Báo cáo hóa học: "Research Article A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions" pptx

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 567147, 20 pages doi:10.1155/2009/567147 Research Article A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions, Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces Somyot Plubtieng 1, 2 and Wanna Sriprad 1, 2 1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand 2 PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand Correspondence should be addressed to Somyot Plubtieng, somyotp@nu.ac.th Received 12 February 2009; Accepted 18 May 2009 Recommended by William A. Kirk We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of J. W. Peng et al. 2008 and many others. Copyright q 2009 S. Plubtieng and W. Sriprad. 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 whose inner product and norm are denoted by ·, · and ·, respectively. Let C be a nonempty closed convex subset of H,andletF be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for F : C × C → R is to find x ∈ C such that F  x, y  ≥ 0, ∀y ∈ C. 1.1 The set of solutions of 1.1 is denoted by EPF. Recently, Combettes and Hirstoaga 1 introduced an iterative scheme of finding the best approximation to t he initial data when EPF is nonempty and proved a strong convergence theorem. Let A : C → H be a nonlinear 2 Fixed Point Theory and Applications map. The classical variational inequality which is denoted by VIA, C is to find u ∈ C such that Au, v − u≥0, ∀v ∈ C. 1.2 The variational inequality has been extensively studied in literature. See, for example, 2, 3 and the references therein. Recall that a mapping T of C into itself is called nonexpansive if Su − Sv≤u − v, ∀u, v ∈ C. 1.3 A mapping f : C → C is called contractive if there exists a constant β ∈ 0, 1 such that fu− fv≤βu − v, ∀u, v ∈ C. 1.4 We denote by FS the set of fixed points of S. Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping; see, for instance, 3– 6 and the references therein. Recently, Plubtieng and Punpaeng 6 introduced the following iterative scheme. Let x 1 ∈ H and let {x n },and{u n } be sequences generated by F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ H, x n1  α n γf  x n    I − α n A  Su n , ∀n ∈ N. 1.5 They proved that if the sequences {α n } and {r n } of parameters satisfy appropriate conditions, then the sequences {x n } and {u n } both converge strongly to the unique solution of the variational inequality  A − γf  z, z − x  ≥ 0, ∀x ∈ F  S  ∩ EP  F  , 1.6 which is the optimality condition for the minimization problem min x∈FS∩EP F 1 2 Ax, x−h  x  , 1.7 where h is a potential function for γf. Let A : H → H be a single-valued nonlinear mapping, and let M : H → 2 H be a set-valued mapping. We consider the following variational inclusion, which is to find a point u ∈ H such that θ ∈ A  u   M  u  , 1.8 where θ is the zero vector in H. The set of solutions of problem 1.8 is denoted by IA, M. If A  0, then problem 1.8 becomes the inclusion problem introduced by Rockafellar 7. Fixed Point Theory and Applications 3 If M  ∂δ C , where C is a nonempty closed convex subset of H and δ C : H → 0, ∞ is the indicator function of C, that is, δ C  x   ⎧ ⎨ ⎩ 0,x∈ C, ∞,x / ∈ C, 1.9 then the variational inclusion problem 1.8 is equivalent to variational inequality problem 1.2. It is known that 1.8 provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, com- plementarity, variational inequalities, optimal control, mathematical economics, equilibria, game theory. Also various types of variational inclusions problems have been extended and generalized see 8 and the references therein. Very recently, Peng et al. 9 introduced the following iterative scheme for finding a common element of the set of solutions to the problem 1.8, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with x 1 ∈ H, define sequence, {x n }, {y n },and{u n } by F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ H, x n1  α n γf  x n    1 − α n  Sy n , y n  J M,λ  u n − λAu n  , ∀n ≥ 0, 1.10 for all n ∈ N, where λ ∈ 0, 2α, {α n }⊂0, 1 and {r n }⊂0, ∞. They proved that under certain appropriate conditions imposed on {α n } and {r n }, the sequences {x n }, {y n }, and {u n } generated by 1.10 converge strongly to z ∈ FT ∩ IA, M ∩ EPF, where z  P FS∩IA,M∩EPF fz. Motivated and inspired by Plubtieng and Punpaeng 6, Peng et al. 9 and Aoyama et al. 10, we introduce an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem 1.8 with multi-valued maximal monotone mapping and inverse-strongly monotone mappings, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with an arbitrary x 1 ∈ H, define sequences {x n }, {y n } and {u n } by F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ H, x n1  α n γf  x n    I − α n B  S n y n , y n  J M,λ  u n − λAu n  , ∀n ≥ 0, 1.11 for all n ∈ N, where λ ∈ 0, 2α, {α n }⊂0, 1,andlet{r n }⊂0, ∞; B be a strongly bounded linear operator on H,and{S n } is a sequence of nonexpansive mappings on H. Under suitable conditions, some strong convergence t heorems for approximating to this common elements are proved. Our results extend and improve some corresponding results in 3, 9 and the references therein. 4 Fixed Point Theory and Applications 2. Preliminaries This section collects some lemmas which will be used in the proofs for the main results in the next section. Let H be a real Hilbert space with inner product ·, · and norm ·, respectively. It is wellknown that for all x, y ∈ H and λ ∈ 0, 1, there holds   λx 1 − λy   2  λ  x  2   1 − λ    y   2 − λ  1 − λ    x − y   2 . 2.1 Let C be a nonempty closed convex subset of H. T hen, for any x ∈ H, there exists a unique nearest point of C, denoted by P C x, such that x − P C x≤x − y for all y ∈ C. Such a P C is called the metric projection from H into C. We know that P C is nonexpansive. It is also known that, P C x ∈ C and x − P C x, P C x − z≥0, ∀x ∈ H and z ∈ C. 2.2 It is easy to see that 2.2 is equivalent to  x − z  2 ≥  x − P C x  2   z − P C x  2 , ∀x ∈ H, z ∈ C. 2.3 For solving the equilibrium problem for a bifunction F : C × C → R, let us assume that F satisfies the following conditions: A1 Fx, x0 for all x ∈ C; A2 F is monotone, that is, Fx, yFy, x ≤ 0 for all x, y ∈ C; A3 for each x, y, z ∈ C, lim t → 0 F  tz   1 − t  x, y  ≤ F  x, y  ; 2.4 A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous. The following lemma appears implicitly in 11 and 1. Lemma 2.1 See 1, 11. Let C be a nonempty closed convex subset of H and let F be a bifunction of C × C in 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. 2.5 Define a mapping T r : H → C as follows: T r  x    z ∈ C : F  z, y   1 r  y − z, z − x  ≥ 0, ∀y ∈ C  , 2.6 Fixed Point Theory and Applications 5 for all z ∈ H. Then, the following hold: 1 T r is single-valued; 2 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  ; 2.7 3 FT r EPF; 4 EPF is closed and convex. We also need the following lemmas for proving our main result. Lemma 2.2 See 12. Let H be a Hilbert space, C a nonempty closed convex subset of H, f : H → H a contraction with coefficient 0 <α<1, and B a strongly positive linear bounded operator with coefficient γ>0.Then: 1 if 0 <γ< γ/α,thenx − y, B − γfx − B − γfy≥γ − γαx − y 2 ,x,y ∈ H. 2 if 0 <ρ<B −1 ,thenI − ρB≤1 − ργ. Lemma 2.3 See 13. Assume {a n } is a sequence of nonnegative real numbers such that a n1 ≤  1 − γ n  a n  δ n ,n≥ 0, 2.8 where {γ n } is a sequence in 0, 1 and {δ n } is a sequence in R 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. Recall that a mapping A : H → H is called α-inverse-strongly monotone, if there exists a positive number α such that Au − Av, u − v≥α  Au − Av  2 , ∀u, v ∈ H. 2.9 Let I be the identity mapping on H. It is well known that if A : H → H is α-inverse- strongly monotone, then A is 1/α-Lipschitz continuous and monotone mapping. In addition, if 0 <λ≤ 2α, then I − λA is a nonexpansive mapping. A set-valued M : H → 2 H is called monotone if for all x, y ∈ H, f ∈ Mx and g ∈ My imply x − y, f − g≥0. A monotone mapping M : H → 2 H is maximal if its graph GM : {x, f ∈ H × H | f ∈ Mx} of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for x, f ∈ H × H, x − y, f − g≥0 for every y, g ∈ GM implies f ∈ Mx. Let the set-valued mapping M : H → 2 H be maximal monotone. We define the resolvent operator J M,λ associated with M and λ as follows: J M,λ  u    I  λM  −1  u  , ∀u ∈ H, 2.10 6 Fixed Point Theory and Applications where λ is a positive number. It is worth mentioning that the resolvent operator J M,λ is single-valued, nonexpansive and 1-inverse-strongly monotone, see for example 14 and that a solution of problem 1.8 is a fixed point of the operator J M,λ I − λA for all λ>0, see for instance 15. Lemma 2.4 See 14. Let M : H → 2 H be a maximal monotone mapping and A : H → H be a Lipschitz-continuous mapping. Then the mapping S  M  A : H → 2 H is a maximal monotone mapping. Remark 2.5 See 9. Lemma 2.4 implies that IA, M is closed and convex if M : H → 2 H is a maximal monotone mapping and A : H → H be an inverse strongly monotone mapping. Lemma 2.6 See 10. Let C be a nonempty closed subset of a Banach space and let {S n } 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 x ∈ C, {S n x} converges strongly to some point of C. Moreover, let S be a mapping from C into itself defined by Sx  lim n →∞ S n x, ∀x ∈ C. 2.11 Then lim n →∞ sup{Sz − S n z : z ∈ C}  0. 3. Main Results We begin this section by proving a strong convergence theorem of an implicit iterative sequence {x n } obtained by the viscosity approximation method for finding a common element of the set of solutions of the variational inclusion, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. Throughout the rest of this paper, we always assume that f is a contraction of H into itself with coefficient β ∈ 0, 1,andB is a strongly positive bounded linear operator with coefficient γ and 0 <γ<γ/β.LetS be a nonexpansive mapping of H into H.LetA : H → H be an α-inverse-strongly monotone mapping, M : H → 2 H be a maximal monotone mapping and let J M,λ be defined as in 2.10.Let{T r n } be a sequence of mappings defined as Lemma 2.1. Consider a sequence of mappings {S n } on H defined by S n x  α n γf  x    I − α n B  SJ M,λ  I − λA  T r n x, x ∈ H, n ≥ 1, 3.1 where {α n }⊂0, B −1 . By Lemma 2.2,wenotethatS n is a contraction. Therefore, by the Banach contraction principle, S n has a unique fixed point x n ∈ H such that x n  α n γf  x n    I − α n B  SJ M,λ  I − λA  T r n x n . 3.2 Theorem 3.1. Let H be a real Hilbert space, let F be a bifunction from H × H → R satisfying (A1)–(A4) and let S be a nonexpansive mapping on H.LetA : H → H be an α-inverse-strongly monotone mapping, M : H → 2 H be a maximal monotone mapping such that Ω : FS ∩ EPF ∩ IA, M /  ∅. Let f be a contraction of H into itself with a constant β ∈ 0, 1 and let B be a strongly Fixed Point Theory and Applications 7 bounded linear operator on H with coefficient γ>0 and 0 <γ<γ/β.Let{x n }, {y n } and {u n } be sequences generated by x 1 ∈ H and F  u n ,y   1 r n y − u n ,u n − x n ≥0, ∀y ∈ H x n  α n γf  x n    I − α n B  Sy n , y n  J M,λ  u n − λAu n  ∀n ≥ 0, 3.3 where λ ∈ 0, 2α, {r n }⊂0, ∞ and {α n }⊂0, 1 satisfy lim n →∞ α n  0 and lim inf n →∞ r n > 0. Then, {x n }, {y n } and {u n } converges strongly to a point z in Ω which solves the variational inequality:   B − γf  z, z − x≤0,x∈ Ω. 3.4 Equivalently, we have z  P Ω I − B  γfz. Proof. First, we assume that α n ∈ 0, B −1 .ByLemma 2.2,weobtainI − α n B≤1 − α n γ.Let v ∈ Ω. Since u n  T r n x n , we have u n − v  T r n x n − T r n v≤x n − v∀n ∈ N. 3.5 We note from v ∈ Ω that v  J M,λ v − λAv.AsI − λA is nonexpansive, we have y n − v  J M,λ  u n − λAu n  − J M,λ  v − λAv   ≤  u n − λAu n  −  v − λAv  ≤u n − v≤x n − v 3.6 for all n ∈ N. Thus, we have x n − v  α n γf  x n    I − α n B  Sy n − v ≤ α n γf  x n  − Bv  I − α n By n − v ≤ α n γf  x n  − Bv   1 − α n γ  x n − v ≤ α n γ  f  x n  − f  v     γf  v  − Bv     1 − α n γ  x n − v ≤ α n γβx n − v  α n γf  v  − Bv   1 − α n γ  x n − v   1 − α n  γ − γβ  x n − v  α n γf  v  − Bv. 3.7 It follows that x n − v≤γfv − Bv/γ − γβ, ∀n ≥ 1. Hence {x n } is bounded and we also obtain that {u n },{y n }, {fx n },{Sy n } and {Au n } are bounded. Next, we show that y n − Sy n →0. Since α n → 0, we note that x n − Sy n   α n γf  x n  − BSy n −→0asn −→ ∞ . 3.8 8 Fixed Point Theory and Applications Moreover, it follows from Lemma 2.1 that  u n − v  2   T r n x n − T r n v  2 ≤T r n x n − T r n v, x n − v  u n − v, x n − v  1 2   u n − v  2   x n − v  2 −  x n − u n  2  , 3.9 and hence u n − v 2 ≤x n − v 2 −x n − u n  2 . Therefore, we have  x n − v  2    α n γfx n I − α n BSy n − v   2    I − α n BSy n − vα n γfx n  − Bv   2 ≤  1 − α n γ  2   Sy n − v   2  2α n γf  x n  − Bv, x n − v ≤  1 − α n γ  2   y n − v   2  2α n γf  x n  − f  v  ,x n − v  2α n γf  v  − Bv, x n − v ≤  1 − α n γ  2  u n − v  2  2α n γf  x n  − f  v  ,x n − v  2α n γf  v  − Bv, x n − v ≤  1 − α n γ  2   x n − v  2 −  x n − u n  2   2α n γβ  x n − v  2  2α n γf  v  − Bvx n − v   1 − 2α n  γ − γβ    α n γ  2   x n − v  2 −  1 − α n γ  2  x n − u n  2  2α n γf  v  − Bvx n − v ≤  x n − v  2  α n γ 2  x n − v  2 −  1 − α n γ  2  x n − u n  2  2α n γf  v  − Bvx n − v, 3.10 and hence  1 − α n γ  2  x n − u n  2 ≤ α n γ 2  x n − v  2  2α n γf  v  − Bvx n − v. 3.11 Since {x n } is bounded and α n → 0, it follows that x n − u n →0asn →∞. Put M  sup n≥1 {γfv − Bvx n − v}.From3.10, it follows by the nonexpansive of J M,λ and the inverse strongly monotonicity of A that  x n − v  2 ≤  1 − α n γ  2   y n − v   2  2α n γβ  x n − v  2  2α n M ≤  1 − α n γ  2  u n − λAu n  − v − λAv  2  2α n γβ  x n − v  2  2α n M ≤  1 − α n γ  2   u n − v  2  λ  λ − 2α   Au n − Av  2   2α n γβ  x n − v  2  2α n M ≤  1−α n γ  2  x n −v  2   1−α n γ  2 λ  λ−2α   Au n −Av  2 2α n γβ  x n −v  2  2α n M   1−2α n  γ −γβ    α n γ  2   x n −v  2   1−α n γ  2 λ  λ−2α   Au n −Av  2 2α n M ≤  x n − v  2  α n γ 2  x n − v  2   1 − α n γ  2 λ  λ − 2α   Au n − Av  2  2α n M, 3.12 Fixed Point Theory and Applications 9 which implies that  1 − α n γ  2 λ  2α − λ   Au n − Av  2 ≤ α n γ 2  x n − v  2  2α n M. 3.13 Since α n → 0, we have Au n −Av→0asn →∞. Since J M,λ is 1–inverse-strongly monotone and I − λA is nonexpansive, we have   y n − v   2   J M,λ  u n − λAu n  − J M,λ  v − λAv   2 ≤  u n − λAu n −  v − λAv  ,y n − v   1 2   u n −λAu n −v−λAv  2    y n −v   2 −   u n −λAu n −v−λAv−y n −v   2  ≤ 1 2   u n − v  2    y n − v   2 −   u n − y n − λAu n − Av   2   1 2   u n −v  2    y n −v   2 −   u n −y n   2 2λ  u n −y n ,Au n −Av  −λ 2  Au n − Av  2  . 3.14 Thus, we have   y n − v   2 ≤  u n − v  2 −   u n − y n   2  2λ  u n − y n ,Au n − Av  − λ 2  Au n − Av  2 . 3.15 From 3.5, 3.10,and3.15, we have  x n − v  2 ≤  1 − α n γ  2   y n − v   2  2α n γβ  x n − v  2  2α n M ≤  1 − α n γ  2   u n − v  2 −   u n − y n   2  2λ  u n − y n ,Au n − Av  − λ 2  Au n − Av  2   2α n γβ  x n − v  2  2α n M ≤  1 − α n γ  2  x n − v  2 −  1 − α n γ  2   u n − y n   2  2  1 − α n γ  2 λ  u n − y n ,Au n − Av  −  1 − α n γ  2 λ 2  Au n − Av  2  2α n γβ  x n − v  2  2α n M   1 − 2α n  γ − γβ    α n γ  2   x n − v  2 −  1 − α n γ  2   u n − y n   2  2  1 − α n γ  2 λ  u n − y n ,Au n − Av  −  1 − α n γ  2 λ 2  Au n − Av  2  2α n M ≤  x n − v  2  α n γ 2  x n − v  2 −  1 − α n γ  2   u n − y n   2  2  1 − α n γ  2 λ  u n − y n ,Au n − Av  −  1 − α n γ  2 λ 2  Au n − Av  2  2α n M. 3.16 Thus, we get  1 − α n γ  2   u n − y n   2 ≤ α n γ 2  x n − v  2  2  1 − α n γ  2 λ  u n − y n ,Au n − Av  −  1 − α n γ  2 λ 2  Au n − Av  2  2α n M. 3.17 10 Fixed Point Theory and Applications Since α n → 0, Au n − Av→0asn →∞, we have u n − y n →0asn →∞. It follows from the inequality y n − Sy n ≤y n − u n   u n − x n   x n − Sy n  that y n − Sy n →0as n →∞. Moreover, we have x n − y n ≤x n − u n   u n − y n →0asn →∞. Put U ≡ SJ M,λ I − λA. Since both S and J M,λ I − λA are nonexpansive, we have U is a nonexpansive mapping on H and t hen we have x n  α n γfx n I − α n BUT r n x n for all n ∈ N. It follows by Theorem 3.1 of Plubtieng and Punpaeng 6 that {x n } converges strongly to z ∈ FU ∩ EPF, where z  P FU∩EP F γf I − Bz and B − γfz, u − z≥0, for all u ∈ FU ∩ EPF. We will show that z ∈ FS ∩ IA, M. Since {x n } converges strongly to z, we also have x n z. Let us show z ∈ FS. Assume z / ∈ FS. Since x n − y n →0and x n z, we have y n zSince z /  Sz, it follows by the Opial’s condition that lim inf n →∞ y n − z < lim inf n →∞ y n − Sz≤lim inf n →∞  y n − Sy n   Sy n − Sz  ≤ lim inf n →∞ y n − z. 3.18 This is a contradiction. Hence z ∈ FS. We now show that z ∈ IA, M. In fact, since A is α–inverse-strongly monotone, A is an 1/α-Lipschitz continuous monotone mapping and DAH. It follows from Lemma 2.4 that MA is maximal monotone. Let p, g ∈ GMA, that is, g − Ap ∈ Mp. Again since y n  J M,λ u n − λAu n , we have u n − λAu n ∈ I  λMy n , that is, 1 λ  u n − y n − λAu n  ∈ M  y n  . 3.19 By the maximal monotonicity of M  A, we have  p − y n ,g− Ap − 1 λ  u n − y n − λAu n   ≥ 0, 3.20 and so  p − y n ,g  ≥  p − y n ,Ap 1 λ  u n − y n − λAu n     p − y n ,Ap− Ay n  Ay n − Au n  1 λ  u n − y n   ≥ 0   p − y n ,Ay n − Au n    p − y n , 1 λ  u n − y n   . 3.21 It follows from u n − y n →0, Au n − Ay n →0andy n zthat lim n →∞  p − y n ,g    p − z, g  ≥ 0. 3.22 Since A  M is maximal monotone, this implies that θ ∈ M  Az, that is, z ∈ IA, M. Hence, z ∈ Ω : FS∩EPF∩IA, M. Since FS∩IA, MFS∩FJ M,λ I−λA ⊂ FU, we have Ω ⊂ FU∩EPF. It implies that z is the unique solution of the variational inequality 3.4. [...]... Wang, D S Shyu, and J.-C Yao, Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems,” Journal of Inequalities and Applications, vol 2008, Article ID 720371, 15 pages, 2008 10 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:... Colao, G Marino, and H.-K Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,” Journal of Mathematical Analysis and Applications, vol 344, no 1, pp 340– 352, 2008 20 Fixed Point Theory and Applications 6 S Plubtieng and R Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis... 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Plubtieng and R Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol 197, no 2, pp 548–558, 2008 4 S.-S Chang, H W Joseph Lee, and C K Chan, A new method for solving equilibrium problem fixed point problem with application to optimization,” Nonlinear Analysis: Theory, Methods & Applications,... to z As in 10, Theorem 4.1 , we can generate a sequence {Sn } of nonexpansive mappings satisfying condition ∞ 1 sup{ Sn 1 z − Sn z : z ∈ K} < ∞ for any bounded K of H by using n convex combination of a general sequence {Tk } of nonexpansive mappings with a common fixed point Corollary 3.3 Let H be a real Hilbert space, let F be a bifunction from H × H → R satisfying (A1 )– (A4 ) and let A : H → H be an α-inverse-strongly... } and {un } converges strongly to a point z in Ω which solves the variational inequality: B − γf z, z − x ≥ 0, x ∈ Ω 3.60 Acknowledgments The first author thank the National Research Council of Thailand to Naresuan University, 2009 for the financial support Moreover, the second author would like to thank the “National Centre of Excellence in Mathematics”, PERDO, under the Commission on Higher Education,... Theory and Applications 11 Now we prove the following theorem which is the main result of this paper Theorem 3.2 Let H be a real Hilbert space, let F be a bifunction from H × H → R satisfying (A1 )– (A4 ) and let {Sn } be a sequence of nonexpansive mappings on H Let A : H → H be an α-inverse-strongly monotone mapping, M : H → 2H be a maximal monotone mapping such that ∞ Ω : n 1 F Sn ∩ EP F ∩ I A, M /... } and {un } converges strongly to z ∈ Ω, where z PΩ f z Fixed Point Theory and Applications If Sn 19 S, A ≡ 0 and M ≡ 0 in Theorem 3.2, we obtain the following corollary Corollary 3.5 see S Plubtieng and R Punpaeng 6 Let H be a real Hilbert space, let F be a bifunction from H × H → R satisfying (A1 )– (A4 ) and let S be a nonexpansive mapping on H such that Ω : F S ∩ EP F / ∅ Let f be a contraction of . 2008, Article ID 720371, 15 pages, 2008. 10 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,”. pp. 877–898, 1976. 8 S. Adly, “Perturbed algorithms and sensitivity analysis for a general class of variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 201, no. 2,. mathematical programming, com- plementarity, variational inequalities, optimal control, mathematical economics, equilibria, game theory. Also various types of variational inclusions problems have