpeleviˇ , An extragradient method for finding saddle points and for other problems, ” c ` Ekonomika i Matematicheskie Metody, vol 12, no 4, pp 747–756, 1976 8 N Nadezhkina and W Takahashi, “Weak convergence... Journal of Optimization Theory and Applications, vol 124, no 3, pp 725–738, 2005 4 F E Browder and W V Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol 20, no 2, pp 197–228, 1967 5 F Liu and M Z Nashed, “Regularization of nonlinear ill-posed variational inequalities and convergence rates,” Set-Valued Analysis, vol 6,... valuable suggestions and comments, and pointing out a major error in the original version of this paper This research was partially supported by the Commission on Higher Education References 1 S D Fl˚ m and A S Antipin, Equilibrium programming using proximal-like algorithms,” a Mathematical Programming, vol 78, no 1, pp 29–41, 1997 2 J.-C Yao and O Chadli, “Pseudomonotone complementarity problems and. .. and variational inequalities,” in Handbook of Generalized Convexity and Generalized Monotonicity, N Hadjisavvas, S Komlosi, and S ´ Schaible, Eds., vol 76 of Nonconvex Optimization and Its Applications, pp 501–558, Springer, New York, NY, USA, 2005 3 L C Zeng, S Schaible, and J C Yao, “Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities,” Journal of Optimization... ∞ for any bounded subset B of C n by using convex combination of a general sequence {Tk } of nonexpansive mappings with a common fixed point Corollary 3.2 Let C be a closed convex subset of a real Hilbert space H Let φ be a bifunction from C × C to R satisfying (A1 )– (A4 ), A : C → H a monotone, L-Lipschitz continuous mapping and let 14 Fixed Point Theory and Applications k {βn } be a family of nonnegative... the graph of any other monotone operator Let I denote the identity operator on H and let S : H → 2H be a maximal monotone operator Then we can define, for each r > 0, a nonexpansive single-valued I rS −1 It is called the resolvent or the proximal mapping of mapping Jr : H → H by Jr I − Jr /r We know that Ar x ∈ SJr x S We also define the Yosida approximation Ar by Ar and Ar x ≤ inf{ y : y ∈ Sx} for all . K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods &amp;. for Equilibrium Problems and Fixed Point Problems of a Countable Family of Nonexpansive Mappings Rabian Wangkeeree Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence. point problems and variational inequality problems, ” Journal of Inequalities and Applications, vol. 2007, Article ID 38752, 12 pages, 2007. 11 S. Takahashi and W. Takahashi, “Viscosity approximation