Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 728028, 43 pages doi:10.1155/2010/728028 Research Article Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities Chaichana Jaiboon1, 2 and Poom Kumam1 1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, KMUTT, Bangkok 10140, Thailand 2 Department of Mathematics, Faculty of Applied Liberal Arts, Rajamangala University of Technology, Rattanakosin, RMUTR, Bangkok 10100, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 31 October 2009; Accepted 1 February 2010 Academic Editor: Jong Kim Copyright q 2010 C. Jaiboon and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new iterative scheme for finding the common element of the set of solutions of the generalized equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inequality problems for a relaxed u, vcocoercive and ξ-Lipschitz continuous mapping in a real Hilbert space. Then, we prove the strong convergence of a common element of the above three sets under some suitable conditions. Our result can be considered as an improvement and refinement of the previously known results. 1. Introduction Variational inequalities introduced by Stampacchia 1 in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences. It is well known that the variational inequalities are equivalent to the fixed point problems. This alternative equivalent formulation has been used to suggest and analyze in variational inequalities. In particular, the solution of the variational inequalities can be computed using the iterative projection methods. It is well known that the convergence of a projection method requires the operator to be strongly monotone and Lipschitz continuous. Gabay 2 has shown that the convergence of a projection method can be proved for cocoercive operators. Note that cocoercivity is a weaker condition than strong monotonicity. Equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization which has been extended and generalized in many directions using novel and innovative technique; see 3, 4. Related to the equilibrium 2 Journal of Inequalities and Applications problems, we also have the problem of finding the fixed points of the nonexpansive mappings. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of a set of the solutions of the equilibrium problems and a set of the fixed points of infinitely finitely many nonexpansive mappings; see 5–7 and the references therein. In this paper, we suggest and analyze a new iterative method for finding a common element of a set of the solutions of generalized equilibrium problems and a set of fixed points of an infinite family of nonexpansive mappings and the set solution of the variational inequality problems for a relaxed u, v-cocoercive mapping in a real Hilbert space. Let H be a real Hilbert space and let E be a nonempty closed convex subset of H and PE is the metric projection of H onto E. Recall that a mapping f : E → E is contraction on E if there exists a constant α ∈ 0, 1 such that fx − fy ≤ αx − y for all x, y ∈ E. A mapping S of E into itself is called nonexpansive if Sx − Sy ≤ x − y for all x, y ∈ E. We denote by FS the set of fixed points of S, that is, FS {x ∈ E : Sx x}. If E ⊂ H is nonempty, bounded, closed, and convex and S is a nonexpansive mapping of E into itself, then FS is nonempty; see, for example, 8. We recalled some definitions as follows. Definition 1.1. Let B : E → H be a mapping. Then one has the following. 1 B is called monotone if Bx − By, x − y ≥ 0, for all x, y ∈ E. 2 B is called v-strongly monotone if there exists a positive real number v such that 2 Bx − By, x − y ≥ vx − y , ∀x, y ∈ E. 1.1 3 B is called ξ-Lipschitz continuous if there exists a positive real number ξ such that Bx − By ≤ ξx − y, ∀x, y ∈ E. 1.2 4 B is called η-inverse-strongly monotone, 9, 10 if there exists a positive real number η such that 2 Bx − By, x − y ≥ ηBx − By , ∀x, y ∈ E. 1.3 If η 1, we say that B is firmly nonexpansive. It is obvious that any η-inversestrongly monotone mapping B is monotone and 1/η-Lipschitz continuous. 5 B is called relaxed u, v-cocoercive if there exists a positive real number u, v such that 2 2 Bx − By, x − y ≥ −uBx − By vx − y , ∀x, y ∈ E. 1.4 For u 0, B is v-strongly monotone. This class of maps is more general than the class of strongly monotone maps. It is easy to see that we have the following implication: v-strongly monotonicity ⇒ relaxed u, v-cocoercivity. Journal of Inequalities and Applications 3 6 A set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H, f ∈ T x and g ∈ T y imply x − y, f − g ≥ 0. A monotone mapping T : H → 2H is maximal if the graph of GT 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 ∈ GT implies f ∈ T x. Let B be a monotone mapping of E into H and let NE w1 be the normal cone to E at w1 ∈ E, that is, NE w1 {w ∈ H : ϑ − w1 , w ≥ 0, ∀ϑ ∈ E}. 1.5 Define T w1 ⎧ ⎨Bw1 NE w1 , if w1 ∈ E, ⎩∅, ∈ E. if w1 / 1.6 Then T is the maximal monotone and 0 ∈ T w1 if and only if w1 ∈ VIE, B; see 11, 12 In addition, let D : E → H be a inverse-strongly monotone mapping. Let F be a bifunction of E × E into R, where R is the set of real numbers. The generalized equilibrium problem for F : E × E → R is to find x ∈ E such that F x, y Dx, y − x ≥ 0, ∀y ∈ E. 1.7 The set of such x ∈ E is denoted by EPF, D, that is, EPF, D x ∈ E : F x, y Dx, y − x ≥ 0, ∀y ∈ E . 1.8 Special Cases I If D ≡ 0 :the zero mapping, then the problem 1.7 is reduced to the equilibrium problem: Find x ∈ E such that F x, y ≥ 0, ∀y ∈ E. 1.9 The set of solutions of 1.9 is denoted by EPF, that is, EPF x ∈ E : F x, y ≥ 0, ∀y ∈ E . 1.10 II If F ≡ 0, the problem 1.7 is reduced to the variational inequality problem: Find x ∈ E such that Dx, y − x ≥ 0, ∀y ∈ E. 1.11 The set of solutions of 1.11 is denoted by VIE, D, that is, VIE, D x ∈ E : Dx, y − x ≥ 0, ∀y ∈ E . 1.12 4 Journal of Inequalities and Applications The generalized equilibrium problem 1.7 is very general in the sense that it includes, as special case, some optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, economics, and others see, e.g., 4, 13. Some methods have been proposed to solve the equilibrium problem and the generalized equilibrium problem; see, for instance, 5, 14–28. Recently, Combettes and Hirstoaga 29 introduced an iterative scheme of finding the best approximation to the initial data when EPF is nonempty and proved a strong convergence theorem. Very recently, Moudafi 24 introduced an itertive method for finding an element of EPF, D ∩ FS, where D : E → H is an inverse-strongly monotone mapping and then proved a weak convergence theorem. For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem for an η-inverse-strongly monotone, Takahashi and Toyoda 30 introduced the following iterative scheme: x0 ∈ E chosen arbitrary, xn1 αn xn 1 − αn SPE xn − τn Bxn , ∀n ≥ 0, 1.13 where B is an η-inverse-strongly monotone mapping, {αn } is a sequence in 0, 1, and {τn } is a sequence in 0, 2η. They showed that if FS ∩ VIE, B is nonempty, then the sequence {xn } generated by 1.13 converges weakly to some z ∈ FS ∩ VIE, B. On the other hand, Shang et al. 31 introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a relaxed u, v-cocoercive mapping in a real Hilbert space. Let S : E → E be a nonexpansive mapping. Starting with arbitrary initial x1 ∈ E, defined sequences {xn } recursively by xn1 αn fxn βn xn γn SPE I − τn Bxn , ∀n ≥ 1. 1.14 They proved that under certain appropriate conditions imposed on {αn }, {βn }, {γn }, and {τn }, the sequence {xn } converges strongly to z ∈ FS ∩ VIE, B, where z PFS∩VIE,B fz. In 2008, S. Takahashi and W. Takahashi 27 introduced the following iterative scheme for finding an element of FS ∩ EFF, D under some mild conditions. Let E be a nonempty closed convex subset of a real Hilbert space H. Let D be an η-inverse-strongly monotone mapping of E into H and let S be a nonexpansive mapping of E into itself such that FS ∩ EPF, D / ∅. Suppose x1 σ ∈ E and let {un }, {yn }, and {xn } by sequences generated by 1 F un , y Dxn , y − un y − un , un − xn ≥ 0, rn yn αn σ 1 − αn un , xn1 βn xn 1 − βn Syn , ∀y ∈ C, 1.15 where {αn } ⊂ 0, 1, {βn } ⊂ 0, 1, and {rn } ⊂ 0, 2η satisfy some parameters controlling conditions. They proved that the sequence {xn } defined by 1.15 converges strongly to a common element of FS ∩ EFF, D. On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, 32–35 and the Journal of Inequalities and Applications 5 references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping in a real Hilbert space H: min x∈E 1 Ax, x − x, b , 2 1.16 where E is the fixed point set of a nonexpansive mapping S on H and b is a given point in H. Assume that A is a strongly positive bounded linear operator on H; that is, there exists a constant γ > 0 such that Ax, x ≥ γx2 , 1.17 ∀x ∈ H. In 2006, Marino and Xu 36 considered the following iterative method: xn1 n γfxn 1 − n ASxn , ∀n ≥ 0. 1.18 They proved that if the sequence { n } of parameters satisfies appropriate conditions, then the sequence {xn } generated by 1.18 converges strongly to the unique of the variational inequality A − γf z, x − z ≥ 0, ∀x ∈ FS, 1.19 which is the optimality condition for the minimization problem min x∈FS 1 Ax, x − hx , 2 1.20 where h is a potential function for γf i.e., h x γfx for x ∈ H. In 2008, Qin et al. 26 proposed the following iterative algorithm: 1 F un , y y − un , un − xn ≥ 0, rn xn1 n γfxn I − ∀y ∈ H, 1.21 n ASPE I − τn Bun , where A is a strongly positive linear bounded operator and B is a relaxed cocoercive mapping of E into H. They prove that if the sequences { n }, {τn }, and {rn } of parameters satisfy appropriate condition, then the sequences {xn } and {un } both converge to the unique solution z of the variational inequality A − γf z, x − z ≥ 0, ∀x ∈ FS ∩ VIE, B ∩ EPF, 1.22 6 Journal of Inequalities and Applications which is the optimality condition for the minimization problem min x∈FS∩VIE,B∩EPF 1 Ax, x − hx , 2 1.23 where h is a potential function for γf i.e., h x γfx for x ∈ H. Furthermore, for finding approximate common fixed points of an infinite family of nonexpansive mappings {Tn } under very mild conditions on the parameters, we need the following definition. Definition 1.2 see 37. Let {Tn }∞ n1 be a sequence of nonexpansive mappings of E into itself be a sequence of nonnegative numbers in 0, 1. For each n ≥ 1, define a and let {μn }∞ n1 mapping Wn of E into itself as follows: Un,n1 I, Un,n μn Tn Un,n1 1 − μn I, Un,n−1 μn−1 Tn−1 Un,n 1 − μn−1 I, .. . Un,k μk Tk Un,k1 1 − μk I, Un,k−1 μk−1 Tk−1 Un,k 1 − μk−1 I, 1.24 .. . Un,2 μ2 T2 Un,3 1 − μ2 I, Wn Un,1 μ1 T1 Un,2 1 − μ1 I. Such a mappings Wn is called the W-mapping generated by T1 , T2 , . . . , Tn and μ1 , μ2 , . . . , μn . It is obvious that Wn is nonexpansive, and if x Tn x, then x Un,k Wn x. On the other hand, Yao et al. 38 introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite family of nonexpansive mappings on E. Starting with an arbitrary initial x1 ∈ H, define sequences {xn } and {un } recursively by xn1 1 F un , y y − un , un − xn ≥ 0, ∀y ∈ H, rn n γfxn βn xn 1 − βn I − n A Wn un , ∀n ≥ 1, 1.25 where { n } is a sequence in 0, 1. It is proved 38 that under certain appropriate conditions imposed on { n } and {rn }, the sequence {xn } generated by 1.25 converges strongly to z P∞n1 FTn ∩EPF I − A γfz. Very recently, Qin et al. 6 introduced an iterative scheme for finding a common fixed points of a finite family of nonexpansive mappings, the set of Journal of Inequalities and Applications 7 solutions of the variational inequality problem for a relaxed cocoercive mapping, and the set of solutions of the equilibrium problems in a real Hilbert space. Starting with an arbitrary initial x1 ∈ H, define sequences {xn } and {un } recursively by 1 y − un , un − xn ≥ 0, F un , y rn xn1 n γfWn xn I − n AWn PE I ∀y ∈ H, − τn Bun , 1.26 ∀n ≥ 1, where B is a relaxed u, v-cocoercive mapping and A is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on { n }, {τn }, and {rn }, the sequences {xn } and {un } generated by 1.26 converge strongly to some point z ∈ ∞ n1 FTn ∩ EPF ∩ VIE, B, which is a unique solution of the variation inequality: A − γf z, x − z ≥ 0, ∀x ∈ ∞ FTn ∩ EPF ∩ VIE, B 1.27 n1 and is also the optimality for some minimization problems. In this paper, motivated by iterative schemes considered in 1.15, 1.25, and 1.26 we will introduce a new iterative process 3.4 below for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the generalized equilibrium problem, and the set of solutions of variational inequality problem for a relaxed u, v-cocoercive mapping in a real Hilbert space. The results obtained in this paper improve and extend the recent ones announced by Yao et al. 38, S. Takahashi and W. Takahashi 27, and Qin et al. 6 and many others. 2. Preliminaries Let H be a real Hilbert space with inner product ·, · and norm · . Let E be a nonempty closed convex subset of H. We denote weak convergence and strong convergence by notations and → , respectively. Recall that the nearest point projection PE from H to E assigns each x ∈ H the unique point in PE x ∈ E satisfying the property x − PE x minx − y. 2.1 y∈E The following characterizes the projection PE . We need some facts tools in a real Hilbert space H which are listed as follows. Lemma 2.1. For any x ∈ H, z ∈ E, z PE x ⇐⇒ x − z, z − y ≥ 0, ∀y ∈ E. 2.2 It is well known that PE is a firmly nonexpansive mapping of H onto E and satisfies PE x − PE y2 ≤ PE x − PE y, x − y , ∀x, y ∈ H. 2.3 8 Journal of Inequalities and Applications Moreover, PE x is characterized by the following properties: PE x ∈ E and for all x ∈ H, y ∈ E, x − PE x, y − PE x ≤ 0. 2.4 Lemma 2.2 see 39. Let H be a Hilbert space, let E be a nonempty closed convex subset of H, and let B be a mapping of E into H. Let p ∈ E. Then for λ > 0, p ∈ VIE, B ⇐⇒ p PE p − λBp , 2.5 where PE is the metric projection of H onto E. It is clear from Lemma 2.2 that variational inequality and fixed point problem are equivalent. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems. Lemma 2.3 see 40. Each Hilbert space H satisfies Opials condition; that is, for any sequence x, the inequality {xn } ⊂ H with xn lim infxn − x < lim infxn − y n→∞ n→∞ 2.6 holds for each y ∈ H with y / x. Lemma 2.4 see 36. Assume that A is a strongly positive linear bounded operator on H with coefficient γ > 0 and 0 < ρ ≤ A−1 . Then I − ρA ≤ 1 − ργ. For solving the equilibrium problem for a bifunction F : E × E → R, let us assume that F satisfies the following conditions: A1 Fx, x 0, for all x ∈ E; A2 F is monotone, that is, Fx, y Fy, x ≤ 0, for all x, y ∈ E; A3 limt↓0 Ftz 1 − tx, y ≤ Fx, y, for all x, y, z ∈ E; A4 for each x ∈ E, y → Fx, y is convex and lower semicontinuous. The following lemma appears implicitly in 4. Lemma 2.5 see 4. Let E be a nonempty closed convex subset of H and let F be a bifunction of E × E into R satisfying (A1)–(A4). Let r > 0 and x ∈ H. Then, there exists z ∈ E such that 1 F z, y y − z, z − x ≥ 0, r The following lemma was also given in 5. ∀y ∈ E. 2.7 Journal of Inequalities and Applications 9 Lemma 2.6 see 5. Assume that F : E × E → R satisfies (A1)–(A4). For r > 0 and x ∈ H, define a mapping Tr : H → E as follows: Tr x 1 z ∈ E : F z, y y − z, z − x ≥ 0, ∀y ∈ E , r 2.8 for all z ∈ H. Then, the following holds: 1 Tr is single-valued; 2 Tr is firmly nonexpansive, that is, for any x, y ∈ H, Tr x − Tr y2 ≤ Tr x − Tr y, x − y ; 2.9 3 FTr EPF; 4 EPF is closed and convex. Remark 2.7. Replacing x with x − rDx ∈ H in 2.7, then there exists z ∈ E, such that 1 F z, y Dx, y − z y − z, z − x ≥ 0, r ∀y ∈ E. 2.10 Lemma 2.8 see 41. Let E be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be nonexpansive mappings of E into itself such that ∞ n1 FTn is nonempty, and let μ1 , μ2 , . . . be real numbers such that 0 ≤ μn ≤ b < 1 for every n ≥ 1. Then, for every x ∈ E and k ∈ N, the limit limn → ∞ Un,k x exists. Using Lemma 2.8, one can define a mapping W of E into itself as follows: Wx lim Wn x lim Un,1 x, n→∞ n→∞ 2.11 for every x ∈ E. Such a W is called the W-mapping generated by T1 , T2 , . . . and μ1 , μ2 , . . .. Throughout this paper, we will assume that 0 ≤ μn ≤ b < 1 for every n ≥ 1. Then, we have the following results. Lemma 2.9 see 41. Let E be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be nonexpansive mappings of E into itself such that ∞ n1 FTn is nonempty, let μ1 , μ2 , . . . be real numbers such that 0 ≤ μn ≤ b < 1 for every n ≥ 1. Then, FW ∞ n1 FTn . Lemma 2.10 see 7. If {xn } is a bounded sequence in E, then limn → ∞ Wxn − Wn xn 0. Lemma 2.11 see 42. Let {xn } and {zn } be bounded sequences in a Banach space X and let {βn } be a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose xn1 1 − βn zn βn xn for all integers n ≥ 0 and lim supn → ∞ zn1 − zn − xn1 − xn ≤ 0. Then, limn → ∞ zn − xn 0. Lemma 2.12. Let H be a real Hilbert space. Then the following inequality holds: 1 x y2 ≤ x2 2y, x y, 2 x y2 ≥ x2 2y, x for all x, y ∈ H. 10 Journal of Inequalities and Applications Lemma 2.13 see 43. Assume that {an } is a sequence of nonnegative real numbers such that an1 ≤ 1 − ln an σn , ∀n ≥ 0, 2.12 where {ln } is a sequence in 0, 1 and {σn } is a sequence in R such that 1 ∞ n1 ln ∞, 2 lim supn → ∞ σn /ln ≤ 0 or ∞ n1 |σn | < ∞. Then limn → ∞ an 0. 3. Main Results In this section, we prove a strong convergence theorem of a new iterative method 3.4 for an infinite family of nonexpansive mappings and relaxed u, v-cocoercive mappings in a real Hilbert space. We first prove the following lemmas. Lemma 3.1. Let H be a real Hilbert space, let E be a nonempty closed convex subset of H, and let D : E → H be η-inverse-strongly monotone. It 0 ≤ rn ≤ 2η, then I − rn D is a nonexpansive mapping in H. Proof. For all x, y ∈ E and 0 ≤ rn ≤ 2η, we have I − rn Dx − I − rn Dy2 x − y − rn Dx − Dy2 2 2 x − y − 2rn x − y, Dx − Dy rn2 Dx − Dy 2 2 ≤ x − y − 2rn ηDx − Dy rn2 Dx − Dy 3.1 2 2 x − y rn rn − 2η Dx − Dy 2 ≤ x − y . So, I − rn D is a nonexpansive mapping of E into H. Lemma 3.2. Let H be a real Hilbert space, let E be a nonempty closed convex subset of H, and let B : E → H be a relaxed u, v-cocoercive and ξ-Lipschitz continuous. If 0 ≤ τn ≤ 2v − uξ2 /ξ2 , v > uξ2 , then I − τn B is a nonexpansive mapping in H. Proof. For any x, y ∈ E and τn ≤ 2v − uξ2 /ξ2 , v > uξ2 . Putting r 1 2τn uξ2 − 2τn v τn2 ξ2 , we obtain 2 v − uξ2 τn ≤ ⇐⇒ τn ξ2 2uξ2 − 2v ≤ 0 ξ2 ⇐⇒ τn2 ξ2 2τn uξ2 − 2τn v ≤ 0 ⇐⇒ 1 τn2 ξ2 2τn uξ2 − 2τn v ≤ 1, 3.2 Journal of Inequalities and Applications 11 that is, r ≤ 1. It follows that I − τn Bx − I − τn By2 x − y − τn Bx − By2 2 2 x − y − 2τn x − y, Bx − By τn2 Bx − By 2 2 2 2 ≤ x − y − 2τn −uBx − By vx − y τn2 Bx − By 2 2 2 2 ≤ x − y 2τn uξ2 x − y − 2τn vx − y τn2 ξ2 x − y 2 1 2τn uξ2 − 2τn v τn2 ξ2 x − y 2 r x − y 2 ≤ x − y , 3.3 for all x, y ∈ E. Thus I − τn Bx − I − τn By ≤ x − y. So, I − τn B is a nonexpansive mapping of E into H. Now, we prove the following main theorem. Theorem 3.3. Let E be a nonempty closed convex subset of a real Hilbert space H, and let F : E×E → R be a bifunction satisfying (A1)–(A4). Let 1 {Tn } be an infinite family of nonexpansive mappings of E into E; 2 D be an η-inverse strongly monotone mappings of E into H; 3 B be relaxed u, v-cocoercive and ξ-Lipschitz continuous mappings of E into H. Assume that Θ : ∞ ∅. Let f : E → E be a contraction mapping n1 FTn ∩ EPF, D ∩ VIE, B / with 0 < α < 1 and let A be a strongly positive linear bounded operator on H with coefficient γ > 0 and 0 < γ < γ/α. Let {xn }, {yn }, {kn }, and {un } be sequences generated by x1 ∈ E chosen arbitrary, xn1 1 F un , y Dxn , y − un y − un , un − xn ≥ 0, ∀y ∈ E, rn yn ϕn un 1 − ϕn Wn PE un − δn Bun , kn αn xn 1 − αn Wn PE yn − λn Byn , n γfWn xn βn xn 1 − βn I − n A Wn PE kn − τn Bkn , ∀n ≥ 1, 3.4 where {Wn } is the sequence generated by 1.24 and { n }, {αn }, {ϕn }, and {βn } are sequences in 0, 1 satisfy the following conditions: C1 limn → ∞ n 0, ∞ n1 n ∞, C2 limn → ∞ αn limn → ∞ ϕn 0, 12 Journal of Inequalities and Applications C3 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1, C4 lim infn → ∞ rn > 0 and limn → ∞ |rn1 − rn | 0, C5 limn → ∞ |λn1 − λn | limn → ∞ |δn1 − δn | limn → ∞ |τn1 − τn | 0, C6 {τn }, {λn }, {δn } ⊂ a, b for some a, b with 0 ≤ a ≤ b ≤ 2v − uξ2 /ξ2 , v > uξ2 , C7 {rn } ⊂ c, d for some c, d with 0 < c < d < 2η. Then, {xn } and {un } converge strongly to a point z ∈ Θ, where z PΘ I − A γfz, which solves the variational inequality A − γf z, x − z ≥ 0, ∀x ∈ Θ, 3.5 which is the optimality condition fot the minimization problem min x∈Θ 1 Ax, x − hx , 2 3.6 where h is a potential function for γf (i.e., h x γfx for x ∈ H). Proof. Since limn → ∞ n 0 by the condition C1 and lim supn → ∞ βn < 1, we may assume, without loss of generality, that n ≤ 1 − βn A−1 . Since A is a strongly positive bounded linear operator on H, then A sup{|Ax, x| : x ∈ H, x 1}. 3.7 Observe that 1 − βn I − nA x, x 1 − βn − ≥ 1 − βn − n Ax, x n A 3.8 ≥ 0, that is to say 1 − βn I − 1 − βn I − nA is positive. It follows that sup 1 − βn I − nA sup 1 − βn − ≤ 1 − βn − nA n Ax, x x, x : x ∈ H, x 1 : x ∈ H, x 1 3.9 n γ. We will divide the proof of Theorem 3.3 into six steps. Step 1. We prove that there exists z ∈ E such that z P∞n1 FTn ∩EPF,D∩VIE,B I − A γfz. Journal of Inequalities and Applications 13 Let I P∞n1 FTn ∩EPF,D∩VIE,B . Note that f is a contraction mapping of E into itself with coefficient α ∈ 0, 1. Then, we have I I − A γf x − I I − A γf y ≤ I − A γf x − I − A γf y ≤ I − Ax − y γ fx − f y ≤ 1 − γ x − y γαx − y 1 − γ − αγ x − y, ∀x, y ∈ H. 3.10 Therefore, II − A γf is a contraction mapping of E into itself. Therefore by the Banach Contraction Mapping Principle guarantee that II − A γf has a unique fixed point, say z ∈ E. That is, z II − A γfz P∞n1 FTn ∩EPF,D∩VIE,B I − A γfz. Step 2. We prove that {xn } is bounded. Since 1 F un , y Dxn , y − un y − un , un − xn ≥ 0, rn ∀y ∈ E, 3.11 we obtain 1 F un , y y − un , un − I − rn Dxn ≥ 0, rn ∀y ∈ E. 3.12 From Lemma 2.6, we have un Trn xn − rn Dxn , for all n ∈ N. For any p ∈ Θ : ∞ n1 FTn ∩ EPF, D ∩ VIE, B, it follows from p ∈ EPF, D that F p, y y − p, Dp ≥ 0, ∀y ∈ E. 3.13 So, we have 1 F p, y y − p, p − p − rn Dp ≥ 0, rn ∀y ∈ E. 3.14 By Lemma 2.6 again, we have p Trn p − rn Dp, for all n ∈ N. If follows that un − p Tr xn − rn Dxn − Tr p − rn Dp n n ≤ xn − rn Dxn − p − rn Dp I − rn Dxn − I − rn Dp ≤ xn − p. 3.15 14 Journal of Inequalities and Applications If we applied Lemma 3.2, we get I − λn B and I − δn B are nonexpansive. Since p ∈ VIE, B and Wn is a nonexpansive, we have p Wn PE p − λn Bp Wn PE p − δn Bp, and we have yn − p ≤ ϕn un − p 1 − ϕn ≤ ϕn un − p 1 − ϕn ϕn un − p 1 − ϕn ≤ ϕn un − p 1 − ϕn ≤ un − p ≤ xn − p. Wn PE un − δn Bun − Wn PE p − δn Bp un − δn Bun − p − δn Bp I − δn Bun − I − δn Bp un − p 3.16 It follows that kn − p ≤ αn xn − p 1 − αn Wn PE yn − λn Byn − Wn PE p − λn Bp ≤ αn xn − p 1 − αn yn − λn Byn − p − λn Bp αn xn − p 1 − αn I − λn Byn − I − λn Bp 3.17 ≤ αn xn − p 1 − αn yn − p ≤ αn xn − p 1 − αn xn − p xn − p, which yields that xn1 − p n γfxn − Ap βn xn − p 1 − βn I − n A Wn PE kn − τn Bkn − p ≤ 1 − βn − n γ PE I − τn Bkn − p βn xn − p n γfxn − Ap ≤ 1 − βn − n γ kn − p βn xn − p n γfxn − Ap ≤ 1 − βn − n γ xn − p βn xn − p n γfxn − Ap ≤ 1− ≤ 1− fxn − f p n γf p − Ap n γα xn − p n γf p − Ap γf p − Ap xn − p γ − αγ n . γ − αγ xn − p xn − p nγ nγ 1 − γ − αγ n nγ 3.18 This in turn implies that γf p − Ap xn − p ≤ max x1 − p, , γ − αγ n ∈ N. 3.19 Journal of Inequalities and Applications 15 Therefore, {xn } is bounded. We also obtain that {un }, {kn }, {yn }, {Bun }, {Bkn }, {Byn }, {Wn un }, {Wn kn }, {Wn yn }, and {fWn xn } are all bounded. Step 3. We claim that limn → ∞ xn1 − xn 0 and limn → ∞ Wn θn − xn 0. From Lemma 2.6, we have un Trn xn − rn Dxn and un1 Trn1 xn1 − rn1 Dxn1 . Let n xn − rn Dxn , we get un Trn n , un1 Trn1 n1 , and so 1 y − un , un − n ≥ 0, F un , y rn ∀y ∈ E, 1 F un1 , y y − un1 , un1 − n1 ≥ 0, rn1 ∀y ∈ E. 3.20 3.21 Putting y un1 in 3.20 and y un in 3.21, we have Fun , un1 Fun1 , un 1 un1 − un , un − n ≥ 0, rn 1 rn1 3.22 un − un1 , un1 − n1 ≥ 0. So, from the monotonicity of F, we get un − n un1 − n1 − ≥ 0, un1 − un , rn rn1 3.23 rn un1 − un , un − un1 un1 − n − un1 − n1 ≥ 0. rn1 3.24 and hence Without loss of generality, let us assume that there exists a real number c such that rn > c > 0 for all n ∈ N. Then, we have 2 un1 − un ≤ rn un1 − un , n1 − n 1 − un1 − n1 rn1 rn ≤ un1 − un n1 − n 1 − un1 − n1 , rn1 3.25 16 Journal of Inequalities and Applications and hence 1 un1 − un ≤ n1 − n |rn1 − rn |un1 − n1 c 1 xn1 − rn1 Dxn1 − xn − rn Dxn |rn1 − rn |un1 − n1 c ≤ xn1 − rn1 Dxn1 − xn − rn1 Dxn |rn1 − rn |Dxn 1 |rn1 − rn |un1 − n1 c 3.26 1 ≤ xn1 − xn |rn1 − rn |Dxn |rn1 − rn |un1 − n1 c ≤ xn1 − xn M1 |rn1 − rn |, where M1 sup{Dxn 1/cun1 − n1 : n ∈ N}. Put θn PE kn − τn Bkn , φn PE yn − λn Byn , and ψn PE un − δn Bun . Since I − τn B, I − λn B, and I − δn B are nonexpansive, then we have the following some estimates: ψn1 − ψn ≤ PE un1 − δn1 Bun1 − PE un − δn Bun ≤ un1 − δn1 Bun1 − un − δn Bun un1 − δn1 Bun1 − un − δn1 Bun δn − δn1 Bun ≤ un1 − δn1 Bun1 − un − δn1 Bun |δn − δn1 |Bun 3.27 I − δn1 Bun1 − I − δn1 Bun |δn − δn1 |Bun ≤ un1 − un |δn − δn1 |Bun . Similarly, we can prove that φn1 − φn ≤ yn1 − yn |λn − λn1 |Byn , 3.28 θn1 − θn ≤ kn1 − kn |τn − τn1 |Bkn . 3.29 Since Ti and Un,i are nonexpansive, we deduce that, for each n ≤ 1, Wn1 ψn − Wn ψn μ1 T1 Un1,2 ψn − μ1 T1 Un,2 ψn ≤ μ1 Un1,2 ψn − Un,2 ψn μ1 μ2 T2 Un1,3 ψn − μ2 T2 Un,3 ψn ≤ μ1 μ2 Un1,3 ψn − Un,3 ψn Journal of Inequalities and Applications 17 .. . ≤ n μi Un1,n1 ψn − Un,n1 ψn i1 ≤ M2 n μi , i1 3.30 where M2 ≥ 0 is a constant such that Un1,n1 ψn − Un,n1 ψn ≤ M2 for all n ≥ 0. Similarly, we can obtain that there exist nonnegative numbers M3 , M4 such that Un1,n1 ϕn − Un,n1 ϕn ≤ M3 , Un1,n1 θn − Un,n1 θn ≤ M4 , n Wn1 φn − Wn φn ≤ M3 μi , Wn1 θn − Wn θn ≤ M4 3.31 and so are i1 n μi . 3.32 i1 Observing that yn1 yn ϕn un 1 − ϕn Wn ψn , ϕn1 un1 1 − ϕn1 Wn1 ψn1 , 3.33 we obtain yn − yn1 ϕn un − un1 1 − ϕn Wn ψn − Wn1 ψn1 Wn1 ψn1 − un1 ϕn1 − ϕn , 3.34 which yields that yn − yn1 ≤ ϕn un − un1 1 − ϕn Wn1 ψn1 − Wn ψn ϕn1 − ϕn Wn1 ψn1 − un1 ≤ ϕn un − un1 1 − ϕn Wn1 ψn1 − Wn1 ψn Wn1 ψn − Wn ψn ϕn1 − ϕn Wn1 ψn1 − un1 ≤ ϕn un − un1 1 − ϕn ψn1 − ψn Wn1 ψn − Wn ψn ϕn1 − ϕn Wn1 ψn1 − un1 ≤ ϕn un − un1 1 − ϕn ψn1 − ψn Wn1 ψn − Wn ψn ϕn1 − ϕn Wn1 ψn1 − un1 . 3.35 18 Journal of Inequalities and Applications Substitution of 3.27 and 3.30 into 3.35 yields that yn − yn1 ≤ ϕn un − un1 1 − ϕn {un1 − un |δn − δn1 |Bun } M2 n μi ϕn1 − ϕn Wn1 ψn1 − un1 i1 un − un1 1 − ϕn |δn − δn1 |Bun M2 n 3.36 μi Wn1 ψn1 − un1 ϕn1 − ϕn i1 n ≤ un − un1 M5 |δn − δn1 | ϕn1 − ϕn M2 μi , i1 where M5 is an appropriate constant such that M5 max{supn≥1 Bun , supn≥1 Wn ψn − un }. Observing that kn αn xn 1 − αn Wn φn , kn1 αn1 xn1 1 − αn1 Wn φn1 , 3.37 we obtain kn − kn1 αn xn − xn1 1 − αn Wn φn − Wn1 φn1 Wn1 φn1 − xn1 αn1 − αn , 3.38 which yields that kn − kn1 ≤ αn xn − xn1 1 − αn Wn φn − Wn1 φn1 |αn1 − αn |Wn1 φn1 − xn1 ≤ αn xn − xn1 1 − αn Wn1 φn1 − Wn1 φn Wn1 φn − Wn φn |αn1 − αn |Wn1 φn1 − xn1 ≤ αn xn − xn1 1 − αn φn1 − φn Wn1 φn − Wn φn |αn1 − αn |Wn1 φn1 − xn1 . 3.39 Journal of Inequalities and Applications 19 Substitution of 3.28 and 3.32 into 3.39 yields that kn − kn1 ≤ αn xn − xn1 1 − αn yn1 − yn |λn − λn1 |Byn M3 n μi |αn1 − αn |Wn1 φn1 − xn1 i1 αn xn − xn1 1 − αn yn1 − yn 1 − αn |λn − λn1 |Byn M3 n μi |αn1 − αn |Wn1 φn1 − xn1 3.40 i1 n ≤ αn xn − xn1 1 − αn yn1 − yn M3 μi i1 M6 |λn − λn1 | |αn1 − αn |, where M6 is an appropriate constant such that M6 max{supn≥1 Byn , supn≥1 Wn φn − xn }. Substituting 3.26 and 3.36 into 3.40, we obtain n μi kn − kn1 ≤ αn xn − xn1 1 − αn un − un1 M5 |δn − δn1 | ϕn1 − ϕn M2 M3 n i1 μi M6 |λn − λn1 | |αn1 − αn | i1 αn xn − xn1 1 − αn un − un1 1 − αn M5 |δn − δn1 | ϕn1 − ϕn n n 1 − αn M2 μi M3 μi M6 |λn − λn1 | |αn1 − αn | i1 i1 ≤ αn xn − xn1 1 − αn {xn1 − xn M1 |rn1 − rn |} n 1 − αn M5 |δn − δn1 | ϕn1 − ϕn 1 − αn M2 μi M3 n i1 μi M6 |λn − λn1 | |αn1 − αn | i1 αn xn − xn1 1 − αn xn1 − xn 1 − αn M1 |rn1 − rn | n 1 − αn M5 |δn − δn1 | ϕn1 − ϕn 1 − αn M2 μi M3 n i1 μi M6 |λn − λn1 | |αn1 − αn | i1 ≤ xn − xn1 M1 |rn1 − rn | M2 n i1 μi M3 n μi i1 M5 |δn − δn1 | ϕn1 − ϕn M6 |λn − λn1 | |αn1 − αn | n n ≤ xn − xn1 M2 μi M3 μi i1 i1 K1 |rn1 − rn | |δn − δn1 | ϕn1 − ϕn |λn − λn1 | |αn1 − αn | , 3.41 where K1 is an appropriate constant such that K1 max{M1 , M5 , M6 }. 20 Journal of Inequalities and Applications Substituting 3.41 into 3.29, we obtain θn1 − θn ≤ kn1 − kn |τn − τn1 |Bkn ≤ xn − xn1 M2 n μi M3 n i1 μi i1 K1 |rn1 − rn | |δn − δn1 | ϕn1 − ϕn |λn − λn1 | |αn1 − αn | |τn − τn1 |Bkn ≤ xn − xn1 M2 n n μi M3 μi i1 i1 K2 |rn1 − rn | |δn − δn1 | ϕn1 − ϕn |λn − λn1 | |αn1 − αn | |τn − τn1 | , 3.42 where K2 is an appropriate constant such that K2 max{supn≥1 Bkn , K1 }. Define xn1 1 − βn zn βn xn , n ≥ 1. 3.43 Observe that from the definition zn , we obtain zn1 − zn n1 γfWn1 xn1 − n1 1 − βn1 n γfWn xn n n1 1 − βn1 I − 1 − βn1 1 − βn I − 1 − βn γfWn1 xn1 − 1 − βn 1 − βn1 AWn θn − n 1 − βn n1 1 − βn1 nA n1 A Wn1 θn1 Wn θn γfWn xn Wn1 θn1 − Wn θn AWn1 θn1 γfWn1 xn1 − AWn1 θn1 Wn1 θn1 − Wn1 θn Wn1 θn − Wn θn . n 1 − βn AWn θn − γfWn xn 3.44 Journal of Inequalities and Applications 21 It follows from 3.32, 3.42, and 3.44 that zn1 − zn − xn1 − xn ≤ n1 1 − βn1 γfWn1 xn1 AWn1 θn1 n AWn θn γfWn xn 1 − βn Wn1 θn1 − Wn1 θn Wn1 θn − Wn θn − xn1 − xn ≤ n1 1 − βn1 γfWn1 xn1 AWn1 θn1 n AWn θn γfWn xn 1 − βn θn1 − θn Wn1 θn − Wn θn − xn1 − xn ≤ n1 1 − βn1 M2 γfWn1 xn1 AWn1 θn1 n AWn θn γfWn xn 1 − βn n n n μi M3 μi M4 μi i1 i1 i1 K2 |rn1 − rn | |δn − δn1 | ϕn1 − ϕn |λn − λn1 | |αn1 − αn | |τn − τn1 | ≤ n1 1 − βn1 3K γfWn1 xn1 AWn1 θn1 n n 1 − βn AWn θn γfWn xn μi i1 K2 |rn1 − rn | |δn − δn1 | ϕn1 − ϕn |λn − λn1 | |αn1 − αn | |τn − τn1 | , 3.45 where K is an appropriate constant such that K max{M2 , M3 , M4 }. It follows from conditions C1, C2, C3, C4, C5, and 0 < μi ≤ b < 1, for all i ≥ 1 lim supzn1 − zn − xn1 − xn ≤ 0. n→∞ 3.46 Hence, by Lemma 2.11, we obtain lim zn − xn 0. 3.47 lim xn1 − xn lim 1 − βn zn − xn 0. 3.48 n→∞ It follows that n→∞ n→∞ 22 Journal of Inequalities and Applications Applying 3.48 and conditions in Theorem 3.3 to 3.26, 3.41, and 3.42, we obtain that lim un1 − un lim kn1 − kn lim θn1 − θn 0. n→∞ n→∞ n→∞ 3.49 From 3.49, C2, C5, and 0 < μi ≤ b < 1, for all i ≥ 1, we also have lim yn1 − yn 0. 3.50 n→∞ Since xn1 n γfWn xn βn xn 1 − βn I − n AWn θn , we have xn − Wn θn ≤ xn − xn1 xn1 − Wn θn xn − xn1 n γfWn xn βn xn 1 − βn I − nA Wn θn − Wn θn xn − xn1 n γfWn xn − AWn θn βn xn − Wn θn ≤ xn − xn1 n 3.51 γfWn xn AWn θn βn xn − Wn θn , that is, xn − Wn θn ≤ 1 n γfWn xn AWn θn . xn − xn1 1 − βn 1 − βn 3.52 By C1, C3, and 3.48 it follows that lim Wn θn − xn 0. n→∞ 3.53 Step 4. We claim that the following statements hold: i limn → ∞ un − θn 0; ii limn → ∞ xn − un 0; iii limn → ∞ Wn θn − θn 0. Since B is relaxed u, v-cocoercive and ξ-Lipschitz continuous mappings, by the assumptions imposed on {τn } for any p ∈ Θ : ∞ n1 FTn ∩ EPF, D ∩ VIE, B, we have Journal of Inequalities and Applications Wn θn − p2 ≤ PE kn − τn Bkn − PE p − τn Bp2 2 ≤ kn − τn Bkn − p − τn Bp 2 kn − p − τn Bkn − Bp 2 2 ≤ kn − p − 2τn kn − p, Bkn − Bp τn2 Bkn − Bp 2 2 ≤ xn − p − 2τn kn − p, Bkn − Bp τn2 Bkn − Bp 2 2 2 2 ≤ xn − p − 2τn −uBkn − Bp vkn − p τn2 Bkn − Bp 23 3.54 2 2 2 2 ≤ xn − p 2τn uBkn − Bp − 2τn vkn − p τn2 Bkn − Bp 2 2 2τn v 2 2 ≤ xn − p 2τn uBkn − Bp − 2 Bkn − Bp τn2 Bkn − Bp ξ 2 2τn v Bkn − Bp2 . xn − p 2τn u τn2 − 2 ξ Similarly, we have Wn φn − p2 ≤ xn − p2 2λn u λ2n − 2λn v Byn − Bp2 , 2 ξ Wn ψn − p2 ≤ xn − p2 2δn u δn2 − 2δn v Bun − Bp2 . 2 ξ 3.55 Observe that xn1 − p2 1 − βn I − n AWn θn − p βn xn − p n γfWn xn − Ap2 2 2 1 − βn I − n AWn θn − p βn xn − p n2 γfWn xn − Ap 2βn n xn − p, γfWn xn − Ap 2 n 1 − βn I − n A Wn θn − p , γfWn xn − Ap 2 2 ≤ 1 − βn − n γ Wn θn − p βn xn − p n2 γfWn xn − Ap 2βn n xn − p, γfWn xn − Ap 2 n 1 − βn I − n A Wn θn − p , γfWn xn − Ap 2 ≤ 1 − βn − n γ Wn θn − p βn xn − p cn 2 2 2 ≤ 1 − βn − n γ Wn θn − p βn2 xn − p 2 1 − βn − n γ βn Wn θn − pxn − p cn 2 2 2 ≤ 1 − βn − n γ Wn θn − p βn2 xn − p 1 − βn − nγ 2 2 βn Wn θn − p xn − p cn 24 Journal of Inequalities and Applications 1− 1− 1− 1− 2 nγ nγ nγ nγ βn − βn2 2 2 βn βn2 Wn θn − p βn2 xn − p Wn θn − p2 xn − p2 cn − 1− nγ 2 βn Wn θn − p 1 − 1 − βn − nγ Wn θn − p2 1 − 2 nγ −2 1− nγ nγ 2 βn xn − p cn 2 βn xn − p cn , 3.56 where cn 2 − Ap 2βn n xn − p, γfxn − Ap 2 n 1 − βn I − n A Wn θn − p , γfxn − Ap. 2 n γfxn 3.57 It follows from condition C1 that lim cn 0. 3.58 n→∞ Substituting 3.54 into 3.56, and using condition C6, we have xn1 − p2 ≤ 1 − nγ 1− 1− nγ nγ 1 − βn − nγ xn − p2 2τn u τn2 − 2τn v Bkn − Bp2 2 ξ 2 βn xn − p cn 2 xn − p 1 − 2 nγ 1 − βn − nγ 2τn v Bkn − Bp2 cn × 2τn u − 2 ξ 2 2τn v Bkn − Bp2 cn . ≤ xn − p 2τn u τn2 − 2 ξ τn2 3.59 It follows that 2 2av 2 ≤ 2τn v − 2τn u − τn2 Bkn − Bp2 Bk − 2bu − b − Bp n 2 2 ξ ξ 2 2 ≤ xn − p − xn1 − p cn xn − p − xn1 − p xn − p xn1 − p cn ≤ xn − xn1 xn − p xn1 − p cn . 3.60 Journal of Inequalities and Applications 25 Since cn → 0 as n → ∞ and 3.48, we obtain lim Bkn − Bp 0. n→∞ 3.61 Note that kn − p2 ≤ αn xn − p 1 − αn Wn φn − p2 2 2 2λn v Byn − Bp2 ≤ αn xn − p 1 − αn xn − p 2λn u λ2n − 2 ξ 2 2λn v Byn − Bp2 , ≤ xn − p 1 − αn 2λn u λ2n − 2 ξ yn − p2 ≤ ϕn un − p 1 − ϕn Wn ψn − p2 2 2 2δn v Bun − Bp2 ≤ ϕn xn − p 1 − ϕn xn − p 2δn u δn2 − 2 ξ 2 2δn v Bun − Bp2 . ≤ xn − p 1 − ϕn 2δn u δn2 − 2 ξ 3.62 3.63 Using 3.56 again, we have xn1 − p2 ≤ 1 − Wn θn − p2 1 − n γ βn xn − p2 cn 2 2 ≤ 1 − n γ 1 − βn − n γ θn − p 1 − n γ βn xn − p cn 2 2 ≤ 1 − n γ 1 − βn − n γ kn − p 1 − n γ βn xn − p cn . nγ 1 − βn − nγ 3.64 Substituting 3.62 into 3.64 and using condition C2 and C6, we have xn1 − p2 ≤ 1 − nγ 1− nγ 1 − βn − nγ xn − p2 1 − αn 2λn u λ2n − 2λn v Byn − Bp2 ξ2 2 βn xn − p cn 2λn v 2 Byn − Bp2 γ − α u λ − 2λ 1 n n n n 2 ξ 2 2 1 − n γ xn − p cn 2 2λn v Byn − Bp2 cn . ≤ xn − p 1 − αn 2λn u λ2n − 2 ξ 1− nγ 1 − βn − 3.65 26 Journal of Inequalities and Applications It follows that 1 − αn 2 2 2av 2τn v 2 2 By By − 2bu − b − Bp ≤ − α − 2τ u − τ − Bp 1 n n n n n ξ2 ξ2 2 2 ≤ xn − p − xn1 − p cn 3.66 ≤ xn − xn1 xn − p xn1 − p cn . Since cn → 0 as n → ∞ and 3.48, we obtain lim Byn − Bp 0. 3.67 lim Bun − Bp 0. 3.68 n→∞ In a similar way, we can prove n→∞ By 2.3, we also have θn − p2 PE kn − τn Bkn − PE p − τn Bp2 2 PE I − τn Bkn − PE I − τn Bp ≤ I − τn Bkn − I − τn Bp, θn − p 1 I − τn Bkn − I − τn Bp2 θn − p2 2 2 −I − τn Bkn − I − τn Bp − θn − p ≤ 1 kn − p2 θn − p2 − kn − θn − τn Bkn − Bp2 2 ≤ 1 xn − p2 θn − p2 − kn − θn 2 − τn2 Bkn − Bp2 2τn kn − θn , Bkn − Bp , 2 3.69 which yields that θn − p2 ≤ xn − p2 − kn − θn 2 2τn kn − θn Bkn − Bp. 3.70 Journal of Inequalities and Applications 27 Substituting 3.70 into 3.56, we have xn1 − p2 ≤ 1 − nγ ≤ 1− nγ ≤ 1− nγ 1− 1 − βn − 1 − βn − 1 − βn − nγ nγ Wn θn − p2 1 − nγ nγ θn − p2 1 − 2 βn xn − p cn nγ xn − p2 − kn − θn 2 2τn kn − θn Bkn − Bp nγ 2 βn xn − p cn 2 βn xn − p cn 2 2 1 − n γ xn − p − 1 − n γ 1 − βn − n γ kn − θn 2 2 1 − n γ 1 − βn − n γ τn kn − θn Bkn − Bp cn 2 ≤ xn − p − 1 − n γ 1 − βn − n γ kn − θn 2 2 1 − n γ 1 − βn − n γ τn kn − θn Bkn − Bp cn . 3.71 It follows that 1− nγ 1 − βn − nγ 2 2 kn − θn 2 ≤ xn − p − xn1 − p 2 1 − n γ 1 − βn − n γ τn kn − θn Bkn − Bp cn ≤ xn − xn1 xn − p xn1 − p 2 1 − n γ 1 − βn − n γ τn kn − θn Bkn − Bp cn . 3.72 Applying xn1 − xn → 0, Bkn − Bp → 0 and cn → 0 as n → ∞ to the last inequality, we have lim kn − θn 0. n→∞ On the other hand, we have Wn θn − p2 ≤ PE kn − τn Bkn − PE p − τn Bp2 2 PE I − τn Bkn − PE I − τn Bp ≤ I − τn Bkn − I − τn Bp, Wn θn − p 1 I − τn Bkn − I − τn Bp2 Wn θn − p2 2 2 −I − τn Bkn − I − τn Bp − Wn θn − p 3.73 28 Journal of Inequalities and Applications 1 kn − p2 Wn θn − p2 − kn − Wn θn − τn Bkn − Bp2 2 1 xn − p2 Wn θn − p2 − kn − Wn θn 2 ≤ 2 2 −τn2 Bkn − Bp 2τn kn − Wn θn , Bkn − Bp , ≤ 3.74 which yields that Wn θn − p2 ≤ xn − p2 − kn − Wn θn 2 2τn kn − Wn θn Bkn − Bp. 3.75 Similarly, we can prove Wn φn − p2 ≤ xn − p2 − yn − Wn φn 2 2λn yn − Wn φn Byn − Bp, Wn ψn − p2 ≤ xn − p2 − un − Wn ψn 2 2δn un − Wn ψn Bun − Bp. 3.76 3.77 Substituting 3.75 into 3.56, we have xn1 − p2 ≤ 1 − n γ 1 − βn − n γ Wn θn − p2 1 − n γ βn xn − p2 cn ≤ 1 − n γ 1 − βn − n γ 2 × xn − p − kn − Wn θn 2 2τn kn − Wn θn Bkn − Bp 2 βn xn − p cn 2 2 1 − n γ xn − p − 1 − n γ 1 − βn − n γ kn − Wn θn 2 2 1 − n γ 1 − βn − n γ τn kn − Wn θn Bkn − Bp cn 2 ≤ xn − p − 1 − n γ 1 − βn − n γ kn − Wn θn 2 2 1 − n γ 1 − βn − n γ τn kn − Wn θn Bkn − Bp cn , 1− nγ 3.78 which yields that 1− 1 − βn − n γ kn − Wn θn 2 2 2 ≤ xn − p − xn1 − p 2 1 − n γ 1 − βn − n γ τn kn − Wn θn Bkn − Bp cn ≤ xn − xn1 xn − p xn1 − p 2 1 − n γ 1 − βn − n γ τn kn − Wn θn Bkn − Bp cn . 3.79 nγ Journal of Inequalities and Applications 29 Applying 3.48 and 3.61 to the last inequality, we have lim kn − Wn θn 0. n→∞ 3.80 Using 3.64 again, we have xn1 − p2 ≤ 1 − kn − p2 1 − n γ βn xn − p2 cn 2 1 − n γ 1 − βn − n γ αn xn − p 1 − αn Wn φn − p nγ 1 − βn − nγ 2 βn xn − p cn 2 2 ≤ 1 − n γ 1 − βn − n γ αn xn − p 1 − αn Wn φn − p 1− nγ 2 1 − n γ βn xn − p cn 2 1 − n γ 1 − βn − n γ αn xn − p 2 2 1 − n γ 1 − βn − n γ 1 − αn Wn φn − p 1 − n γ βn xn − p cn 2 ≤ 1 − n γ 1 − βn − n γ αn xn − p 1 − n γ 1 − βn − n γ 1 − αn 2 2 × xn − p − yn − Wn φn 2λn yn − Wn φn Byn − Bp 2 1 − n γ βn xn − p cn 2 1 − n γ 1 − βn − n γ αn xn − p 2 1 − n γ 1 − βn − n γ 1 − αn xn − p 2 − 1 − n γ 1 − βn − n γ 1 − αn yn − Wn φn 1 − n γ 1 − βn − n γ 1 − αn 2λn yn − Wn φn Byn − Bp 2 1 − n γ βn xn − p cn 2 1 − n γ 1 − βn − n γ xn − p 2 − 1 − n γ 1 − βn − n γ 1 − αn yn − Wn φn 1 − n γ 1 − βn − n γ 1 − αn 2λn yn − Wn φn Byn − Bp 2 1 − n γ βn xn − p cn 2 2 2 1 − n γ xn − p − 1 − n γ 1 − βn − n γ 1 − αn yn − Wn φn 1 − n γ 1 − βn − n γ 1 − αn 2λn yn − Wn φn Byn − Bp cn 2 2 ≤ xn − p − 1 − n γ 1 − βn − n γ 1 − αn yn − Wn φn 1 − n γ 1 − βn − n γ 1 − αn 2λn yn − Wn φn Byn − Bp cn , 3.81 30 Journal of Inequalities and Applications which implies that 1− nγ 1 − βn − nγ 2 1 − αn yn − Wn φn 2 2 ≤ xn − p − xn1 − p 2 1 − n γ 1 − βn − n γ 1 − αn λn yn − Wn φn Byn − Bp cn ≤ xn − xn1 xn − p xn1 − p 2 1 − n γ 1 − βn − n γ 1 − αn λn yn − Wn φn Byn − Bp cn . 3.82 From 3.48 and 3.67, we obtain lim yn − Wn φn 0. n→∞ 3.83 By using the same argument, we can prove that lim un − Wn ψn 0. n→∞ 3.84 Note that kn − Wn φn αn xn − Wn φn , yn − Wn ψn ϕn un − Wn ψn . 3.85 Since αn → 0 and ϕn → 0 as n → ∞, respectively, we also have lim kn − Wn φn lim yn − Wn ψn 0. n→∞ n→∞ 3.86 On the other hand, we observe un − θn ≤ un − Wn ψn Wn ψn − yn yn − Wn φn Wn φn − kn kn − θn . 3.87 Applying 3.73, 3.83, 3.84, and 3.86, we have lim un − θn 0. n→∞ 3.88 Journal of Inequalities and Applications 31 On the other hand, we have kn − p2 ≤ αn xn − p2 1 − αn Wn φn − p2 2 2 ≤ αn xn − p 1 − αn φn − p 2 2 ≤ αn xn − p 1 − αn yn − p 2 2 ≤ αn xn − p 1 − αn ϕn un − p 1 − ϕn Wn ψn − p 2 2 ≤ αn xn − p 1 − αn ϕn un − p 1 − ϕn ψn − p 2 2 ≤ αn xn − p 1 − αn ϕn un − p 1 − ϕn un − p 3.89 2 2 αn xn − p 1 − αn un − p 2 2 αn xn − p 1 − αn Trn I − rn Dxn − p 2 2 ≤ αn xn − p 1 − αn I − rn Dxn − p 2 2 2 ≤ αn xn − p 1 − αn xn − p − rn 2η − rn Dxn − Dp 2 2 xn − p − 1 − αn rn 2η − rn Dxn − Dp . Substituting 3.89 into 3.64 and using conditions C2 and C7, we have xn1 − p2 ≤ 1 − kn − p2 1 − n γ βn xn − p2 cn 2 2 ≤ 1 − n γ 1 − βn − n γ xn − p − 1 − αn rn 2η − rn Dxn − Dp nγ 1− nγ 1 − βn − nγ 2 βn xn − p cn 2 1 − βn − n γ 1 − αn rn 2η − rn Dxn − Dp 2 2 1 − n γ xn − p cn 2 ≤ xn − p − 1 − αn rn 2η − rn Dxn − Dp. 1− nγ 3.90 This implies that 2 2 1 − αn rn 2η − rn Dxn − Dp ≤ xn − p − xn1 − p cn . 3.91 In view of the restrictions C2 and C7, we obtain that lim Dxn − Dp 0. n→∞ 3.92 32 Journal of Inequalities and Applications Let p ∈ Θ : ∞ n1 FTn ∩ EPF, D ∩ VIE, B. Since un Trn xn − rn Dxn and Trn is firmly nonexpansive Lemma 2.6, then we obtain un − p2 Tr xn − rn Dxn − Tr p − rn Dp2 n n ≤ Trn xn − rn Dxn − Trn p − rn Dp , un − p xn − rn Dxn − p − rn Dp , un − p ≤ 1 xn − rn Dxn − p − rn Dp 2 un − p2 2 2 −xn − rn Dxn − p − rn Dp − un − p 3.93 1 xn − p2 un − p2 − xn − un − rn Dxn − Dp 2 2 1 xn − p2 un − p2 − xn − un 2 2rn xn − un , Dxn − Dp 2 2 −rn2 Dxn − Dp . So, we obtain un − p2 ≤ xn − p2 − xn − un 2 2rn xn − un Dxn − Dp. 3.94 Therefore, we have xn1 − p2 1 − Wn θn − p2 1 − n γ βn xn − p2 cn 2 2 ≤ 1 − n γ 1 − βn − n γ θn − p 1 − n γ βn xn − p cn 2 2 1 − n γ 1 − βn − n γ θn − un un − p 1 − n γ βn xn − p cn 2 ≤ 1 − n γ 1 − βn − n γ θn − un 2 un − p 2θn − un , un − p nγ 1− 1 − βn − nγ 2 βn xn − p cn nγ 2 θn − un 2 1 − n γ 1 − n γ − βn un − p 2 2 1 − n γ 1 − βn − n γ θn − un un − p 1 − n γ βn xn − p cn ≤ 1 − n γ 1 − βn − n γ θn − un 2 2 1 − n γ 1 − βn − n γ xn − p − xn − un 2 2rn xn − un Dxn − Dp ≤ 1− nγ 2 1− 1 − βn − nγ nγ 1 − βn − nγ θn − un un − p 1 − nγ 2 βn xn − p cn Journal of Inequalities and Applications 33 2 1 − n γ 1 − βn − n γ θn − un 2 1 − n γ 1 − βn − n γ xn − p − 1 − n γ 1 − βn − n γ xn − un 2 1 − n γ 1 − βn − n γ 2rn xn − un Dxn − Dp 2 2 1 − n γ 1 − βn − n γ θn − un un − p 1 − n γ βn xn − p cn 2 2 1 − n γ xn − p − 1 − n γ 1 − βn − n γ xn − un 2 1 − n γ 1 − βn − n γ θn − un 2 1 − n γ 1 − βn − n γ 2rn xn − un Dxn − Dp 2 1 − n γ 1 − βn − n γ θn − un un − p cn 2 xn − p2 − 1 − n γ 1 − βn − n γ xn − un 2 1 − 2 nγ nγ 1 − n γ 1 − βn − n γ θn − un 2 1 − n γ 1 − βn − n γ 2rn xn − un Dxn − Dp 2 1 − n γ 1 − βn − n γ θn − un un − p cn 2 2 2 ≤ xn − p n γ xn − p 1 − n γ 1 − βn − n γ θn − un 2 − 1 − n γ 1 − βn − n γ xn − un 2 1 − n γ 1 − βn − n γ 2rn xn − un Dxn − Dp 2 1 − n γ 1 − βn − n γ θn − un un − p cn . 3.95 It follows that 1− 1 − βn − n γ xn − un 2 2 2 2 2 ≤ xn − p − xn1 − p n γ xn − p 1 − n γ 1 − βn − n γ θn − un 2 1 − n γ 1 − βn − 2 1 − n γ 1 − βn − n γ θn − un un − p cn 2 2 ≤ xn − xn1 xn − p xn1 − p n γ xn − p nγ 1 − n γ 1 − βn − n γ θn − un 2 1 − n γ 1 − βn − 2 1 − n γ 1 − βn − n γ θn − un un − p cn . nγ 2rn xn − un Dxn − Dp nγ 2rn xn − un Dxn − Dp 3.96 Using n → 0, cn → 0 as n → ∞, 3.48, 3.88, and 3.92, we obtain lim xn − un 0. n→∞ 3.97 34 Journal of Inequalities and Applications Since lim infn → ∞ rn > 0, we obtain xn − un lim 1 xn − un 0. lim n→∞ rn n → ∞ rn 3.98 xn − θn ≤ xn − un un − θn , 3.99 Note that and thus from 3.88 and 3.97, we have lim xn − θn 0. 3.100 Wn θn − θn ≤ Wn θn − xn xn − θn . 3.101 n→∞ Observe that Applying 3.53 and 3.100, we obtain lim Wn θn − θn 0. 3.102 n→∞ Let W be the mapping defined by 2.11. Since {θn } is bounded, applying Lemma 2.10 and 3.102, we have Wθn − θn ≤ Wθn − Wn θn Wn θn − θn −→ 0 as n −→ ∞. 3.103 Step 5. We claim that lim supn → ∞ A − γfz, z − xn ≤ 0, where z is the unique solution of the variational inequality A − γfz, x − z ≥ 0, for all x ∈ Θ. Since z PΘ I − A γfz is a unique solution of the variational inequality 3.5, to show this inequality, we choose a subsequence {θni } of {θn } such that lim A − γf z, z − θni lim sup A − γf z, z − θn . i→∞ n→∞ 3.104 Since {θni } is bounded, there exists a subsequence {θnij } of {θni } which converges weakly to w. From Wθn − θn → 0, we w ∈ E. Without loss of generality, we can assume that θni w. Next, We show that w ∈ Θ, where Θ : ∞ obtain Wθni n1 FTn ∩ EPF, D ∩ VIE, B. a First, we prove w ∈ EPF, D. Since un Trn xn − rn Dxn , we know that 1 F un , y Dxn , y − un y − un , un − xn ≥ 0, rn ∀y ∈ E. 3.105 Journal of Inequalities and Applications 35 From A2, we also have Dxn , y − un 1 y − un , un − xn ≥ −F un , y ≥ F y, un . rn 3.106 Replacing n by ni , we have Dxni , y − uni un − xni y − uni , i rni ≥ F y, uni . 3.107 For any t with 0 < t ≤ 1 and y ∈ E, let ϕt ty 1 − tz. Since y ∈ E and z ∈ E, we have ϕt ∈ E. So, from 3.107 we have un − xni F ϕt , uni ϕt − uni , Dϕt ≥ ϕt − uni , Dϕt − Dxni , ϕt − uni − ϕt − uni , i rni ≥ ϕt − uni , Dϕt − Duni ϕt − uni , Duni − Dxni − ϕt − uni , uni − xni F ϕt , uni . rni 3.108 Since D is Lipschitz continuous, from 3.97, we have Duni − Dxni → 0 as i → ∞. Further, from the monotonicity of D, we get that ϕt − uni , Dϕt − Duni ≥ 0. 3.109 It follows from A4 and 3.108 that ϕt − z, Dϕt ≥ F ϕt , z . 3.110 From A1, A4, and 3.110, we also have 0 F ϕt , ϕt ≤ tF ϕt , y 1 − tF ϕt , z ≤ tF ϕt , y 1 − t ϕt − z, Dϕt 3.111 tF ϕt , y 1 − tt y − z, Dϕt , and hence F ϕt , y 1 − t y − z, Dϕt ≥ 0. 3.112 36 Journal of Inequalities and Applications Letting t → ∞ in the above inequality, we have, for each y ∈ E, F z, y y − z, Dz ≥ 0. 3.113 Thus z ∈ EPF, D. . b Next, we show that w ∈ ∞ n1 FT n ∈ FW. Since xn − θn → By Lemma 2.9, we have FW ∞ n1 FTn . Assume w / w i → ∞ and w / Ww, and it follows by the Opial’s condition 0, we know that θni Lemma 2.3 that lim infθni − w < lim infθni − Ww i→∞ i→∞ ≤ lim infθni − Wθni Wθni − Ww i→∞ 3.114 < lim infθni − w, i→∞ that is a contradiction. Thus, we have w ∈ FW ∞ n1 FTn . c Finally, Now we prove that w ∈ VIE, B. Define, T w1 ⎧ ⎨Bw1 NE w1 , if w1 ∈ E, ⎩∅, ∈ E. if w1 / 3.115 Since B is relaxed u, v-cocoercive and condition C6, we have 2 2 2 Bx − By, x − y ≥ −uBx − By vx − y ≥ v − uξ2 x − y ≥ 0, 3.116 which yields that B is monotone. Then, T is maximal monotone. Let w1 , w2 ∈ GT . Since w2 − Bw1 ∈ NE w1 and θn ∈ E, we have w1 − θn , w2 − Bw1 ≥ 0. On the other hand, from θn PE kn − τn Bkn , we have w1 − θn , θn − kn − τn Bkn ≥ 0, 3.117 and hence θn − kn Bkn w1 − θn , τn ≥ 0. 3.118 Journal of Inequalities and Applications 37 Therefore, we have w1 − θni , w ≥ w1 − θni , Bw1 θni − kni ≥ w1 − θni , Bw1 − w1 − θni , Bkni τni θni − kni w1 − θni , Bw1 − Bkni − τni θni − kni w1 − θni , Bv − Bθni w1 − θni , Bθni − Bkni − w1 − θni , τni θni − kni ≥ w1 − θni , Bθni − Bkni − w1 − θni , , τni 3.119 which implies that w1 − w, w2 ≥ 0. 3.120 Since T is maximal monotone, we have w ∈ T −1 0 and hence w ∈ VIE, B. That is, w ∈ Θ, where Θ : ∞ n1 FTn ∩ EPF, D ∩ VIE, B. Since z PΘ I − A γfz, it follows that lim sup A − γf z, z − xn lim sup A − γf z, z − θn n→∞ n→∞ lim A − γf z, z − θni i→∞ 3.121 A − γf z, z − w ≤ 0. On the other hand, we have A − γf z, z − Wn θn A − γf z, xn − Wn θn A − γf z, z − xn ≤ A − γf zxn − Wn θn A − γf z, z − xn . 3.122 From 3.53 and 3.121, we obtain that lim supγfz − Az, Wn θn − z ≤ 0. n→∞ 3.123 Step 6. Finally, we show that {xn } and {un } converge strongly to z PΘ I − A γfz. 38 Journal of Inequalities and Applications Indeed, from 3.4 and Lemma 2.4, we obtain 2 xn1 − z2 n γfWn xn βn xn 1 − βn I − n AWn θn − z 2 1 − βn I − n AWn θn − z βn xn − z n γfWn xn − Az 2 2 1 − βn I − n AWn θn − z βn xn − z n2 γfWn xn − Az 2βn n xn − z, γfWn xn − Az 2 n 1 − βn I − n A Wn θn − z, γfWn xn − Az 2 2 ≤ 1 − βn − n γ Wn θn − z βn xn − z n2 γfWn xn − Az 2βn n γxn − z, fWn xn − fz 2βn n xn − z, γfz − Az 2 1 − βn γ n Wn θn − z, fWn xn − fz 2 1 − βn n Wn θn − z, γfz − Az − 2 n2 AWn θn − z, γfz − Az, 2 2 ≤ 1 − βn − n γ Wn θn − z βn xn − z n2 γfWn xn − Az 2βn n γxn − zfWn xn − fz 2βn n xn − z, γfz − Az 2 1 − βn γ n Wn θn − zfWn xn − fz 2 1 − βn n Wn θn − z, γfz − Az − 2 n2 AWn θn − z, γfz − Az, 2 2 ≤ 1 − βn − n γ θn − z βn xn − z n2 γfWn xn − Az 2βn n γxn − zfWn xn − fz 2βn n xn − z, γfz − Az 2 1 − βn γ n θn − zfWn xn − fz 2 1 − βn n Wn θn − z, γfz − Az − 2 n2 AWn θn − z, γfz − Az ≤ 1 − βn − nγ xn − z βn xn − z 2 2 n γfWn xn 2 − Az 2βn n γαxn − z2 2βn n xn − z, γfz − Az 2 1 − βn γ n αxn − z2 2 1 − βn n Wn θn − z, γfz − Az − 2 n2 AWn θn − z, γfz − Az 2 2 1 − n γ 2βn n γα 2 1 − βn γ n α xn − z2 n2 γfWn xn − Az 2βn n xn − z, γfz − Az 2 1 − βn n Wn θn − z, γfz − Az − 2 n2 AWn θn − z, γfz − Az 2 xn − z2 γ 2 n2 xn − z2 n2 γfWn xn − Az 2βn n xn − z, γfz − Az 2 1 − βn n Wn θn − z, γfz − Az 2 n2 AWn θn − zγfz − Az ≤ 1 − 2 γ − αγ n Journal of Inequalities and Applications 39 1 − 2 γ − αγ n xn − z2 2 n n γ 2 xn − z2 γfWn xn − Az 2AWn θn − zγfz − Az 2βn xn − z, γfz − Az 2 1 − βn Wn θn − z, γfz − Az . 3.124 Since {xn }, {fWn xn }, and {Wn θn } are bounded, we can take a constant M > 0 such that 2 γ 2 xn − z2 γfWn xn − Az 2AWn θn − zγfz − Az ≤ M 3.125 for all n ≥ 0. It then follows that xn1 − z2 ≤ 1 − ln xn − z2 3.126 n σn , where ln 2 γ − αγ n , σn n M 2βn xn − z, γfz − Az 2 1 − βn Wn θn − z, γfz − Az. 3.127 Using C1, 3.121, and 3.123, we get ln → 0, ∞ n1 ln ∞ and lim supn → ∞ σn /ln ≤ 0. Applying Lemma 2.13 to 3.126, we conclude that xn → z in norm. Finally, noticing un − z Trn xn − rn Dxn − Trn z − rn Dz ≤ xn − z, we also conclude that un → z in norm. This completes the proof. Corollary 3.4. Let E be a nonempty closed convex subset of a real Hilbert space H. Let F : E×E → R be a bifunction satisfying (A1)–(A4), let B : E → H be relaxed u, v-cocoercive and ξ-Lipschitz continuous mappings, and let {Tn } be an infinite family of nonexpansive mappings of E into itself ∅. Let f be a contraction mapping of E into itself such that Θ : ∞ n1 FTn ∩ EPF ∩ VIE, B / with α ∈ 0, 1. Let {xn }, {yn }, {kn }, and {un } be sequences generated by x1 ∈ E chosen arbitrary, 1 F un , y y − un , un − xn ≥ 0, ∀y ∈ E, rn yn ϕn un 1 − ϕn Wn PE un − δn Bun , kn αn xn 1 − αn Wn PE yn − λn Byn , xn1 n fWn xn βn xn γn Wn PE kn − τn Bkn , 3.128 ∀n ≥ 1, where {Wn } is the sequence generated by 1.24 and { n }, {αn }, {ϕn }, and {βn } are sequences in 0, 1 and {rn } is a real sequence in 0, ∞ satisfying the following conditions: C1 βn γn 1, C2 limn → ∞ n 0, ∞ n1 n n ∞, 40 Journal of Inequalities and Applications C4 limn → ∞ αn limn → ∞ ϕn 0, C5 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1, C6 limn → ∞ |λn1 − λn | limn → ∞ |δn1 − δn | limn → ∞ |τn1 − τn | 0, C7 {τn }, {λn }, {δn } ⊂ a, b for some a, b with 0 ≤ a ≤ b ≤ 2v − uξ2 /ξ2 , v > uξ2 . Then, {xn } and {un } converge strongly to a point z ∈ Θ, where z PΘ fz. Proof. Put A I, γ ≡ 1, γn 1− n −βn , D 0 :the zero mapping and { n } 0 in Theorem 3.3. Then yn vn un , and for any η > 0, we see that 2 Dx − Dy, x − y ≥ ηDx − Dy , 3.129 ∀x, y ∈ E. Let {rn } be a sequence satisfying the restriction: c ≤ rn ≤ d, where c, d ∈ 0, ∞. Then we can obtain the desired conclusion easily from Theorem 3.3. Corollary 3.5. Let E be a nonempty closed convex subset of a real Hilbert space H. Let {Tn } be an infinite family of nonexpansive mappings of E into itself and let B : E → H be relaxed u, v ∅. Let f : cocoercive and ξ-Lipschitz continuous mappings such that Θ : ∞ n1 FTn ∩ VIE, B / E → E be a contraction mapping with 0 < α < 1 and let A be a strongly positive linear bounded operator on H with coefficient γ > 0 and 0 < γ < γ/α. Let {xn },{yn }, and {kn } be sequences generated by xn1 x1 ∈ E chosen arbitrary, yn ϕn xn 1 − ϕn Wn PE xn − δn Bxn , kn αn xn 1 − αn Wn PE yn − λn Byn , n γfWn xn βn xn 1 − βn I − n A Wn PE kn − τn Bkn , 3.130 ∀n ≥ 1, where {Wn } is the sequence generated by 1.24 and { n }, {αn }, {ϕn }, and {βn } are sequences in 0, 1 satisfying the following conditions: C1 limn → ∞ n 0, ∞ n1 n ∞, C2 limn → ∞ αn limn → ∞ ϕn 0, C3 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1, C4 limn → ∞ |λn1 − λn | limn → ∞ |δn1 − δn | limn → ∞ |τn1 − τn | 0, C5 {τn }, {λn }, {δn } ⊂ a, b for some a, b with 0 ≤ a ≤ b ≤ 2v − uξ2 /ξ2 , v > uξ2 . Then, {xn } converges strongly to a point z ∈ Θ, where z PΘ I − A γfz, which solves the variational inequality A − γf z, x − z ≥ 0, ∀x ∈ Θ, 3.131 Journal of Inequalities and Applications 41 which is the optimality condition fot the minimization problem min x∈Θ 1 Ax, x − hx , 2 3.132 where h is a potential function for γf (i.e., h x γfx for x ∈ H). Proof. Put D 0, Fx, y 0 for all x, y ∈ E and rn 1 for all n ∈ N in Theorem 3.3. Then, we have un PC xn xn . So, by Theorem 3.3, we can conclude the desired conclusion easily. Acknowledgments The authors would like to express their thanks to the Faculty of Science KMUTT Research Fund for their financial support. The first author was supported by the Faculty of Applied Liberal Arts RMUTR Research Fund and King Mongkut’s Diamond scholarship for fostering special academic skills by KMUTT. The second author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5180034. Moreover, the authors are extremely grateful to the referees for their helpful suggestions that improved the content of the paper. References 1 G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus Academy of Sciences, vol. 258, pp. 4413–4416, 1964. 2 D. Gabay, “Applications of the method of multipliers to variational inequalities,” in Augmented Lagrangian Methods, M. Fortin and R. Glowinski, Eds., pp. 299–331, North-Holland, Amsterdam, The Netherlands, 1983. 3 M. A. Noor and W. Oettli, “On general nonlinear complementarity problems and quasi-equilibria,” Le Matematiche, vol. 49, no. 2, pp. 313–331, 1994. 4 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. 5 S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997. 6 X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1033–1046, 2008. 7 Y. Yao, Y.-C. Liou, and J.-C. Yao, “Convergence 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. 8 W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama, Yokohama, Japan, 2000. 9 R. U. Verma, “Generalized system for relaxed cocoercive variational inequalities and projection methods,” Journal of Optimization Theory and Applications, vol. 121, no. 1, pp. 203–210, 2004. 10 R. U. Verma, “General convergence analysis for two-step projection methods and applications to variational problems,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1286–1292, 2005. 11 R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970. 12 H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp. 341–350, 2005. 13 A. Moudafi and M. Théra, “Proximal and dynamical approaches to equilibrium problems,” in Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), vol. 477 of Lecture Notes in Economics and Mathematical Systems, pp. 187–201, Springer, Berlin, Germany, 1999. 42 Journal of Inequalities and Applications 14 X. Qin, Y. J. Cho, and S. M. Kang, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 99–112, 2010. 15 Y. J. Cho, X. Qin, and S. M. Kang, “Some results for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Computational Analysis and Applications, vol. 11, no. 2, pp. 294–316, 2009. 16 Y. J. Cho, X. Qin, and J. I. Kang, “Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 4203–4214, 2009. 17 C. S Hu and G. Cai, “Viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1792–1808, 2010. 18 N.-J. Huang, H.-Y. Lan, and K. L. Teo, “On the existence and convergence of approximate solutions for equilibrium problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2007, Article ID 17294, 14 pages, 2007. 19 C. Jaiboon and P. Kumam, “Strong convergence theorems for solving equilibrium problems and fixed point problems of ξ-strict pseudo-contraction mappings by two hybrid projection methods,” Journal of Computational and Applied Mathematics. In press. 20 C. Jaiboon and P. Kumam, “A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2009, Article ID 374815, 32 pages, 2009. 21 C. Jaiboon, W. Chantarangsi, and P. Kumam, “A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 199–215, 2010. 22 A. Kangtunyakarn and S. Suantai, “A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4448–4460, 2009. 23 Q.-Y. Liu, W.-Y. Zeng, and N.-J. Huang, “An iterative method for generalized equilibrium problems, fixed point problems and variational inequality problems,” Fixed Point Theory and Applications, vol. 2009, Article ID 531308, 20 pages, 2009. 24 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. 25 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 720371, 15 pages, 2008. 26 X. Qin, M. Shang, and Y. Su, “A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3897–3909, 2008. 27 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008. 28 W.-Y. Zeng, N.-J. Huang, and C.-W. Zhao, “Viscosity approximation methods for generalized mixed equilibrium problems and fixed points of a sequence of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 714939, 15 pages, 2008. 29 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, pp. 117–136, 2005. 30 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. 31 M. Shang, Y. Su, and X. Qin, “Strong convergence theorem for nonexpansive mappings and relaxed cocoercive mappings,” International Journal of Applied Mathematics and Mechanics, vol. 3, no. 4, pp. 24– 34, 2007. 32 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998. Journal of Inequalities and Applications 43 33 I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001. 34 H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. 35 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. 36 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. 37 S.-S. Chang, H. W. J. Lee, and C. K. Chan, “A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3307–3319, 2009. 38 Y. Yao, M. A. Noor, and Y.-C. Liou, “On iterative methods for equilibrium problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 497–509, 2009. 39 Y. J. Cho and X. Qin, “Generalized systems for relaxed cocoercive variational inequalities and projection methods in Hilbert spaces,” Mathematical Inequalities & Applications, vol. 12, no. 2, pp. 365– 375, 2009. 40 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967. 41 K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001. 42 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. 43 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.