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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 417089, 15 pages doi:10.1155/2008/417089 Research Article A New Hybrid Iterative Algorithm for Fixed-Point Problems, Variational Inequality Problems, and Mixed Equilibrium Problems Yonghong Yao,1 Yeong-Cheng Liou,2 and Jen-Chih Yao3 1 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan 3 Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan 2 Correspondence should be addressed to Jen-Chih Yao, yaojc@math.nsysu.edu.tw Received 29 August 2007; Accepted 6 February 2008 Recommended by Tomonari Suzuki We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. This study, proves a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems. Copyright q 2008 Yonghong Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping f : C → C is called contractive if there exists a constant α ∈ 0, 1 such that fx − fy ≤ αx − y for all x, y ∈ C. A mapping T : C → C is said to be nonexpansive if T x − T y ≤ x − y for all x, y ∈ C. Denote the set of fixed points of T by FT . Let ϕ : C → R be a real-valued function and Θ : C × C → R be an equilibrium bifunction, that is, Θu, u 0 for each u ∈ C. The mixed equilibrium problem for short, MEP is to find x∗ ∈ C such that MEP: Θ x∗ , y ϕy − ϕ x∗ ≥ 0 ∀y ∈ C. 1.1 In particular, if ϕ ≡ 0, this problem reduces to the equilibrium problem for short, EP, which is to find x∗ ∈ C such that 1.2 EP: Θ x∗ , y ≥ 0 ∀y ∈ C. 2 Fixed Point Theory and Applications Denote the set of solutions of MEP by Ω. The mixed equilibrium problems include fixedpoint problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases see, e.g., 1–5. Some methods have been proposed to solve the MEP and EP see, e.g., 5–14. In 1997, Combettes and Hirstoaga 13 introduced an iterative method of finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi 8 introduced another iterative scheme for finding a common element of the set of solutions of EP and the set of fixed-point points of a nonexpansive mapping. Yao et al. 12 considered an iterative scheme for finding a common element of the set of solutions of EP and the set of common fixed points of an infinite nonexpansive mappings. Very recently, Zeng and Yao 14 considered a new iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Their results extend and improve many results in the literature. Let A of C into H be a nonlinear mapping. It is well known that the variational inequality problem is to find u ∈ C such that Au, v − u ≥ 0 ∀v ∈ C. 1.3 The set of solutions of the variational inequality problem is denoted by V IC, A. A mapping A : C → H is called β-inverse-strongly monotone if there exists a positive real number β such that Au − Av, u − v ≥ βAu − Av2 ∀u, v ∈ C. 1.4 Recently, some authors have proposed new iterative algorithms to approximate a common element of the set of fixed points of a nonxpansive mapping and the set of solutions of the variational inequality. For the details, see 15, 16 and the references therein. Motivated by the recent works, in this paper we introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. We prove a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems. 2. Preliminaries Let H be a real Hilbert space with inner product ·, · and norm ·. Let C be a nonempty closed convex subset of H. Then for any x ∈ H, there exists a unique nearest point in C, denoted by PC x such that x − PC x ≤ x − y ∀y ∈ C. 2.1 Such a PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping and satisfies 2 x − y, PC x − PC y ≥ PC x − PC y ∀x, y ∈ H. 2.2 Yonghong Yao et al. 3 Moreover, PC is characterized by the following properties: x − PC x, y − PC x ≤ 0, 2 2 x − y2 ≥ x − PC x y − PC x ∀x ∈ H, y ∈ C. 2.3 It is clear that u ∈ V IC, A ⇐⇒ u PC u − λAu ∀λ > 0. 2.4 In this paper, for solving the mixed equilibrium problems for an equilibrium bifunction Θ : C × C → R, we assume that Θ satisfies the following conditions: H1 Θ is monotone, that is, Θx, y Θy, x ≤ 0 for all x, y ∈ C; H2 for each fixed y ∈ C, x → Θx, y is concave and upper semicontinuous; H3 for each x ∈ C, y → Θx, y is convex. A mapping η : C × C → H is called Lipschitz continuous if there exists a constant λ > 0 such that ηx, y ≤ λx − y ∀x, y ∈ C. 2.5 A diﬀerentiable function K : C → R on a convex set C is called: i η-convex if Ky − Kx ≥ K x, ηy, x ∀x, y ∈ C, 2.6 where K is the Fréchet derivative of K at x; ii η-strongly convex if there exists a constant σ > 0 such that σ x − y2 Ky − Kx − K x, ηy, x ≥ 2 ∀x, y ∈ C. 2.7 Let C be a nonempty closed convex subset of a real Hilbert space H, ϕ : C → R be a realvalued function, and Θ : C × C → R be an equilibrium bifunction. Let r be a positive number. For a given point x ∈ C, the auxiliary problem for MEP consists of finding y ∈ C such that Θy, z ϕz − ϕy 1 K y − K x, ηz, y ≥ 0 ∀z ∈ C. r 2.8 Let Sr : C → C be the mapping such that for each x ∈ C, Sr x is the solution set of the auxiliary problem MEP, that is, Sr x y ∈ C : Θy, z ϕz − ϕy 1 K y − K x, ηz, y ≥ 0 ∀z ∈ C r We first need the following important and interesting result. ∀x ∈ C. 2.9 4 Fixed Point Theory and Applications Lemma 2.1 see 14. Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ : C → R be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)–(H3). Assume that i η : C × C → H is Lipschitz continuous with constant λ > 0 such that a ηx, y ηy, x 0 for all x, y ∈ C, b η·, · is aﬃne in the first variable, c for each fixed y ∈ C, x → ηy, x is sequentially continuous from the weak topology to the weak topology; ii K : C → R is η-strongly convex with constant σ > 0 and its derivative K is sequentially continuous from the weak topology to the strong topology; iii for each x ∈ C, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈ C \ Dx , 1 Θ y, zx ϕzx − ϕy K y − K x, η zx , y < 0. r 2.10 Then there hold the following: i Sr is single-valued; ii Sr is nonexpansive if K is Lipschitz continuous with constant ν > 0 such that σ ≥ λν and ∀ x1 , x2 ∈ C × C, K x1 − K x2 , η u1 , u2 ≥ K u1 − K u2 , η u1 , u2 2.11 where ui Sr xi for i 1, 2; iii FSr Ω; vi Ω is closed and convex. We also need the following lemmas for proving our main results. Lemma 2.2 see 17. 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 2.3 see 18. Assume {an } is a sequence of nonnegative real numbers such that an1 ≤ 1 − γn an δn , where {γn } is a sequence in 0, 1 and {δn } is a sequence such that 1 ∞ n1 γn ∞; 2 lim supn→∞ δn /γn ≤ 0 or ∞ n1 |δn | < ∞. Then limn→∞ an 0. 3. Iterative algorithm and strong convergence theorems In this section, we first introduce a new iterative algorithm. Consequently, we will establish a strong convergence theorem for this iteration algorithm. To be more specific, let T1 , T2 , . . . be Yonghong Yao et al. 5 infinite mappings of C into itself and let ξ1 , ξ2 , . . . be real numbers such that 0 ≤ ξi ≤ 1 for every i ∈ N. For any n ∈ N, define a mapping Wn of C into itself as follows: Un,n1 I, Un,n ξn Tn Un,n1 1 − ξn I, Un,n−1 ξn−1 Tn−1 Un,n 1 − ξn−1 I, .. . Un,k ξk Tk Un,k1 1 − ξk I, 3.1 Un,k−1 ξk−1 Tk−1 Un,k 1 − ξk−1 I, .. . Un,2 ξ2 T2 Un,3 1 − ξ2 I, Wn Un,1 ξ1 T1 Un,2 1 − ξ1 I. Such Wn is called the W-mapping generated by Tn , Tn−1 , . . . , T2 , T1 and ξn , ξn−1 , . . . , ξ2 , ξ1 . For the iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to 19. We have the following crucial Lemmas 3.1 and 3.2 concerning Wn which can be found in 20. Now we only need the following similar version in Hilbert spaces. Lemma 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∞ n1 FTn is nonempty, and let ξ1 , ξ2 , . . . be real numbers such that 0 < ξi ≤ b < 1 for any i ∈ N. Then for every x ∈ C and k ∈ N, the limit limn→∞ Un,k x exists. Lemma 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∞ is nonempty, and let ξ1 , ξ2 , . . . be real n1 FTn numbers such that 0 < ξi ≤ b < 1 for any i ∈ N. Then FW ∞ n1 FTn . The following remark 12 is important to prove our main results. Remark 3.3. Using Lemma 3.1, one can define a mapping W of C into itself as Wx limn→∞ Wn x limn→∞ Un,1 x for every x ∈ C. If {xn } is a bounded sequence in C, then we have lim Wxn − Wn xn 0. n→∞ 3.2 Throughout this paper, we will assume that 0 < ξi ≤ b < 1 for every i ∈ N. Now we introduce the following iteration algorithm. Algorithm 3.4. Let r > 0 be a constant. Let ϕ : C → R be a lower semicontinuous and convex functional and let Θ : C × C → R be an equilibrium bifunction. Let A : C → H be a βinverse-strongly monotone mapping and Wn be the W-mapping defined by 3.1. Let f be a 6 Fixed Point Theory and Applications contraction of C into itself with coeﬃcient α ∈ 0, 1 and given x0 ∈ C arbitrarily. Suppose that the sequences {xn } and {yn } are generated iteratively by 1 Θ zn , x ϕx − ϕ zn K zn − K xn , η x, zn ≥ 0 ∀x ∈ C, r yn PC zn − λn Azn , xn1 αn f Wn xn βn xn γn Wn PC yn − λn Ayn ∀n ≥ 0, 3.3 where {αn }, {βn }, and {γn } are three sequences in 0, 1, and {λn } is a sequence in 0, 2β. Now we study the strong convergence of the hybrid iterative algorithm 3.3. Theorem 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ : C → R be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)–(H3) and let T1 , T2 , . . . be an infinite family of nonexpansive mappings of C into itself. Let A : C → H be a β-inverse-strongly monotone mapping such that ∩∞ n1 FTn ∩V IA, C∩ Ω ∅ . Suppose {α }, {β }, and {γ } are three sequences in 0, 1 with α β / n n n n n γn 1, n ≥ 0. Assume that i η : C × C → H is Lipschitz continuous with constant λ > 0 such that a ηx, y ηy, x 0 for all x, y ∈ C, b η·, · is aﬃne in the first variable, c for each fixed y ∈ C, x → ηy, x is sequentially continuous from the weak topology to the weak topology; ii K : C → R is η-strongly convex with constant σ > 0 and its derivative K is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν; iii for each x ∈ C; there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈ C \ Dx , 1 Θ y, zx ϕ zx − ϕy K y − K x, η zx , y < 0; 3.4 r iv limn→∞ αn 0, ∞ n0 αn ∞, 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1, λn ∈ a, b ⊂ 0, 2β, and limn→∞ λn1 − λn 0. Let f be a contraction of C into itself and given x0 ∈ C arbitrarily. Then the sequence {xn } generated by 3.3 converges strongly to x∗ PΓ fx∗ , where Γ ∩∞ n1 FTn ∩ V IA, C ∩ Ω provided that Sr is firmly nonexpansive. Proof. We first note that f is a contraction with coeﬃcient α ∈ 0, 1. Then PΓ fx − PΓ fy ≤ fx − fy ≤ αx − y for all x, y ∈ C. Therefore PΓ f is a contraction of C into itself which implies that there exists a unique element x∗ ∈ C such that x∗ PΓ fx∗ . Next we divide the following proofs into several steps. Step 1 {xn }, {yn }, and {zn } are bounded. Let x∗ ∈ Γ. From the definition of Sr , we know that zn Sr xn . It follows that zn − x∗ Sr xn − Sr x∗ ≤ xn − x∗ . 3.5 Yonghong Yao et al. 7 For all x, y ∈ C and λn ∈ 0, 2β, we note that I − λn A x − I − λn A y 2 x − y − λn Ax − Ay2 x − y2 − 2λn Ax − Ay, x − y λ2n Ax − Ay2 ≤ x − y2 λn λn − 2β Ax − Ay2 , 3.6 which implies that I − λn A is nonexpansive. Set un PC yn − λn Ayn for all n ≥ 0. From 2.4, we have that x∗ PC x∗ − λn Ax∗ . It follows from 3.6 that yn − x∗ PC zn − λn Azn − PC x∗ − λn Ax∗ ≤ zn − λn Azn − x∗ − λn Ax∗ ≤ zn − x∗ ≤ xn − x∗ , un − x∗ PC yn − λn Ayn − PC x∗ − λn Ax∗ ≤ yn − λn Ayn − x∗ − λn Ax∗ ≤ yn − x∗ ≤ xn − x∗ . Hence we obtain that xn1 − x∗ αn f Wn xn − x∗ βn xn − x∗ γn Wn un − x∗ ≤ αn f Wn xn − x∗ βn xn − x∗ γn Wn un − x∗ ≤ αn f Wn xn − f x∗ αn f x∗ − x∗ βn xn − x∗ γn un − x∗ ≤ αn βxn − x∗ αn f x∗ − x∗ βn xn − x∗ γn xn − x∗ ∗ f x − x∗ 1 − βαn 1 − 1 − βαn xn − x∗ 1−β f x∗ − x∗ f x∗ − x∗ ≤ max xn − x∗ , ≤ max x0 − x∗ , . 1−β 1−β 3.7 3.8 Therefore {xn } is bounded, so are {yn } and {zn }. Step 2 xn1 − xn → 0. Setting xn1 βn xn 1 − βn Vn for all n ≥ 0. It follows that xn2 − βn1 xn1 xn1 − βn xn − 1 − βn1 1 − βn αn1 f Wn1 xn1 γn1 Wn1 un1 αn f Wn xn γn Wn un − 1 − βn1 1 − βn αn1 f Wn1 xn1 αn f Wn xn γn1 − Wn1 un1 − Wn1 un 1 − βn1 1 − βn 1 − βn1 γn1 γn γn Wn1 un − Wn1 un − Wn un , 1 − βn1 1 − βn 1 − βn Vn1 − Vn 3.9 8 Fixed Point Theory and Applications which implies that Vn1 − Vn ≤ αn1 f Wn1 xn1 Wn1 un 1 − βn1 αn f Wn xn Wn1 un 1 − βn γn1 un1 − un γn Wn1 un − Wn un . 1 − βn1 1 − βn 3.10 Now we estimate un1 − un and Wn1 un − Wn un . From 3.1, since Ti and Un,i are nonexpansive, we have Wn1 un − Wn un ξ1 T1 Un1,2 un − ξ1 T1 Un,2 un ≤ ξ1 Un1,2 un − Un,2 un ξ1 ξ2 T2 Un1,3 un − ξ2 T2 Un,3 un ≤ ξ1 ξ2 Un1,3 un − Un,3 un 3.11 ≤ ··· ≤ ξ1 ξ2 · · · ξn Un1,n1 un − Un,n1 un ≤M n ξi , i1 where M is a constant such that sup{Un1,n1 un − Un,n1 un , n ≥ 0} ≤ M. At the same time, we observe that yn1 − yn PC zn1 − λn1 Azn1 − PC zn − λn Azn ≤ zn1 − λn1 Azn1 − zn − λn Azn zn1 − λn1 Azn1 − zn − λn1 Azn λn − λn1 Azn ≤ zn1 − λn1 Azn1 − zn − λn1 Azn λn − λn1 Azn ≤ zn1 − zn λn − λn1 Azn , un1 − un PC yn1 − λn1 Ayn1 − PC yn − λn Ayn ≤ yn1 − λn1 Ayn1 − yn − λn Ayn yn1 − λn1 Ayn1 − yn − λn1 Ayn λn − λn1 Ayn ≤ yn1 − yn λn − λn1 Ayn ≤ zn1 − zn λn − λn1 Ayn Azn . 3.12 Yonghong Yao et al. 9 Since zn Sr xn and zn1 Sr xn1 , from the nonexpansivity of Sr , we get zn1 − zn ≤ xn1 − xn . Substituting 3.11–3.13 into 3.10, we have Vn1 − Vn − xn1 − xn ≤ αn1 f Wn1 xn1 Wn1 un 1 − βn1 n αn f Wn xn Wn1 un M ξi 1 − βn i1 λn − λn1 Ayn Azn . Since αn → 0, λn1 − λn → 0, and ξi ∈ a, b, we have lim sup Vn1 − Vn − xn1 − xn ≤ 0. n→∞ Hence by Lemma 2.2, we have 3.14 3.15 lim Vn − xn 0. 3.16 lim xn1 − xn 0. 3.17 n→∞ Consequently, 3.13 n→∞ Step 3 un − Wun → 0. Note that xn1 − xn αn fWn xn − xn γn Wn un − xn . Then we have xn − Wn un ≤ 1 xn − xn1 αn f Wn xn − xn −→ 0. γn 3.18 For x∗ ∈ Γ, noting that Sr is firmly nonexpansive, we have zn − x∗ 2 Sr xn − Sr x∗ 2 ≤ S r x n − S r x ∗ , xn − x ∗ z n − x ∗ , xn − x ∗ and hence 3.19 1 zn − x∗ 2 xn − x∗ 2 − xn − zn 2 , 2 zn − x∗ 2 ≤ xn − x∗ 2 − xn − zn 2 . 3.20 So, we have xn1 − x∗ 2 ≤ αn f Wn xn − x∗ 2 βn xn − x∗ 2 γn Wn un − x∗ 2 2 2 2 ≤ αn f Wn xn − x∗ βn xn − x∗ γn un − x∗ 2 2 2 ≤ αn f Wn xn − x∗ βn xn − x∗ γn zn − x∗ 2 2 2 2 ≤ αn f Wn xn − x∗ βn xn − x∗ γn xn − x∗ − xn − zn 2 2 2 ≤ αn f Wn xn − x∗ xn − x∗ − γn xn − zn , 3.21 10 Fixed Point Theory and Applications that is, xn − zn 2 ≤ 1 αn f Wn xn − x∗ 2 xn − x∗ 2 − xn1 − x∗ 2 γn ≤ 2 1 αn f Wn xn − x∗ xn1 − xn xn − x∗ xn1 − x∗ γn 3.22 −→ 0. From 3.6, we obtain that xn1 − x∗ 2 ≤ αn f Wn xn − x∗ 2 βn xn − x∗ 2 γn yn − x∗ 2 2 2 2 ≤ αn f Wn xn − x∗ βn xn − x∗ γn zn − λn Azn − x∗ − λn Ax∗ 2 2 2 2 ≤ αn f Wn xn −x∗ βn xn −x∗ γn zn −x∗ λn λn − 2β Azn − Ax∗ 2 2 2 ≤ αn f Wn xn − x∗ xn − x∗ γn ab − 2βAzn − Ax∗ . 3.23 Then we have 2 2 2 2 −γn ab − 2βAzn − Ax∗ ≤ αn f Wn xn − x∗ xn − x∗ − xn1 − x∗ −→ 0, 3.24 which implies that lim Azn − Ax∗ 0. n→∞ 3.25 We note that yn − x∗ 2 PC zn − λn Azn − PC x∗ − λn Ax∗ 2 ≤ zn − λn Azn − x∗ − λn Ax∗ , yn − x∗ 1 zn − λn Azn − x∗ − λn Ax∗ 2 yn − x∗ 2 2 2 − zn − λn Azn − x∗ − λn Ax∗ − yn − x∗ 1 zn − x∗ 2 yn − x∗ 2 − zn − yn − λn Azn − Ax∗ 2 2 1 zn −x∗ 2 yn −x∗ 2 − zn −yn 2 2λn Azn −Ax∗ , zn −yn −λ2n Azn −Ax∗ 2 . 2 3.26 ≤ Then we derive yn − x∗ 2 ≤ zn − x∗ 2 − zn − yn 2 2λn Azn − Ax∗ , zn − yn − λ2n Azn − Ax∗ 2 2 2 ≤ xn − x∗ − zn − yn 2λn Azn − Ax∗ , zn − yn . 3.27 Yonghong Yao et al. 11 Hence xn1 − x∗ 2 2 2 2 2 ≤ αn f Wn xn −x∗ βn xn −x∗ γn xn −x∗ − zn −yn 2λn Azn −Ax∗ , zn −yn , 3.28 which implies that zn −yn ≤ 2 2 2 1 αn f Wn xn −x∗ xn −x∗ − xn1 −x∗ 2γn λn Azn −Ax∗ zn −yn −→ 0. γn 3.29 Since yn − un PC zn − λn Azn − PC yn − λn Ayn ≤ zn − yn , we have Wun − un ≤ Wun − Wn un Wn un − un ≤ Wun − Wn un Wn un − xn xn − zn zn − yn yn − un ≤ Wun − Wn un Wn un − xn xn − zn 2zn − yn . Combining the above inequality, 3.18–3.29, and Remark 3.3, we have lim Wun − un 0. n→∞ 3.30 3.31 Step 4 limn→∞ fx∗ − x∗ , xn − x∗ , where x∗ PΓ fx∗ . To show the above inequality, we can choose a subsequence {uni } of {un } such that lim f x∗ − x∗ , unj − x∗ lim sup f x∗ − x∗ , un − x∗ . 3.32 j→∞ n→∞ Since {unj } is bounded, there exists a subsequence {unji } of {unj } which converges weakly to w. Without loss of generality, we can assume that unj w. From Wun − un → 0, we obtain Wunj w. First, we show w ∈ FW ∩∞ ∈ FW. Since unj w and n1 FTn . Assume that ω / w/ Ww, by Opial’s condition, we have lim infun − w < lim infun − Ww j→∞ j j→∞ j ≤ lim inf unj − Wunj Wunj − Ww j→∞ 3.33 ≤ lim infunj − w, j→∞ which is a contradiction. Hence we get w ∈ FW. By the same argument as that in the proof of [21, Theorem 3.1], we can prove that w ∈ V IA, C; and by the same argument as that in the proof of [14, Theorem 4.1], we also can prove that w ∈ Ω. Hence w ∈ Γ. Since x∗ PΓ fx∗ ∈ Γ and un − xn → 0, we have lim sup f x∗ − x∗ , xn − x∗ lim f x∗ − x∗ , xnj − x∗ n→∞ j→∞ lim f x∗ − x∗ , unj − x∗ f x∗ − x∗ , w − x∗ ≤ 0. j→∞ 3.34 12 Fixed Point Theory and Applications Step 5 xn → x∗ , where x∗ PΓ fx∗ . From 3.3, we have xn1 − x∗ 2 ≤ βn xn − x∗ γn Wn un − x∗ 2 2αn f Wn xn − x∗ , xn1 − x∗ 2 ≤ βn xn − x∗ γn un − x∗ 2αn f Wn xn − f x∗ , xn1 − x∗ 2αn f x∗ − x∗ , xn1 − x∗ 2 2 ≤ 1 − αn xn − x∗ 2αn f Wn xn − f x∗ xn1 − x∗ 2αn f x∗ − x∗ , xn1 − x∗ 2 2 ≤ 1 − αn xn − x∗ 2ααn xn − x∗ xn1 − x∗ 2αn f x∗ − x∗ , xn1 − x∗ 2 2 2 2 ≤ 1−αn xn −x∗ ααn xn −x∗ xn1 −x∗ 2αn f x∗ −x∗ , xn1 −x∗ , 3.35 that is, αn xn1 −x∗ 2 ≤ 1− 21−α αn xn −x∗2 21−α αn xn −x∗2 1 f x∗ −x∗ , xn1 −x∗ . 1−ααn 1−ααn 21−α 1−α 3.36 It is easy to see that ∞ n0 21 lim sup n→∞ − α/1 − ααn αn ∞ and αn xn − x∗ 2 1 f x∗ − x∗ , xn1 − x∗ 21 − α 1−α ≤ 0. 3.37 Applying Lemma 2.3 and 3.34 to 3.36, we conclude that xn → x∗ as n → ∞. This completes the proof. Concerning Sr , we give the following remark. Remark 3.6. For each x1 , x2 ∈ C, we denote u1 Sr x1 and u2 Sr x2 . Then for all y ∈ C, we have r Θ u1 , y ϕy − ϕ u1 K u1 − K x1 , η y, u1 ≥ 0, 3.38 r Θ u2 , y ϕy − ϕ u2 K u2 − K x2 , η y, u2 ≥ 0. 3.39 Taking y u2 in 3.38 and y u1 in 3.39, and adding up these two inequalities, we obtain r Θ u1 , u2 ϕ u2 − ϕ u1 K u1 − K x1 , η u2 , u1 r Θ u2 , u1 ϕ u1 − ϕ u2 K u2 − K x2 , η u1 , u2 ≥ 0. 3.40 Note that ηu1 , u2 ηu2 , u1 0 and Θu1 , u2 Θu2 , u1 ≤ 0. Hence from 3.40, we deduce K x1 − K u1 , η u1 , u2 K u2 − K x2 , η u1 , u2 ≥ 0, 3.41 Yonghong Yao et al. 13 which implies that K x1 − K x2 , η u1 , u2 ≥ K u1 − K u2 , η u1 , u2 . 3.42 Since K : C → H is η-strongly monotone with constant μ > 0, then from 3.42, we conclude that 2 K x1 − K x2 , η u1 , u2 ≥ μu1 − u2 . 3.43 Take Kx x2 /2, ηy, x y − x, and μ 1. Then from 3.43, we have 2 x1 − x2 , u1 − u2 ≥ u1 − u2 . 3.44 This indicates that Sr is firmly nonexpansive. Corollary 3.7. Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ : C → R be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction satisfying conditions (H1)–(H3). Let A : C → H be a β-inverse-strongly monotone mapping such that V IA, C ∩ Ω / ∅. Suppose {αn }, {βn }, and {γn } are three sequences in 0, 1 with αn βn γn 1, n ≥ 0. Assume that i η : C × C → H is Lipschitz continuous with constant λ > 0 such that a ηx, y ηy, x 0 for all x, y ∈ C, b η·, · is aﬃne in the first variable, c for each fixed y ∈ C, x → ηy, x is sequentially continuous from the weak topology to the weak topology; ii K : C → R is η-strongly convex with constant σ > 0 and its derivative K is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν; iii for each x ∈ C, there exist a bounded subset Dx ⊂ C and zx ∈ C such that, for any y ∈ C\Dx , 1 Θ y, zx ϕ zx − ϕy K y − K x, η zx , y < 0; r 3.45 14 Fixed Point Theory and Applications ∞ iv limn→∞ αn 0, n0 αn ∞, 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1, λn ∈ a, b ⊂ 0, 2β, and limn→∞ λn1 − λn 0. Let f be a contraction of C into itself and given x0 ∈ C arbitrarily. Let the sequences {xn }, {yn }, and {zn } be generated iteratively by 1 Θ zn , x ϕx − ϕ zn K zn − K xn , η x, zn ≥ 0 r yn PC zn − λn Azn , xn1 αn f xn βn xn γn PC yn − λn Ayn ∀x ∈ C, 3.46 ∀n ≥ 0. Then the sequence {xn } generated by 3.46 converges strongly to x∗ PΓ fx∗ , where Γ V IA, C∩ Ω provided that Sr is firmly nonexpansive. Proof. Take Tn x x for all n 1, 2, . . . , and for all x ∈ C in 3.1. Then Wn x x for all x ∈ C. The conclusion follows immediately from Theorem 3.5. This completes the proof. Corollary 3.8. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be an infinite family of nonexpansive mappings of C into itself. Let A : C → H be a β-inverse-strongly monotone mapping such that ∩∞ / ∅. Suppose {αn }, {βn }, and {γn } are three n1 FTn ∩ V IA, C sequences in 0, 1 with αn βn γn 1, n ≥ 0. Assume that i limn→∞ αn 0 and ∞ n0 αn ∞; ii 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1; iii λn ∈ a, b ⊂ 0, 2β and limn→∞ λn1 − λn 0. 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