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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 531308, 20 pages doi:10.1155/2009/531308 Research Article An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems Qing-you Liu, 1 Wei-you Zeng, 2 and Nan-jing Huang 2 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Nan-jing Huang, nanjinghuang@hotmail.com Received 11 January 2009; Accepted 28 May 2009 Recommended by Fabio Zanolin We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. 2007. Copyright q 2009 Qing-you Liu 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. 1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H and let Φ : C × C → R be a bifunction, where R is the set of real numbers. Let Ψ : C → H be a nonlinear mapping. The generalized equilibrium problem GEP for Φ : C × C → R and Ψ : C → H is to find u ∈ C such that Φ  u, v   Ψu, v − u≥0 ∀v ∈ C. 1.1 The set of solutions for the problem 1.1 is denoted by Ω,thatis, Ω { u ∈ C : Φ  u, v   Ψu, v − u≥0, ∀v ∈ C } . 1.2 2 Fixed Point Theory and Applications If Ψ0in1.1, then GEP1.1 reduces to the classical equilibrium problem EP and Ω is denoted by EPΦ,thatis, EP  Φ   { u ∈ C : Φ  u, v  ≥ 0, ∀v ∈ C } . 1.3 If Φ0in1.1, then GEP1.1 reduces to the classical variational inequality and Ω is denoted by VIΨ,C,thatis, VI  Ψ,C   { u ∗ ∈ C : Ψu ∗ ,v− u ∗ ≥0, ∀v ∈ C } . 1.4 It is well known that GEP1.1 contains as special cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalities see, e.g., 1–6 and the reference therein. A mapping A : C → H is called α-inverse-strongly monotone 7, if there exists a positive real number α such that Ax − Ay, x − y≥α   Ax − Ay   2 1.5 for all x, y ∈ C. It is obvious that any α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. A mapping S : C → C is called nonexpansive if Sx − Sy≤x − y 1.6 for all x,y ∈ C. We denote by FS the set of fixed points of S,thatis,FS{x ∈ C : x  Sx}.IfC ⊂ H is bounded, closed and convex and S is a nonexpansive mappings of C into itself, then FS is nonempty see 8. In 1997, Fl ˚ am and Antipin 9 introduced an iterative scheme of finding the best approximation to initial data when EPΦ is nonempty and proved a strong convergence theorem. In 2003, Iusem and Sosa 10 presented some iterative algorithms for solving equi- librium problems in finite-dimensional spaces. They have also established the convergence of the algorithms. Recently, Huang et al. 11 studied the approximate method for solving the equilibrium problem and proved the strong convergence theorem for approximating the solutions of the equilibrium problem. On the other hand, for finding an element of FS ∩ VIA, C, Takahashi and Toyoda 12 introduced the following iterative scheme: x n1  α n x n   1 − α n  SP C  x n − λ n Ax n  ,n 0, 1, 2, , 1.7 where x 0 ∈ C, P C is metric projection of H onto C, {α n } is a sequence in 0, 1 and {λ n } is a sequence in 0, 2α. Further, Iiduka and Takahashi 13 introduced the following iterative scheme: x n1  α n u  β n x n  γ n SP C  x n − λ n Ax n  , 1.8 where u, x 0 ∈ C, and proved the strong convergence theorems for iterative scheme 1.8 under some conditions on parameters. In 2007, S. Takahashi and W. Takahashi 14 introduced an Fixed Point Theory and Applications 3 iterative scheme by the viscosity approximation method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result 3 and Wittmann’s result 15. Tada and W. Takahashi 16 introduced the Mann type iterative algorithm for finding a common element of the set of solutions of the EP Φ and the set of common fixed points of nonexpansive mapping and obtained the weak convergence of the Mann type iterative algorithm. Yao et al. 17 introduced an iteration process for finding a common element of the set of solutions of the EPΦ and the set of common fixed points of infinitely many nonexpansive mappings in Hilbert spaces. They proved a strong-convergence theorem under mild assumptions on parameters. Very recently, Moudafi 18 proposed an iterative algorithm for finding a common element of Ω∩FS, where Ψ : C → H is an α-inverse-strongly monotone mapping, and obtained a weak convergence theorem. There are some related works, we refer to 19–22 and the references therein. Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative process for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP1.1, and the solution set of the variational inequality problem for an α-inverse-strongly monotone mapping in real Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. 17. 2. Preliminaries Let H be a real Hilbert space with inner product ·, · and norm ·,andletC be a closed convex subset of H. Then, 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∀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 and satisfies P C x − P C y, x − y≥   P C x − P C y   2 2.2 for all x, y ∈ H. Furthermore, P C x ∈ C is characterized by the following properties: x − P C x, y − P C x≤0,  x − P C x  2    y − P C x   2 ≤   x − y   2 2.3 for all x ∈ H and y ∈ C.Itiseasytoseethat u ∈ VI  A, C  ⇐⇒ u  P C  u − λAu  , 2.4 where λ>0 is a parameter in R. 4 Fixed Point Theory and Applications A set-valued mapping T : H → 2 H is called monotone if for all x,y ∈ H, p ∈ Tx and q ∈ Ty imply x − y, p − q≥0. A monotone mapping T : H → 2 H is maximal if the graph GT of T is not properly contained in the graph of any other monotone mappings. It is known that a monotone mapping T is maximal if and only if for x, p ∈ H × H, x − y,p − q≥0 for all y,q ∈ GT implies p ∈ Tx.LetA : C → H be a monotone, L-Lipschitz continuous mappings and let N C u be the normal cone to C at u ∈ C,thatis,N C u  {w ∈ H : u − v, w≥ 0, ∀v ∈ C}. Define Tu  ⎧ ⎨ ⎩ Au  N C u, u ∈ C, ∅,u / ∈ C. 2.5 Then T is the maximal monotone and 0 ∈ Tu if and only if u ∈ VIA, C;see23. Let {T n } ∞ n1 be a sequence of nonexpansive mappings of C into itself and let {π n } ∞ n1 be a sequence of nonnegative numbers in 0, 1. For any n ≥ 1, define a mapping S n of C into itself as follows: U n,n1  I, U n,n  π n T n U n,n1   1 − π n  I, U n,n−1  π n−1 T n−1 U n,n   1 − π n−1  I, . . . U n,k  π k T k U n,k1   1 − π k  I, U n,k−1  π k−1 T k−1 U n,k   1 − π k−1  I, . . . U n,2  π 2 T 2 U n,3   1 − π 2  I, S n  U n,1  π 1 T 1 U n,2   1 − π 1  I. 2.6 Such a mapping S n is called the S-mapping generated by T n ,T n−1 , ,T 1 and π n ,π n−1 , ,π 1 see 24. It is obvious that S n is nonexpansive and if x  T n x then x  U n,k  S n x. Lemma 2.1 see 24. Let C be a nonempty closed convex subset of a real Hilbert space H.Let {T n } ∞ n1 be a sequence of nonexpansive mappings of C into itself such that  ∞ n1 FT n  /  ∅ and let {π n } ∞ n1 be a sequence in 0,σ for some σ ∈ 0, 1. Then, for every x ∈ C and k ∈ N  {1, 2, },the limit lim n →∞ U n,k x exists. Remark 2.2 see 17. It can be known from Lemma 2.1 that if D is a nonempty bounded subset of C, then for >0, there exists n 0 ≥ 1 such that for all n>n 0 sup x∈D U n,k x − U k x≤. 2.7 Fixed Point Theory and Applications 5 Using Lemma 2.1, one can define a mapping S of C into itself as follows: Sx  lim n →∞ S n x  lim n →∞ U n,1 x 2.8 for every x ∈ C. Such a mapping S is called the S-mapping generated by T 1 ,T 2 , and π 1 ,π 2 , Since S n is nonexpansive, S : C → C is also nonexpansive. If {x n } is a bounded sequence in C, then we put D  {x n : n ≥ 0}. Hence, it is clear from Remark 2.2 that for an arbitrary >0 there exists N 0 ∈ N such that for all n>N 0 S n x n − Sx n   U n,1 x n − U 1 x n ≤sup x∈D U n,1 x − U 1 x≤. 2.9 This implies that lim n →∞ S n x n − Sx n   0. 2.10 Since T i and U n,i are nonexpansive, we deduce that, for each n ≥ 1, S n1 x n − S n x n   π 1 T 1 U n1,2 x n − π 1 T 1 U n,2 x n  ≤ π 1 U n1,2 x n − U n,2 x n   π 1 π 2 T 2 U n1,3 x n − π 2 T 2 U n,3 x n  ≤ π 1 π 2 U n1,3 x n − U n,3 x n  . . . ≤ n  i1 π i U n1,n1 x n − U n,n1 x n  ≤ M n  i1 π i 2.11 for some constant M ≥ 0. Lemma 2.3 see 24. Let C be a nonempty closed convex subset of a real Hilbert space H.Let {T n } ∞ n1 be a sequence of nonexpansive mappings of C into itself such that  ∞ n1 FT n  /  ∅, and let {π n } ∞ n1 be a sequence in 0,σ for some σ ∈ 0, 1. Then, FS  ∞ n1 FT n . For solving the generalized equilibrium problem, we assume that the bifunction Φ : C × C → R satisfies the following conditions: a1Φu, u0 for all u ∈ C; a2Φis monotone, that is, Φu, vΦv,u ≤ 0 for all u, v ∈ C; a3 for each u, v, w ∈ C, lim t↓0 Φtw 1 − tu, v ≤ Φu, v; a4 for each u ∈ C, v → Φu, v is convex and lower semicontinuous. 6 Fixed Point Theory and Applications The following lemma appears implicitly in 1. Lemma 2.4 see 1. Let C be a nonempty closed convex subset of H, and let Φ be a bifunction from C × C into R satisfying (a1)–(a4). Let r>0 and x ∈ H. Then, there exists u ∈ C such that Φ  u, v   1 r v − u, u − x≥0 ∀v ∈ C. 2.12 The following lemma was also given in 3. Lemma 2.5 see 3. Assume that Φ : C × C → R satisfies (a1)–(a4). For r>0, define a mapping T r : H → C as follows: T r  x    u ∈ C : Φ  u, v   1 r  v − u, u − x  ≥ 0, ∀v ∈ C  2.13 for all x ∈ H. Then, the following hold: b1 T r is single-valued; b2 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; b3 FT r EPΦ; b4 EPΦ is closed and convex. Remark 2.6. Replacing x with x − rΨx ∈ H in 2.12, then there exists u ∈ C such that Φ  u, v   Ψx, v − u  1 r v − u, u − x≥0 ∀v ∈ C. 2.14 The following lemmas will be useful for proving the convergence result of this paper. Lemma 2.7 see 25. Let {x n } and {z n } be bounded sequences in Banach space E, and let {β n } be a sequence in 0, 1. Suppose x n1   1 − β n  z n  β n x n 2.15 for all integers n ≥ 1.If 0 < lim inf n →∞ β n ≤ lim sup n →∞ β n < 1, lim sup n →∞  z n1 − z n −x n1 − x n   ≤ 0, 2.16 then lim n →∞ z n − x n   0. Lemma 2.8 see 26. Assume {a n } is a sequence of nonnegative real numbers such that a n1 ≤  1 − α n  a n  δ n ,n≥ 1, 2.17 Fixed Point Theory and Applications 7 where {α n } is a sequence in (0,1) and {δ n } is a sequence in R such that 1  ∞ n1 α n  ∞; 2 lim sup n →∞ δ n /α n  ≤ 0 or  ∞ n1 |δ n | < ∞. Then lim n →∞ a n  0. 3. Main Results In this section, we deal with an iterative scheme by the approximation method for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP1.1, and the solution set of the variational inequality problem for an α-inverse-strongly monotone mapping in real Hilbert spaces. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H.LetΦ be a bifunction from C × C into R satisfying (a1)–(a4), Ψ : C → H an inverse-strongly monotone mapping with constant φ>0, A : C → H an inverse-strongly monotone mapping with constant >0, f : C → C a contraction mapping with constant α ∈ 0, 1.LetS n : C → C be a S-mapping generated by T 1 ,T 2 , and π 1 ,π 2 , and  ∞ n1 FT n  ∩ Ω ∩ VIA, C /  ∅, where sequence {T n } is nonexpansive and {π n } is a sequence in 0,σ for some σ ∈ 0, 1. For x 1 ∈ C, suppose that {x n }, {y n }, and {u n } are generated by Φ  u n ,v    Ψx n ,v− u n   1 r n  v − u n ,u n − x n  ≥ 0, ∀v ∈ 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 y n 3.1 for all n ∈ N,where{α n }, {β n }, and {γ n } are three sequences in 0, 1, {λ n } is a sequence in 0,b for some 0 <b<2 and {r n }⊂0,d for some 0 <d<2φ satisfying i α n  β n  γ n  1; ii lim n →∞ α n  0 and  ∞ n1 α n  ∞; iii 0 < lim inf n →∞ β n ≤ lim sup n →∞ β n < 1; iv lim inf n →∞ λ n > 0 and lim n →∞ |λ n1 − λ n |  0; v lim inf n →∞ r n > 0 and lim n →∞ |r n1 − r n |  0. Then {x n }, {y n }, and {u n } converge strongly to the point z 0 ∈  ∞ n1 FT n  ∩ Ω ∩ VIA, C,where z 0  P  ∞ n1 FT n ∩Ω∩VIA,C fz 0 . Proof. For any x, y ∈ C and r ∈ 0, 2φ, we have    I − rΨ  x −  I − rΨ  y   2     x − y  − r  Ψx − Ψy    2    x − y   2 − 2rx − y, Ψx − Ψy  r 2   Ψx − Ψy   2 ≤   x − y   2  r  r − 2φ    Ψx − Ψy   2 ≤   x − y   2 , 3.2 8 Fixed Point Theory and Applications which implies that I − rΨ is nonexpansive. Remark 2.6 implies that the sequences {u n } and {x n } are well defined. In view of the iterative sequence 3.1, we have 0 ≤ Φ  u n ,v   Ψx n ,v− u n   1 r n v − u n ,u n − x n  Φ  u n ,v   1 r n  v − u n ,u n −  x n − r n Ψx n   . 3.3 It follows from Lemma 2.5 that u n  T r n x n − r n Ψx n  for all n ≥ 1. Let z ∗ ∈  ∞ n1 FT n  ∩ Ω ∩ VIA, C. For each n ≥ 1, we have z ∗  S n z ∗ T r n z ∗ − r n Ψz ∗ .ByLemma 2.5,  u n − z ∗  2   T r n  x n − r n Ψx n  − T r n  z ∗ − r n Ψz ∗   2 ≤u n − z ∗ ,  x n − r n Ψx n  −  z ∗ − r n Ψz ∗    1 2   u n − z ∗  2    x n − r n Ψx n  −  z ∗ − r n Ψz ∗   2 −   u n − z ∗  −  x n − r n Ψx n  −  z ∗ − r n Ψz ∗   2  3.4 and so 3.2 implies that  u n − z ∗  2 ≤   x n − r n Ψx n  −  z ∗ − r n Ψz ∗   2 −   u n − x n  − r n  Ψz ∗ − Ψx n   2 ≤  x n − z ∗  2 −   u n − x n  − r n  Ψz ∗ − Ψx n   2 ≤  x n − z ∗  2 . 3.5 For z ∗ ∈ VIA, C, we have z ∗  P C z ∗ − λ n Az ∗  from 2.4. Since P C is a nonexpansive mapping and A is an inverse-strongly monotone mapping with constant >0, by 3.1,we have   y n − z ∗   2   P C  u n − λ n Au n  − P C  z ∗ − λ n Az ∗   2 ≤   u n − λ n Au n  −  z ∗ − λ n Az ∗   2 ≤  u n − z ∗  2  λ n  λ n − 2   Au n − Az ∗  2 ≤  u n − z ∗  2 . 3.6 Thus, 3.5 and 3.6 imply that y n − z ∗ ≤u n − z ∗ ≤x n − z ∗ , 3.7 Fixed Point Theory and Applications 9 and so x n1 − z ∗   α n f  x n   β n x n  γ n S n y n − z ∗  ≤ α n f  x n  − z ∗   β n x n − z ∗   γ n S n y n − z ∗  ≤ α n  f  x n  − f  z ∗    f  z ∗  − z ∗    β n x n − z ∗   γ n S n y n − S n z ∗  ≤ α n  αx n − z ∗   f  z ∗  − z ∗    β n x n − z ∗   γ n y n − z ∗  ≤ α n  αx n − z ∗   f  z ∗  − z ∗    β n x n − z ∗   γ n x n − z ∗    1 − α n  1 − α  x n − z ∗   α n  1 − α  f  z ∗  − z ∗  1 − α ≤ max  x 1 − z ∗ , f  z ∗  − z ∗  1 − α  . 3.8 This implies that {x n } is bounded. Therefore, {u n }, {y n }, {Ψx n }, {Au n }, and {S n y n } are also bounded. From u n  T r n x n − r n Ψx n  and u n1  T r n1 x n1 − r n1 Ψx n1 , we have Φ  u n ,v    Ψx n ,v− u n   1 r n  v − u n ,u n − x n  ≥ 0 ∀v ∈ C, 3.9 Φ  u n1 ,v   Ψx n1 ,v− u n1   1 r n1 v − u n1 ,u n1 − x n1 ≥0 ∀v ∈ C. 3.10 Putting v  u n1 in 3.9 and v  u n in 3.10,weget Φ  u n ,u n1   Ψx n ,u n1 − u n   1 r n u n1 − u n ,u n − x n ≥0, Φ  u n1 ,u n   Ψx n1 ,u n − u n1   1 r n1 u n − u n1 ,u n1 − x n1 ≥0. 3.11 Adding the above two inequalities, the monotonicity of Φ implies that Ψx n1 − Ψx n ,u n − u n1    u n − u n1 , u n1 − x n1 r n1 − u n − x n r n  ≥ 0, 3.12 and so 0 ≤  u n − u n1 ,r n  Ψx n1 − Ψx n   r n r n1  u n1 − x n1  −  u n − x n     u n1 − u n ,u n − u n1   1 − r n r n1  u n1   x n1 − r n Ψx n1  −  x n − r n Ψx n  − x n1  r n r n1 x n1    u n1 − u n ,u n − u n1   1 − r n r n1   u n1 − x n1    x n1 − r n Ψx n1  −  x n − r n Ψx n   . 3.13 10 Fixed Point Theory and Applications It follows from 3.2 that  u n1 − u n  2 ≤u n1 − u n       1 − r n r n1     u n1 − x n1   x n1 − x n   , 3.14 and hence u n1 − u n ≤     1 − r n r n1     u n1 − x n1   x n1 − x n . 3.15 From 3.1, y n1 − y n   P C  u n1 − λ n1 Au n1  − P C  u n − λ n Au n   ≤  u n1 − λ n1 Au n1  −  u n − λ n Au n   ≤  u n1 − λ n1 Au n1  −  u n − λ n Au n1    | λ n1 − λ n | Au n  ≤u n1 − u n   | λ n1 − λ n | Au n . 3.16 Putting z n  α n 1 − β n f  x n   γ n 1 − β n S n y n , 3.17 we have x n1  β n x n   1 − β n  z n . 3.18 Obviously, we get z n1 − z n       α n1 1 − β n1 f  x n1   γ n1 1 − β n1 S n1 y n1 −  α n 1 − β n f  x n   γ n 1 − β n S n y n      ≤ α n1 1 − β n1 f  x n1  − f  x n        α n1 1 − β n1 − α n 1 − β n     f  x n    γ n1 1 − β n1 S n1 y n1 − S n y n       γ n1 1 − β n1 − γ n 1 − β n     S n y n  ≤ α n1 1 − β n1 αx n1 − x n       α n1 1 − β n1 − α n 1 − β n     f  x n        α n 1 − β n − α n1 1 − β n1     S n y n    1 − α n1 1 − β n1  S n1 y n1 − S n y n . 3.19 [...]... method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics, vol 12, no 6, pp 1401–1432, 2008 20 J.-W Peng, Y 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... theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, vol 2007, Article ID 64363, 12 pages, 2007 18 A Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems, Journal of Nonlinear and Convex Analysis, vol 9, no 1, pp 37–43, 2008 19 J.-W Peng and J.-C Yao, “A new hybrid-extragradient method for. .. 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Takahashi 14 Fixed Point Theory and Applications 19 Our Theorem 3.1 improves and extends Theorem 3.5 of Yao et al 17 in the following aspects: 1 the equilibrium problem is extended to the generalized equilibrium problem; 2 our iterative process 3.1 is different from Yao et al iterative process 3.60 because there are a project operator and an α-inverse-strongly monotone mapping; 3 our iterative process . Corporation Fixed Point Theory and Applications Volume 2009, Article ID 531308, 20 pages doi:10.1155/2009/531308 Research Article An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and. Peng, Y. 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,. cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalities see, e.g., 1–6 and the reference therein. A