báo cáo hóa học: " Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings" potx

Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 RESEARCH Open Access Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings Jian-Wen Peng Correspondence: jwpeng6@yahoo com.cn School of Mathematics, Chongqing Normal University, Chongqing 400047, PR China Abstract In this article, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, and the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in Hilbert spaces We obtain some strong convergence theorems and weak convergence theorems The results in this article generalize, improve, and unify some well-known convergence theorems in the literature Keywords: Generalized equilibrium problem, Extragradient method, Hybrid method, Nonex-pansive mapping, Strong convergence, Weak convergence Introduction Let H be a real Hilbert space with inner product 〈.,.〉 and induced norm ||·|| Let C be a nonempty closed convex subset of H Let F be a bifunction from C × C to R and let B : C ® H be a nonlinear mapping, where R is the set of real numbers Moudafi [1], Moudafi and Thera [2], Peng and Yao [3,4], Takahashi and Takahashi [5] considered the following generalized equilibrium problem: Find x ∈ C Such that F(x, y) + Bx, y − x ≥ 0, ∀y ∈ C (1:1) The set of solutions of (1.1) is denoted by GEP(F, B) If B = 0, the generalized equilibrium problem (1.1) becomes the equilibrium problem for F : C ì C đ R, which is to find x Î C such that F(x, y) ≥ for all y ∈ C (1:2) The set of solutions of (1.2) is denoted by EP(F) 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 [1-7] Recall that a mapping S : C ® C is nonexpansive if there holds that ||Sx − Sy|| ≤ ||x − y|| for all x, y ∈ C We denote the set of fixed points of S by Fix(S) © 2011 Peng; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Let the mapping A : C ® H be monotone and k-Lipschitz-continuous The variational inequality problem is to find x Ỵ C such that Ax, y − x ≥ for all y Ỵ C The set of solutions of the variational inequality problem is denoted by V I(C, A) Several algorithms have been proposed for finding the solution of problem (1.1) Moudafi [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a weak convergence theorem Moudafi and Thera [2] introduced an auxiliary scheme for finding a solution of problem (1.1) in a Hilbert space and obtained a weak convergence theorem Peng and Yao [3,4] introduced some iterative schemes for finding a common element of the set of solutions of problem (1.1), the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping and obtain both strong convergence theorems, and weak convergence theorems for the sequences generated by the corresponding processes in Hilbert spaces Takahashi and Takahashi [5] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a strong convergence theorem Some methods also have been proposed to solve the problem (1.2); see, for instance, [8-19] and the references therein Takahashi and Takahashi [9] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a non-expansive mapping, and proved a strong convergence theorem in a Hilbert space Su et al [10] introduced and researched an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an a-inverse-strongly monotone mapping in a Hilbert space Tada and Takahashi [11] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and obtained both strong convergence and weak convergence theorems Plubtieng and Punpaeng [12] introduced an iterative processes based on the extragradient method for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of variational inequality problem for an a-inverse-strongly monotone mapping Chang et al [13] introduced an iterative processes based on the extragradient method for finding the common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of variational inequality problem for an a-inverse-strongly monotone mapping Yao et al [14] and Ceng and Yao [15] introduced some iterative viscosity approximation schemes for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely nonexpansive mappings in a Hilbert space Colao et al [16] introduced an iterative viscosity approximation scheme for finding a common element of the set of solutions of problem (1.2) Page of 19 Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page of 19 and the set of fixed points of a family of finitely nonexpansive mappings in a Hilbert space We observe that the algorithms in [13-16] involves the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings which is an effective tool in nonlinear analysis (see [20,21]) However, the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings is too completed to use for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely (finitely) nonexpansive mappings It is natural to raise and to give an answer to the following question: Can one construct algorithms for finding a common element of the set of solutions of a generalized equilibrium problem (an equilibrium problem), the common set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of a variational inequality without the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings? In this article, we will give a positive answer to this question Recently, OHaraa et al [22] introduced and researched an iterative approach for finding a nearest point of infinitely many nonexpansive mappings in a Hilbert spaces without using the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings Inspired by the ideas in [1-6,8-16,22] and the references therein, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz–continuous mapping without using the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings We obtain both strong convergence theorems and weak convergence theorems for the sequences generated by the corresponding processes The results in this article generalize, improve, and unify some well-known convergence theorems in the literature Preliminaries Let H be a real Hilbert space with inner product 〈·,·〉 and norm ||·|| Let C be a nonempty closed convex subset of H Let symbols ® and ⇀ denote strong and weak convergences, respectively In a real Hilbert space H, it is well known that λx + (1 − λ)y =λ x + (1 − λ) y − λ(1 − λ) x − y for all x, y Ỵ H and l Ỵ [0, 1] For any x Ỵ H, there exists the unique nearest point in C, denoted by PC(x), such that ||x - PC(x)|| ≤ ||x - y|| for all y Ỵ C The mapping PC is called the metric projection of H onto C We know that PC is a nonexpansive mapping from H onto C It is also known that PCx Ỵ C and x − PC (x), PC (x) − y ≥ (2:1) for all x Ỵ H and y Ỵ C It is easy to see that (2.1) is equivalent to x−y ≥ x − PC (x) for all x Ỵ H and y Ỵ C + y − PC (x) (2:2) Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 A mapping A of C into H is called monotone if Ax − Ay, x − y ≥ for all x, y Ỵ C A mapping A of C into H is called a-inverse-strongly monotone if there exists a positive real number a such that x − y, Ax − Ay ≥ α Ax − Ay for all x, y Ỵ C A mapping A : C ® H is called k-Lipschitz-continuous if there exists a positive real number k such that Ax − Ay ≤ k x − y for all x, y Ỵ C It is easy to see that if A is a-inverse-strongly monotone, then A is monotone and Lipschitz-continuous The converse is not true in general The class of a-inverse-strongly monotone mappings does not contain some important classes of mappings even in a finite-dimensional case For example, if the matrix in the corresponding linear complementarity problem is positively semidefinite, but not positively definite, then the mapping A will be monotone and Lipschitz-continuous, but not ainverse-strongly monotone (see [23]) Let A be a monotone mapping of C into H In the context of the variational inequality problem, the characterization of projection (2.1) implies the following: u ∈ VI(C, A) ⇒ u = PC (u − λAu), λ > and u = PC (u − λAu) for some λ > ⇒ u ∈ VI(C, A) It is also known that H satisfies the Opial’s condition [24], i.e., for any sequence {xn} ⊂ H with xn ⇀ x, the inequality lim inf xn − x < lim inf xn − y n→∞ n→∞ holds for every y Ỵ H with x ≠ y A set-valued mapping T : H ® 2H is called monotone if for all x, y Ỵ H, f Ỵ Tx and g Ỵ Ty imply 〈x - y, f - g〉 ≥ A monotone mapping T : H ® 2H is maximal if its graph 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) Ỵ H × H, 〈x - y, f - g〉 ≥ for every (y, g) Ỵ G(T) implies f Ỵ Tx Let A be a monotone, k-Lipschitz-continuous mapping of C into H and NCv be normal cone to C at v Ỵ C, i.e., NCv = {w Ỵ H : 〈v - u, w〉 ≥ 0, ∀u Ỵ C} Define Tv = Av + NC v if v ∈ C, ∅ if v ∈ C / Then, T is maximal monotone and Ỵ Tv if and only if v Î V I(C, A) (see [25]) For solving the equilibrium problem, let us assume that the bifunction F satisfies the following condition: (A1) F(x, x) = for all x Î C; (A2) F is monotone, i.e., F(x, y) + F(y, x) ≤ for any x, y Ỵ C; Page of 19 Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page of 19 (A3) for each x, y, z Ỵ C, lim F(tz + (1 − t)x, y) ≤ F(x, y); t↓0 (A4) for each x Î C, y ↦ F(x, y) is convex and lower semicontinuous We recall some lemmas which will be needed in the rest of this article Lemma 2.1.[7] Let C be a nonempty closed convex subset of H, let F be a bifunction from C × C to R satisfying (A1)-(A4) Let r >0 and x Ỵ H Then, there exists z Ỵ C such that F(z, y) + y − z, z − x ≥ 0, r for all y ∈ C Lemma 2.2.[8] Let C be a nonempty closed convex subset of H, let F be a bi-function from C × C to R satisfying (A1)-(A4) For r >0 and x Ỵ H, define a mapping Tr : H ® C as follows: Tr (x) = {z ∈ C : F(z, y) + y − z, z − x ≥ 0, ∀y ∈ C} r for all x Ỵ H Then, the following statements hold: (1) Tr is single-valued; (2) Tr is firmly nonexpansive, i.e., for any x, y Ỵ H, Tr (x) − Tr (y) ≤ Tr (x) − Tr (y), x − y ; (3) F(Tr) = EP (F); (4) EP(F) is closed and convex The main results We first show a strong convergence of an iterative algorithm based on extragradient and hybrid methods which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space Theorem 3.1 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 B be an a-inverse-strongly monotone mapping of C into H Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ GEP(F, B) = ∅ Assume that for all i Ỵ i=1 {1, 2, } and for any bounded subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {un}, {yn} and {zn} be sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ ⎪ F(u , y) + Bx , y − u + y − u , u − x ≥ 0, ∀y ∈ C, ⎪ ⎪ n n n n n n ⎪ r ⎪ ⎪ ⎨ y = (1 − γ )u + γ P (u n− λ Au ), n n n n C n n n ⎪ zn = (1 − αn − βn )xn + αn yn + βn Sn PC (un − λn Ayn ), ⎪ ⎪ C = {z ∈ C : ||z − z||2 ≤ ||x − z||2 + (3 − 3γ + α )b2 ||Au ||2 }, ⎪ n ⎪ n n n n n ⎪ ⎪ ⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0}, ⎪ ⎪ ⎩ xn+1 = PCn Qn x (3:1) Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page of 19 ), {rn} ⊂ [d, e] for some 4k d, e Ỵ (0, 2a), and {an}, {bn}, {gn} are three sequences in [0, 1] satisfying the conditions: for every n = 1, 2, where {ln} ⊂ [a, b] for some a, b ∈ (0, (i) an + bn ≤ for all n Ỵ N; lim (ii) n→∞ αn = 0; (iii) lim inf βn > 0; n→∞ lim (iv) n→∞ γn = and γn > for all n Ỵ N; Then, {xn}, {un}, {yn} and {zn} converge strongly to w = PΩ(x) Proof It is obvious that Cn is closed, and Qn is closed and convex for every n = 1, 2, Since Cn = {z ∈ H : zn − xn + zn − xn , xn − z ≤ (3 − 3γn + αn )b2 Aun }, we also have that Cn is convex for every n = 1, 2, It is easy to see that 〈xn - z, x xn〉 ≥ for all z Ỵ Qn and by (2.1), xn = PQ n x Let tn = PC(un - lnAyn) for every n = 1, 2, Let u Ỵ Ω and let {Trn } >be a sequence of mappings defined as in Lemma 2.2 Then u = PC (u − λn Au) = Trn (u − rn Bu) From un = Trn (xn − rn Bxn ) ∈ C and the ainverse strongly monotonicity of B, we have un − u = Trn (xn − rn Bxn ) − Trn (u − rn Bu) ≤ xn − rn Bxn − (u − rn Bu) ≤ xn − u 2 2 − 2rn xn − u, Bxn − Bu + rn Bxn − Bu ≤ xn − u − 2rn α Bxn − Bu = xn − u 2 + rn Bxn − Bu + rn (rn − 2α) Bxn − Bu (3:2) 2 ≤ xn − u From (2.2), the monotonicity of A, and u Ỵ V I(C, A), we have tn − u 2 ≤ un − λn Ayn − u − un − λn Ayn − tn = un − u − un − tn + 2λn Ayn , u − tn = un − u − un − tn + 2λn ( Ayn − Au, u − yn + Au, u − yn + Ayn , yn − tn ) ≤ un − u ≤ un − u = un − u 2 − un − tn − un − yn − un − yn 2 + 2λn Ayn , yn − tn − un − yn , yn − tn − yn − tn − y n − tn 2 + 2λn Ayn , yn − tn + un − λn Ayn − yn , tn − yn Further, Since yn = (1 - gn)un + gnPC(un - lnAun) and A is k-Lipschitz-continuous, we have un − λn Ayn − yn , tn − yn = un − λn Aun − yn , tn − yn + λn Aun − λn Ayn , tn − yn ≤ un − λn Aun − (1 − γn )un − γn PC (un − λn Aun ), tn − yn + λn Aun − Ayn tn − y n ≤ γn un − λn Aun − PC (un − λn Aun ), tn − yn − (1 − γn )λn Aun , tn − yn + λn k un − yn tn − yn Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page of 19 In addition, from the definition of PC, we have un − λn Aun − PC (un − λn Aun ), tn − yn = un − λn Aun − PC (un − λn Aun ), tn − (1 − γn )un − γn PC (un − λn Aun ) = (1 − γn ) un − λn Aun − PC (un − λn Aun ), tn − un +γn un − λn Aun − PC (un − λn Aun ), tn − PC (un − λn Aun ) ≤ (1 − γn ) un − λn Aun − PC (un − λn Aun ) tn − un ≤ (1 − γn )λn un − Aun − un ( tn − yn + yn − un ) ≤ (1 − γn )λn Aun ( tn − yn + yn − un ) It follows from b < tn − u 4k, ≤ un − u and (3.2) that γn > − un − yn +2(1 − γn )b Aun ≤ un − u 2 − un − yn +(1 − γn )(b Aun = un − u 2 + 2γn (1 − γn )b Aun ( tn − yn + yn − un ) tn − yn + 2bk un − yn − yn − tn 2 − yn − tn + (1 − γn )(2b2 Aun + tn − y n − (γn − bk) un − yn tn − y n + tn − yn ) + bk( un − yn + tn − y n + (1 − 2γn + bk) tn − yn ≤ un − u + 3(1 − γn )b2 Aun ≤ xn − u 2 + yn − un 2 ) (3:3) ) + 3(1 − γn )b Aun 2 + 3(1 − γn )b2 Aun In addition, from u Î V I(C, A) and (3.2), we have yn − u = (1 − γn )(un − u) + γn (PC (un − λn Aun ) − u) ≤ (1 − γn ) un − u + γn PC (un − λn Aun ) − PC (u) ≤ (1 − γn ) un − u ≤ (1 − γn ) un − u 2 + γn un − λn Aun − u + γn [ un − u 2 2 − 2λn Aun , un − u + λ2 Aun ] n ≤ un − u + b2 Aun ≤ xn − u (3:4) + b2 Aun Therefore, from (3.2) to (3.4) and zn = (1 - an - bn)xn + anyn + bnSntn and u = Snu, we have zn − u = (1 − αn − βn )xn + αn yn + βn Sn tn − u ≤ (1 − αn − βn ) xn − u + αn yn − u + βn Sn tn − u ≤ (1 − αn − βn ) xn − u + αn yn − u ≤ (1 − αn − βn ) xn − u + αn [ un − u +βn [ un − u ≤ xn − u 2 + β n tn − u 2 + b2 Aun ] (3:5) + 3(1 − γn )b2 Aun ] + (3 − 3γn + αn )b2 Aun ], for every n = 1, 2, and hence u Ỵ Cn So, Ω ⊂ Cn for every n = 1, 2, Next, let us show by mathematical induction that xn is well defined and Ω ⊂ Cn ∩ Qn for every n = 1, 2, For n = we have x1 = x Ỵ C and Q1 = C Hence, we obtain Ω ⊂C1 ∩ Q1 Suppose that xk is given and Ω ⊂ Ck ∩ Qk for some k Ỵ N Since Ω is nonempty, Ck ∩ Qk is a nonempty closed convex subset of H Hence, there exists a unique element xk+1 Ỵ Ck ∩ Qk such that xk+1 = PCk ∩Qk x It is also obvious that there holds 〈xk+1 - z, x - xk +1〉 ≥ for every z Ỵ Ck ∩ Qk Since Ω ⊂ Ck ∩ Qk, we have 〈xk+1 - z, x - xk+1〉 ≥ for every z Ỵ Ω and hence Ω ⊂ Qk+1 Therefore, we obtain Ω ⊂ Ck+1 ∩ Qk+1 Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page of 19 Let l0 = PΩx From xn+1 = PCn ∩Qn x and l0 v Ω ⊂ Cn ∩ Qn, we have xn+1 − x ≤ l0 − x (3:6) for every n = 1, 2, Therefore, {xn} is bounded From (3.2) to (3.5) and the lipschitz continuity of A, we also obtain that {un}, {yn}, {Aun}, {tn} and {zn} are bounded Since xn+1 Ỵ Cn ∩ Qn ⊂ Cn and xn = PQ n x, we have xn − x ≤ xn+1 − x for every n = 1, 2, It follows from (3.6) that limn®∞ ||xn - x|| exists Since xn = PQ n x and xn+1 Ỵ Qn, using (2.2), we have xn+1 − xn ≤ xn+1 − x − xn − x for every n = 1, 2, This implies that lim xn+1 − xn = n→∞ Since xn+1 Ỵ Cn, we have ||zn - xn+1||2 ≤ ||xn - xn+1||2 + (3 - 3gn + an)b2||Aun||2 and hence it follows from limn®∞ gn = and limn®∞ an = that limn®∞ ||zn - xn+1|| = Since ||xn − zn || ≤ ||xn − xn+1 || + ||xn+1 − zn || for every n = 1, 2, , we have ||xn - zn|| ® For u Ỵ Ω, from (3.5), we obtain ||zn − u||2 − ||xn − u||2 ≤ (−αn − βn )||xn − u||2 + αn ||yn − u||2 + βn ||Sn tn − u||2 ≤ (3 − 3γn + αn )b2 ||Aun ||2 Since limn®∞ gn = and limn®∞ an = 0, {xn}, {yn}, {Aun}, and {zn} are bounded, we have lim βn (||Sn tn − u||2 − ||xn − u||2 ) = n→∞ By lim inf n®∞ bn > 0, we get lim ||Sn tn − u||2 − ||xn − u||2 = n→∞ From (3.3) and u = Snu, we have lim ||Sn tn − u||2 − ||xn − u||2 ≤ lim ||tn − u||2 − ||xn − u||2 n→∞ n→∞ ≤ lim 3(1 − γn )b2 ||Aun ||2 = n→∞ Thus, limn®∞ ||tn - u||2 - ||xn - u||2 = From (3.3) and (3.2), we have (γn − bk)||un − yn ||2 + (2γn − − bk)||tn − yn ||2 ≤ ||xn − u||2 − ||tn − u||2 + 3(1 − γn )b2 ||Aun ||2 It follows that lim (γn − bk)||un − yn ||2 + (2γn − − bk)||tn − yn ||2 = n→∞ Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page of 19 1 and 2γn − − bk > Conse2 quently, limn®∞ ||un - yn|| = limn®∞ ||tn - yn|| = Since A is Lipschitz-continuous, we have limn®∞ ||Atn - Ayn|| = It follows from ||un - tn|| ≤ ||un - yn|| + ||tn - yn|| that limn®∞ ||un - tn|| = We rewrite the definition of zn as The assumptions on gn and ln imply that γn − bk > zn − xn = αn (yn − xn ) + βn (Sn tn − xn ) From limn®∞ ||zn - xn|| = 0, limn®∞ an = 0, the boundedness of {xn}, {yn} and lim infn®∞ bn > we infer that limn®∞ ||Sntn - xn|| = By (3.2)-(3.5), we have ||zn − u||2 ≤ (1 − αn − βn )||xn − u||2 + αn [||un − u||2 + b2 ||Aun ||2 ] + βn [||un − u||2 + 3(1 − γn )b2 ||Aun ||2 ] ≤ (1 − αn − βn )||xn − u||2 + αn [||xn − u||2 + rn (rn − 2α)||Bxn − Bu||2 + b2 ||Aun ||2 ] + βn [||xn − u||2 + rn (rn − 2α)||Bxn − Bu||2 + 3(1 − γn )b2 ||Aun ||2 ] (3:7) ≤ ||xn − u||2 + (αn + βn )rn (rn − 2α)||Bxn − Bu||2 + (3βn − 3βn γn + αn )b2 ||Aun ||2 ] Hence, we have (αn + βn )d(2α − e)||Bxn − Bu||2 ≤ (αn + βn )rn (2α − rn )||Bxn − Bu||2 ≤ ||xn − u||2 − ||zn − u||2 + (3βn − 3βn γn + αn )b2 ||Aun ||2 ≤ (||xn − u|| + ||zn − u||)||xn − zn || + (3βn − 3βn γn + αn )b2 ||Aun ||2 Since lim n®∞ a n = 1, lim infn®∞ b n > 0, lim n®∞ g n = 1, ||x n - z n || ® and the sequences {xn} and {zn} are bounded, we obtain ||Bxn - Bu|| ® For u Ỵ Ω, we have, from Lemma 2.2, ||un − u||2 = ||Trn (xn − rn Bxn ) − Trn (u − rn Bu)||2 ≤ Trn (xn − rn Bxn ) − Trn (u − rn Bu), xn − rn Bxn − (u − rn Bu) = {||un − u||2 + ||xn − rn Bxn − (u − rn Bu)||2 − ||xn − rn Bxn − (u − rn Bu) − (un − u)||2 } ≤ {||un − u||2 + ||xn − u||2 − ||xn − rn Bxn − (u − rn Bu) − (un − u)||2 } 2 = {||un − u||2 + ||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un − rn ||Bxn − Bu||2 } Hence, ||un − u||2 ≤ ||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un − rn ||Bxn − Bu||2 ≤ ||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un Then, by (3.5), we have ||zn − u||2 ≤ (1 − αn − βn )||xn − u||2 + αn [||un − u||2 + b2 ||Aun ||2 ] + βn [||un − u||2 + 3(1 − γn )b2 ||Aun ||2 ] ≤ (1 − αn − βn )||xn − u||2 + αn [(||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un ) + b2 ||Aun ||2 ] + βn [(||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un ) + 3(1 − γn )b2 ||Aun ||2 ] ≤ ||xn − u||2 + (−αn − βn )||xn − un ||2 + 2rn (αn + βn )||Bxn − Bu|| ||xn − un || + (3βn − 3βn γn + αn )b2 ||Aun ||2 Hence, (αn + βn )||xn − un ||2 ≤ ||xn − u||2 − ||zn − u||2 + 2rn (αn + βn )||Bxn − Bu|| ||xn − un || + (3βn − 3βn γn + αn )b2 ||Aun ||2 ≤ (||xn − u|| + ||zn − u||)||xn − zn || + 2rn (αn + βn )||Bxn − Bu|| ||xn − un || + (3βn − 3βn γn + αn )b2 ||Aun ||2 Since limn®∞ an = 0, lim infn®∞ bn > 0, limn®∞ gn = 1, ||xn - zn|| ® 0, ||Bxn - Bu|| ® and the sequences {xn}, {un} and {zn} are bounded, we obtain ||xn - un|| ® From ||zn tn|| ≤ ||zn - xn||+||xn - un||+||un - tn||, we have ||zn - tn|| ® Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page 10 of 19 From ||tn - xn|| ≤ ||tn - un|| + ||xn - un||, we also have ||tn - xn|| ® Since zn = (1 - an - bn)xn + anyn + bnSntn, we have bn(Sntn - tn) = (1 - an - bn)(tn xn) + an(tn - yn) + (zn - tn) Then βn ||Sn tn − tn || ≤ (1 − αn − βn )||tn − xn || + αn ||tn − yn || + ||zn − tn || and hence ||Sntn - tn|| ® At the same time, observe that for all i Î {1, 2, }, ||Si tn − tn || ≤ ||Si tn − Si (Sn tn )|| + ||Si (Sn tn ) − Sn tn || || + ||Sn tn − tn || ≤ 2||Sn tn − tn || + sup ||Si (Sn x) − Sn x|| x∈K It follows from (3.8) and the condition (*) that for all i Î {1, 2, }, lim ||Si tn − tn || = (3:9) n→∞ As {xn} is bounded, there exists a subsequence {xni } of {xn} such that xni ⇀ w From || xn - un|| ® 0, we obtain that uni ⇀ w From ||un - tn|| ® 0, we also obtain that tni ⇀ w Since {uni} ⊂ C and C is closed and convex, we obtain w Î C First, we show w Î GEP(F, B) By un = Trn (xn − rn Bxn ) ∈ C, we know that F(un , y) + Bxn , y − un + y − un , un − xn ≥ 0, ∀y ∈ C rn It follows from (A2) that Bxn , y − un + y − un , un − xn ≥ F(y, un ), ∀y ∈ C rn Hence, Bxni , y − uni + y − uni , uni − xni ≥ F(y, uni ), ∀y ∈ C rni (3:10) For t with < t ≤ and y Ỵ C, let yt = ty + (1 - t)w Since y Ỵ C and w Ỵ C, we obtain yt Ỵ C So, from (3.10) we have yt − uni , Byt ≥ yt − uni , Byt − yt − uni , Bxni un − xni − yt − uni , i + F(yt , uni ) rni = yt − uni , Byt − Buni + yt − uni , Buni − Bxni un − xni + F(yt , uni ) − yt − uni , i rni Since ||uni − xni || → 0, we have ||Buni − Bxni || → Further, from the inverse-strongly monotonicity of B, we have yt − uni , Byt − Buni ≥ Hence, from (A4), and uni uni −xni rni →0 w, we have yt − w, Byt ≥ F(yt , w), as i ® ∞ From (A1), (A4) and (3.11), we also have =F(yt , yt ) ≤ tF(yt , y) + (1 − t)F(yt , w) ≤ tF(yt , y) + (1 − t) yt − w, Byt = tF(yt , y) + (1 − t)t y − w, Byt (3:11) Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page 11 of 19 and hence ≤ F(yt , y) + (1 − t) y − w, Byt Letting t ® 0, we have, for each y Ỵ C, F(w, y) + y − w, Bw ≥ This implies that w Ỵ GEP(F, B) / i=1 We next show that w ∈ ∩∞ Fix(Si ) Assume w ∈ ∩∞ Fix(Si ) Since tni i=1 w = Si0 w for some i0 Ỵ {1, 2, } from the Opial condition, we have lim inf ||tni − w|| i→∞ w and < lim inf ||tni − Si0 w|| i→∞ ≤ lim inf{||tni − Si0 tni || + ||Si0 tni − Si0 w||} i→∞ ≤ lim inf ||tni − w|| i→∞ This is a contradiction Hence, we get w ∈ ∩∞ Fix(Si ) i=1 Finally we show w Ỵ V I(C, A) Let Tv = Av + NC v if v ∈ C, ∅ if v ∈ C where NCv is the normal cone to C at v Ỵ C We have already mentioned that in this case the mapping T is maximal monotone, and Ỵ Tv if and only if v Ỵ V I(C, A) Let (v, g) Ỵ G(T) Then Tv = Av + NCv and hence g - Av Ỵ NCv Hence, we have 〈v - t, g - Av〉 ≥ for all t Ỵ C On the other hand, from tn = PC(un lnAyn) and v Ỵ C, we have un − λn Ayn − tn , tn − v ≥ and hence v − tn , tn − un + Ayn ≥ λn Therefore, we have v − tni , g ≥ v − tni , Av tni − uni + Ayni λni tn − uni = v − tni , Av − Ayni − i λni tn − uni = v − tni , Av − Atni + Atni − Ayni − i λni ≥ v − tni , Av − v − tni , = v − tni , Av − Atni + v − tni , Atni − Ayni − v − tni , ≥ v − tni , Atni − Ayni − v − tni , tni − uni λni tni − uni λni Hence, we obtain 〈v - w, g〉 ≥ as i ® ∞ Since T is maximal monotone, we have w Ỵ T-10 and hence w Ỵ V I(C, A) This implies that w Ỵ Ω Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page 12 of 19 From l0 = PΩx, w Ỵ Ω and (3.6), we have ||l0 − x|| ≤ ||w − x|| ≤ lim inf ||xni − x|| ≤ lim sup ||xni − x|| ≤ ||l0 − x|| i→∞ i→∞ Hence, we obtain lim ||xni − x|| = ||w − x|| i→∞ w − x, we have xni − x → w − x, and hence xni → w Since xn = PQ n x From xni − x and l0 Ỵ Ω ⊂ Cn ∩ Qn ⊂ Qn, we have −||l0 − xni ||2 ≤ l0 − xni , xni − x + l0 − xni , x − l0 ≥ l0 − xni , x − l0 As i ® ∞, we obtain - ||l0 - w||2 ≥ 〈l0 - w, x - l0〉 ≥ by l0 = PΩx and w Î Ω Hence, we have w = l0 This implies that xn ® l0 It is easy to see un ® l0, yn ® l0 and zn ® l0 The proof is now complete By combining the arguments in the proof of Theorem 3.1 and those in the proof of Theorem 3.1 in [3], we can easily obtain the following weak convergence theorem for an iterative algorithm based on the extragradient method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space 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 to R satisfying (A1)-(A4) Let A be a monotone, and k-Lipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone mapping of C into H Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ GEP(F, B) = ∅ Assume that for all i Ỵ i=1 {1, 2, } and for any bounded subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {un} and {yn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ ⎨ y − un , un − xn ≥ 0, F(un , y) + Bxn , y − un + rn ⎪ ⎪ yn = PC (un − λn Aun ), ⎪ ⎪ ⎩x n+1 = βn xn + (1 − βn )Sn PC (un − λn Ayn ) ∀y ∈ C, (3:12) for every n = 1, 2, If {ln} ⊂ [a, b] for some a, b ∈ (0, ), {bn} ⊂ [δ, ε] for some δ, ε k Ỵ (0, 1) and {rn} ⊂ [d, e] for some d, e Î (0, 2a) Then, {xn}, {un} and {yn} converge weakly to w ẻ , where w = limnđ Pxn Applications By Theorems 3.1 and 3.2, we can obtain many new and interesting convergence theorems in a real Hilbert space We give some examples as follows: Let A = 0, by Theorems 3.1 and 3.2, respectively, we obtain the following results 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 to R satisfying (A1)-(A4) Let B be an a-inversestrongly monotone mapping of C into H Let S , S , be a family of infinitely Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page 13 of 19 nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ GEP(F, B) = ∅ i=1 Assume that for all i Ỵ {1, 2, } and for any bounded subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {un} {yn}, and {zn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ ⎪ F(u , y) + Bx , y − u + y − u , u − x ≥ 0, ∀y ∈ C, ⎪ ⎪ n n n n n n ⎪ ⎨ rn zn = (1 − αn − βn )xn + αn un + βn Sn un , ⎪ C = {z ∈ C : ||z − z||2 ≤ ||x − z||2 }, ⎪ n ⎪ n n ⎪ ⎪ Q = {z ∈ C : x − z, x − x ≥ 0}, ⎪ n ⎪ n n ⎪ ⎩ xn+1 = PCn ∩Qn x for every n = 1, 2, where {rn} ⊂ [d, e] for some d, e Ỵ (0, 2a), and {an}, {bn} are sequences in [0, 1] satisfying the conditions: (i) an + bn ≤ for all n Ỵ N; (ii) lim αn = 0; n→∞ (iii) lim inf βn > for all n Ỵ N; n→∞ Then, {xn}, {un}, and {zn} converge strongly to w = P∑(x) Theorem 4.2 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 B be an a-inversestrongly monotone mapping of C into H Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ GEP(F, B) = ∅ Assume i=1 that for all i Ỵ {1, 2, } and for any bounded subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn} and {un} be sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎨ y − un , un − xn ≥ 0, F(un , y) + Bxn , y − un + ⎪ rn ⎪ ⎩x = β x + (1 − β )S u n+1 n n n ∀y ∈ C, n n for every n = 1, 2, If {bn} ⊂ [δ, ε] for some δ, ε Ỵ (0, 1) and {rn} ⊂ [d, e] for some d, e Ỵ (0, 2a) Then, {xn} and {un} converge weakly to w Î ∑, where w = limn®∞ P∑xn Theorem 4.3 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 kLipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone mapping of C into H Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ GEP(F, B) = ∅ Assume that for all i Ỵ i=1 {1, 2, } and for any bounded subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page 14 of 19 Let {xn}, {un}, {yn}, and {zn} be sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ F(u , y) + Bx , y − u + y − u , u − x ≥ 0, ⎪ n n n n n n ⎪ rn ⎪ ⎪ ⎪ yn = PC (un − λn Aun ), ⎨ z = (1 − βn )xn + βn Sn PC (un − λn Ayn ), ⎪ n ⎪ Cn = {z ∈ C : ||zn − z||2 ≤ ||xn − z||2 , ⎪ ⎪ ⎪ ⎪ Q = {z ∈ C : x − z, x − x ≥ 0}, ⎪ n n n ⎪ ⎩ xn+1 = PCn ∩Qn x ∀y ∈ C, ), {rn} ⊂ [d, e] for some 4k d, e Ỵ (0, 2a), and {bn} is a sequence in [0, 1] satisfying lim inf βn > Then, {x n}, n→∞ for every n = 1, 2, where {ln} ⊂ [a, b] for some a, b ∈ (0, {un}, {yn}, and {zn} converge strongly to w = PΩ(x) Proof Putting gn = and an = 0, by Theorem 3.1, we obtain the desired result Let B = 0, by Theorems 3.1, 3.2, and 4.3, we obtain the following results Theorem 4.4 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 kLipschitz-continuous mapping of C into H Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ EP(F) = ∅ i=1 Assume that for all i Ỵ {1, 2, }, and for any bounded subset K of C, there holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {un}, {yn}, and {zn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ F(un , y) + r1 y − un , un − xn ≥ 0, ∀y ∈ C, ⎪ ⎪ n ⎪ ⎪ yn = (1 − γn )un + γn PC (un − λn Aun ), ⎨ z = (1 − αn − βn )xn + αn yn + βn Sn PC (un − λn Ayn ), ⎪ n ⎪ Cn = {z ∈ C : ||zn − z||2 ≤ ||xn − z||2 + (3 − 3γn + αn )b2 ||Aun ||2 }, ⎪ ⎪ ⎪ ⎪ Q = {z ∈ C : x − z, x − x ≥ 0}, ⎪ n n n ⎪ ⎩ xn+1 = PCn ∩Qn x ), {rn} ⊂ [d, +∞) for 4k some d >0, and {an}, {bn}, {gn} are three sequences in [0, 1] satisfying the following conditions: for every n = 1, 2, where {ln} ⊂ [a, b] for some a, b ∈ (0, (i) an + bn ≤ for all n Ỵ N; lim (ii) n→∞ αn = 0; (iii) lim inf βn > 0; n→∞ lim (iv) n→∞ γn = and γn > for all n Ỵ N; Then, {xn}, {un}, {yn} and {zn} converge strongly to w = PΛ(x) 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 to R satisfying (A1)-(A4) Let A be a monotone and kLipschitz-continuous mapping of C into H Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ EP(F) = ∅ i=1 Assume that for all i Î {1, 2, } and for any bounded subset K of C, thenthere holds Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 lim sup ||Sn x − Si (Sn x)|| = Page 15 of 19 ( ) n→∞ x∈K Let {xn}, {un}, and {yn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ ⎨ y − un , un − xn ≥ 0, ∀y ∈ C, F(un , y) + rn ⎪ ⎪ yn = PC (un − λn Aun ), ⎪ ⎪ ⎩x = β x + (1 − β )S P (u − λ Ay ) n+1 n n n n C n n n for every n = 1, 2, If {ln} ⊂ [a, b] for some a, b ∈ (0, ),{bn} ⊂ [δ, ε], for some δ, ε k Ỵ (0, 1) and {rn} ⊂ [d, +∞] for some d > 0, then {xn}, {un} and {yn} converge weakly to w Ỵ Λ, where w = limn®∞ PΛxn Theorem 4.6 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 Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ EP(F) = ∅ i=1 Assume that for all i Ỵ {1, 2, } and for any bounded subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {un} {yn}, and {zn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ ⎪ F(u , y) + y − u , u − x ≥ 0, ∀y ∈ C, ⎪ ⎪ n n n n ⎪ rn ⎪ ⎪ ⎨ y = P (u − λ Au ), n C n n n ⎪ zn = (1 − βn )xn + βn Sn PC (un − λn Ayn ), ⎪ ⎪ C = {z ∈ C : ||z − z||2 ≤ ||x − z||2 , ⎪ n ⎪ n n ⎪ ⎪ ⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0}, ⎪ ⎪ ⎩ xn+1 = PCn ∩Qn x ), {rn} ⊂ [d, +∞) and 4k for some d >0, and {bn} is a sequence in [0, 1] satisfying lim inf βn > Then, {xn}, n→∞ for every n = 1, 2, where {ln} ⊂ [a, b] for some a, b ∈ (0, {un}, {yn}, and {zn} converge strongly to w = PΛ(x) Let B = and F(x, y) = for x, y Ỵ C, by Theorems 3.1 and 4.3, we obtain the following results Theorem 4.7 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 Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) = ∅ Assume that for all i Ỵ {1, 2, } and for any bounded i=1 subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {yn}, and {zn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ yn = (1 − γn )xn + γn PC (xn − λn Axn ), ⎪ ⎪ ⎨ zn = (1 − αn − βn )xn + αn yn + βn Sn PC (xn − λn Ayn ), 2 2 ⎪ ⎪ Cn = {z ∈ C : ||zn − z|| ≤ ||xn − z|| + (3 − 3γn + αn )b ||Axn || }, ⎪ ⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0}, ⎪ ⎪ ⎩ xn+1 = PCn ∩Qn x Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page 16 of 19 for every n = 1, 2, where {ln} ⊂ [a, b] for some a, b ∈ (0, ), and {an}, {bn}, {gn}are 4k three sequences in [0, 1] satisfying the following conditions: (i) an + bn ≤ for all n Ỵ N; lim (ii) n→∞ αn = 0; (iii) lim inf βn > 0; n→∞ lim (iv) n→∞ γn = and γn > for all n Ỵ N; Then, {xn}, {yn}, and {zn} converge strongly to w = PΓ(x) Theorem 4.8 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 Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) = ∅ Assume that for all i Ỵ {1, 2, } and for any bounded i=1 subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {yn}, and {zn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎪ ⎪ yn = PC (xn − λn Axn ), ⎪ ⎪ ⎨ zn = (1 − βn )xn + βn Sn PC (xn − λn Ayn ), Cn = {z ∈ C : ||zn − z||2 ≤ ||xn − z||2 , ⎪ ⎪ ⎪ ⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0}, ⎪ ⎪ ⎩ xn+1 = PCn ∩Qn x ), and {b n } is a 4k sequence in [0, 1] satisfying lim inf βn > Then, {xn}, {yn}, and {zn} converge strongly n→∞ for every n = 1, 2, where {l n } ⊂ [a, b] for some a, b ∈ (0, to w = PΓ(x) Let F(x, y) = for x, y Ỵ C, then by Theorem 3.2 and the proof of Theorem 4.7 in [3], we obtain the following result Theorem 4.9 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 B be an ainverse-strongly monotone mapping of C into H Let S1, S2, be a family of infinitely nonexpansive mappings of C into itself such that ∞ = ∩i=1 Fix(Si ) ∩ VI(C, A) ∩ VI(C, B) = ∅ Assume that for all i Î {1, 2, } and for any bounded subset K of C, thenthere holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K ( ) Let {xn}, {un}, and {yn} be the sequences generated by ⎧ ⎪ x1 = x ∈ C, ⎪ ⎨ un = PC (xn − rn Bxn ), ⎪ yn = PC (un − λn Aun ), ⎪ ⎩ xn+1 = αn xn + (1 − αn )Sn PC (un − λn Ayn ) Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 Page 17 of 19 for every n = 1, 2, if {ln} ⊂ [a, b] for some a, b ∈ (0, ), {bn} ⊂ [δ, ε] for some δ, ε k Ỵ (0, 1) and {rn} ⊂ [d, e] for some d, e Ỵ (0, 2a) Then, {xn} and {un} converge weakly to w Ỵ Ξ, where w = limn®∞ PΞxn Remark 4.1 (i) For all n ≥ 1, let Sn = S be a nonexpansive mapping, by Theorems 3.2, 4.2, 4.7, 4.8, and 4.9 we recover Theorem 3.1 in [5], Theorem 3.1 in [1], Theorem in [26], Theorem 3.1 in [23], and Theorem 4.7 in [3] In addition, let A = 0, by Theorems 4.6 and 4.5, respectively, we recover Theorems 3.1 and 4.1 in [11] (ii) For all n ≥ 1, let Sn = S be a nonexpansive mapping, by Theorems 3.1, 4.3, and 4.4, respectively, we recover Theorems 4.3, 4.4, and 4.7 in [4] with some modified conditions on F (iii) Theorems 3.1, 3.2, 4.3-4.7 also improve the main results in [10,12,13] because the inverse strongly monotonicity of A has been replaced by the monotonicity and Lipschitz continuity of A The following result illustrates that there are the nonexpansive mappings S1, S2 , satisfying the condition (*) Lemma 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let T be a nonexpansive mapping of C into itself such that Fix(T) ≠ ∅ If we define n−1 j Sn (x) = T x for n Ỵ {1, 2, }, and x Ỵ C, then the following results hold: j=0 n (a) For any bounded subset K of C, there holds lim sup ||Sn x − T(Sn x)|| = n→∞ x∈K (b) ∩∞ Fix(Si ) = Fix(T) i=1 (c) for all i Ỵ {1, 2, } and for any bounded subset K of C, there holds lim sup ||Sn x − Si (Sn x)|| = n→∞ x∈K Proof (a) It is due to Bruck [27,28] (please also see Lemma 3.1 in [22]) (b) It follows from (a) that ∩∞ Fix(Si ) ⊆ Fix(T) i=1 Moreover, it is obvious that ∩∞ Fix(Si ) ⊇ Fix(T) Hence, ∩∞ Fix(Si ) = Fix(T) i=1 i=1 (c) It can be proved by mathematical induction In fact, it is clear that this conclusion holds for i = Assume that the conclusion holds for i = m, that is, for any bounded subset K of C, there holds lim sup ||Sn x − Sm (Sn x)|| = (4:1) n→∞ x∈K We now prove that the conclusion also holds for i = m + In fact, we observe that lim sup ||Sn x − Sm+1 (Sn x)|| ≤ lim sup ||Sn x − Sm (Sn x)|| + lim sup ||Sm (Sn x) − Sm+1 (Sn x)|| n→∞ x∈K n→∞ x∈K ⎡ ⎤ m−1 (4:2) j ⎣ ||T m (Sn x)|| + ⎦ ≤ lim sup ||Sn x − Sm (Sn x)|| + lim sup ||T (Sn x)|| n→∞ x∈K n→∞ x∈K m+1 m(m + 1) n→∞ x∈K j=0 Peng Fixed Point Theory and Applications 2011, 2011:12 http://www.fixedpointtheoryandapplications.com/content/2011/1/12 It is easy to verify that S1, S2, are nonexpansive mappings It follows from (4.1) and (4.2) that for any bounded subset K of C, there holds lim sup ||Sn x − Sm+1 (Sn x)|| = n→∞ x∈K From Lemma 4.1, we know that by Theorems 3.1 and 3.2, respectively, we can obtain the following results Theorem 4.10 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 B be an a-inverse-strongly monotone mapping of C into H Let T be a nonexpansive mapping of C into itself such that Θ = Fix(T)∩VI(C, A)∩GEP(F, B) ≠ ∅ Let {ln} ⊂ [a, b] for some a, b ∈ (0, ), {rn} ⊂ [d, e] 4k and for some d, e Î (0, 2a), and {an}, {bn}, and {gn} be three sequences in [0, 1] satisfying the following conditions: (i) an + bn ≤ for all n Ỵ N; (ii) lim αn = 0; n→∞ (iii) lim inf βn > 0; n→∞ n−1 j lim (iv) n→∞ γn = and γn > for all n Ỵ N; If we define Sn (x) = T x for n Î j=0 n {1, 2, }, and x Î C, then the sequences {xn}, {un}, {yn}, and {zn} generated by algorithm (3.1) converge strongly to w = PΘ(x) Theorem 4.11 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 B be an a-inverse-strongly monotone mapping of C into H, and T be a nonexpansive mapping of C into itself such that Θ = Fix(T)∩VI(C, A)∩GEP(F, B) ≠ ∅ Assume that {ln} ⊂ [a, b] for some a, b ∈ (0, ) {bn} k ⊂ [δ, ε] for some δ, ε Ỵ (0, 1), and {r n } ⊂ [d, e] some d, e Ỵ (0, 2a) If we define n−1 j Sn (x) = T x for n Ỵ {1, 2, } and x Ỵ C, then the sequences {xn}, {un}, and {yn} j=0 n generated by algorithm (3.12) converge weakly to w Ỵ Θ, where w = limn®∞ PΘxn Competing interests The authors declare that they have no competing interests Acknowledgements This research was supported by the National Natural Science Foundation of China, the Natural Science Foundation of Chongqing (Grant No CSTC, 2009BB8240), and the Special Fund of Chongqing Key Laboratory (CSTC) The author is grateful to the referees for their detailed comments and helpful suggestions, which have improved the presentation of this article Received: 30 November 2010 Accepted: 29 June 2011 Published: 29 June 2011 References Moudafi, A: Weak convergence theorems for nonexpansive mappings and equilibrium Problems J Nonlinear Convex Anal 9, 37–43 (2008) Moudafi, A, Thera, M: Proximal and dynamical approaches to equilibrium 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Point Theory and Applications 2011 2011:12 Submit your manuscript to a journal and benefit from: Convenient online submission Rigorous peer review Immediate publication on acceptance Open access: articles freely available online High visibility within the field Retaining the copyright to your article Submit your next manuscript at springeropen.com Page 19 of 19 ... method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for. .. finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational... this article as: Peng: Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings Fixed Point Theory and Applications 2011 2011:12