CUBO A Mathematical Journal Vol.16, No¯ 01, (09–20). March 2014 Viscosity approximation methods with a sequence of contractions Koji Aoyama† Department of Economics, Chiba University, Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan. aoyama@le.chiba-u.ac.jp Yasunori Kimura‡ Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan. yasunori@is.sci.toho-u.ac.jp ABSTRACT The aim of this paper is to prove that, in an appropriate setting, every iterative se- quence generated by the viscosity approximation method with a sequence of contrac- tions is convergent whenever so is every iterative sequence generated by the Halpern type iterative method. Then, using our results, we show some convergence theorems for variational inequality problems, zero point problems, and fixed point problems. RESUMEN La meta de este art́ıculo es probar en un marco de trabajo adecuado que cada sucesión iterativa generada por el método de aproximación de viscosidad con una sucesión cualquiera de contracciones es convergente como lo es cada sucesión iterativa gener- ada por el método iterativo del tipo Halpern. Aśı, usando nuestro resultado mostramos algunos teoremas de convergencia para problemas de desigualdades variacionales, prob- lemas de punto cero y problemas de punto fijo. Keywords and Phrases: Viscosity approximation method, nonexpansive mapping, fixed point, hybrid steepest descent method. 2010 AMS Mathematics Subject Classification: 47H09, 47J20, 47H10. 10 Koji Aoyama & Yasunori Kimura CUBO 16, 1 (2014) 1 Introduction Let C be a nonempty closed convex subset of a Hilbert space. This paper is devoted to the study of strong convergence of a sequence {yn} in C defined by an arbitrary point y1 ∈ C and yn+1 = λnfn(yn) + (1 − λn)Tnyn (1.1) for n ∈ N, where λn is a real number in [0, 1], fn is a contraction on C, and Tn is a nonexpansive mapping on C for n ∈ N. In particular, our main interest is the relationship between convergence of such a sequence {yn} and a sequence {xn} defined by an arbitrary point x1 ∈ C and xn+1 = λnu + (1 − λn)Tnxn (1.2) for n ∈ N, where u is a point in C. In §3, using the technique developed in [21], we prove that their convergence are equivalent under some assumptions. Then, as applications of our convergence results in §3, we discuss strong convergence of the sequences generated by the hybrid steepest descent method [30] and we give another proof of Iemoto and Takahashi’s theorem [17] in §4. Moreover, we show one generalization of Ceng, Petruşel, and Yao’s theorem [13] in §5. The iterative method defined by (1.1) is based on the viscosity approximation method due to Moudafi [19]. He considered the fixed point problem of a single nonexpansive mapping and proved strong convergence of sequences generated by the viscosity approximation methods; see also Xu [28] and Suzuki [21]. The iterative method defined by (1.2) is called the Halpern type iterative method; see Halpern [16], Wittmann [25], and Shioji and Takahashi [20]; see also [1,5,7]. 2 Preliminaries Throughout the present paper, H denotes a real Hilbert space with the inner product 〈 · , · 〉 and the norm ‖ · ‖, C a nonempty closed convex subset of H, I the identity mapping on H, and N the set of positive integers. A mapping S: C → H is said to be lipschitzian if there exists a constant η ≥ 0 such that ‖Sx − Sy‖ ≤ η ‖x − y‖ for all x, y ∈ C. In this case, S is called an η-lipschitzian mapping. In particular, an η-lipschitzian mapping is said to be nonexpansive if η = 1; an η-lipschitzian mapping is said to be an η-contraction if 0 ≤ η < 1. It is known that Fix(S) is closed and convex if S: C → H is nonexpansive, where Fix(S) denotes the set of fixed points of S. The metric projection of H onto C is denoted by PC and we know that PC is nonexpansive. We also know the following; see [22]. Lemma 2.1. Let x ∈ H and z ∈ C. Then z = PC(x) if and only if 〈y − z, x − z〉 ≤ 0 for all y ∈ C. Let {Sn} be a sequence of nonexpansive mappings of C into H. We say that {Sn} satisfies the condition (Z) if every weak cluster point of {xn} is a common fixed point of {Sn} whenever {xn} CUBO 16, 1 (2014) Viscosity approximation methods with a sequence . . . 11 is a bounded sequence in C and xn − Snxn → 0; see [1, 3, 8–11]. We say that {Sn} satisfies the condition (R) if lim n→∞ sup y∈D ‖Sn+1y − Sny‖ = 0 for every nonempty bounded subset D of C; see [1,5]. We say that {Sn} is stable on a nonempty subset D of C if {Snz : n ∈ N} is a singleton for every z ∈ D. We need the following lemmas: Lemma 2.2. Let C1 and C2 be nonempty closed convex subsets of H, {Sn} a sequence of non- expansive mappings of C1 into H, and {Tn} a sequence of nonexpansive mappings of C2 into H. Suppose that {Sn} and {Tn} satisfy the condition (R), C1 ⊃ Tn(C2) for every n ∈ N, and {Tn} has a common fixed point. Then {SnTn} satisfies the condition (R). Proof. Let D be a nonempty bounded subset of C2. Then it is clear that each SnTn is nonexpansive and ‖Sn+1Tn+1y − SnTny‖ ≤ ‖Sn+1Tn+1y − SnTn+1y‖ + ‖SnTn+1y − SnTny‖ ≤ ‖Sn+1Tn+1y − SnTn+1y‖ + ‖Tn+1y − Tny‖ (2.1) for all y ∈ D and n ∈ N. Let z be a common fixed point of {Tn}. Then it is obvious that ‖Tny‖ ≤ ‖Tny − Tnz‖ + ‖z‖ ≤ ‖y − z‖ + ‖z‖ for all y ∈ D and n ∈ N. This shows that D′ = {Tny : n ∈ N, y ∈ D} is a bounded subset of C1. Since {Sn} and {Tn} satisfy the condition (R), it follows from (2.1) that sup y∈D ‖Sn+1Tn+1y − SnTny‖ ≤ sup y ′∈D′ ‖Sn+1y ′ − Sny ′‖ + sup y∈D ‖Tn+1y − Tny‖ → 0. Therefore, {SnTn} satisfies the condition (R). Lemma 2.3. Let {Sn} be a sequence of nonexpansive mappings of C into H and {γn} a sequence in [0, 1] such that γn+1 − γn → 0. Suppose that {Sn} satisfies the condition (R) and {Sn} has a common fixed point. Then {γnI + (1 − γn)Sn} satisfies the condition (R). Proof. Set Un = γnI + (1 − γn)Sn for n ∈ N. Let D be a nonempty bounded subset of C. Then it is clear that each Un is nonexpansive and ‖Un+1y − Uny‖ ≤ |γn+1 − γn| ‖y − Sny‖ + |1 − γn+1| ‖Sn+1y − Sny‖ ≤ |γn+1 − γn| ‖y − Sny‖ + ‖Sn+1y − Sny‖ (2.2) for all y ∈ D and n ∈ N. Let z be a common fixed point of {Sn}. Then it is obvious that ‖y − Sny‖ ≤ ‖y − z‖ + ‖Snz − Sny‖ ≤ 2 ‖y − z‖ (2.3) 12 Koji Aoyama & Yasunori Kimura CUBO 16, 1 (2014) for all y ∈ D and n ∈ N. Since {Sn} satisfies the condition (R) and γn+1 − γn → 0, it follows from (2.2) and (2.3) that sup y∈D ‖Un+1y − Uny‖ ≤ 2 |γn+1 − γn| ‖y − z‖ + sup y∈D ‖Sn+1y − Sny‖ → 0. Therefore, {Un} satisfies the condition (R). A set-valued mapping A of H into H, which is denoted by A ⊂ H×H, is said to be a monotone operator if 〈x − y, x′ − y′〉 ≥ 0 for all (x, x′), (y, y′) ∈ A. A monotone operator A ⊂ H × H is said to be maximal if A = B whenever B ⊂ H × H is a monotone operator such that A ⊂ B. Let A ⊂ H × H be a maximal monotone operator. It is known that (I + ρA)−1 is a single-valued mapping of H onto dom(A) = {x ∈ H : Ax 6= ∅} for all ρ > 0. Such a mapping (I + ρA)−1 is called the resolvent of A and denoted by Jρ. It is also known that the resolvent Jρ is nonexpansive and Fix(Jρ) = A −10 = {x ∈ H : Ax 3 0}; see [22] for more details. A mapping A : H → H is said to be strongly monotone if there is a constant κ > 0 such that 〈x − y, Ax − Ay〉 ≥ κ ‖x − y‖ 2 for all x, y ∈ H. In this case, A is called a κ-strongly monotone mapping. The following lemma is well known; see, for example, [4]. Lemma 2.4. Let κ and η be positive real numbers such that η2 < 2κ. Let F be a nonempty closed convex subset of H and A : H → H a κ-strongly monotone and η-lipschitzian mapping. Then the following hold: (1) κ ≤ η, 0 ≤ 1 − 2κ + η2 < 1 and I − A is a θ-contraction, where θ = √ 1 − 2κ + η2. (2) There exists a unique point z ∈ F such that 〈y − z, Az〉 ≥ 0 for all y ∈ F, and moreover, z is the unique fixed point of PF(I − A). The following lemma is well known; see [7,18,24,26,27]. Lemma 2.5. Let {�n} be a sequence of nonnegative real numbers, {γn} a sequence of real num- bers, and {λn} a sequence in [0, 1]. Suppose that �n+1 ≤ (1 − λn)�n + λnγn for every n ∈ N, lim supn→∞ γn ≤ 0, and ∑ ∞ n=1 λn = ∞. Then �n → 0. 3 Viscosity approximation method with a sequence of con- tractions In this section, we deal with the viscosity approximation method due to Moudafi [19] in order to find a common fixed point of a sequence of nonexpansive mappings. In particular, we focus on the viscosity approximation method with a sequence of contractions. We first investigate the relationship between this method and the Halpern type iterative method (Theorem 3.1). Then, by using known results (Theorems 3.3 and 3.5), we show convergence theorems (Theorems 3.4 and 3.6). CUBO 16, 1 (2014) Viscosity approximation methods with a sequence . . . 13 Using the technique in [21], we can prove the following: Theorem 3.1. Let H be a Hilbert space, C a nonempty closed convex subset of H, {Tn} a sequence of nonexpansive self-mappings of C, F a nonempty closed convex subset of C, θ a nonnegative real number with θ < 1, and {λn} a sequence in [0, 1] such that ∑ ∞ n=1 λn = ∞. Then the following are equivalent: (1) For any (x, u) ∈ C × C, the sequence {xn} defined by x1 = x and xn+1 = λnu + (1 − λn)Tnxn (3.1) for n ∈ N converges strongly to PF(u). (2) For any y ∈ C and any sequence {fn} of θ-contractions on C which is stable on F, the sequence {yn} defined by y1 = y and yn+1 = λnfn(yn) + (1 − λn)Tnyn (3.2) for n ∈ N converges strongly to w, where w is the unique fixed point of PF ◦ f1. Proof. We first show that (1) implies (2). Let {fn} be a sequence of θ-contractions on C which is stable on F, w the fixed point of a contraction PF ◦ f1, and y ∈ C. Let {xn} be a sequence defined by x1 = y and xn+1 = λnf1(w) + (1 − λn)Tnxn for n ∈ N. Then xn → PF ( f1(w) ) = w by (1). Since Tn is nonexpansive and fn is a θ-contraction, it follows from f1(w) = fn(w) that ‖xn+1 − yn+1‖ = ∥ ∥(1 − λn)(Tnxn − Tnyn) + λn ( f1(w) − fn(yn) ) ∥ ∥ ≤ (1 − λn) ‖Tnxn − Tnyn‖ + λn ‖fn(w) − fn(yn)‖ ≤ (1 − λn) ‖xn − yn‖ + λnθ ‖w − yn‖ ≤ (1 − λn) ‖xn − yn‖ + λnθ(‖w − xn‖ + ‖xn − yn‖) ≤ ( 1 − (1 − θ)λn ) ‖xn − yn‖ + (1 − θ)λn θ 1 − θ ‖xn − w‖ for every n ∈ N. Since ∑ ∞ n=1 (1 − θ)λn = ∞ and xn → w, Lemma 2.5 shows that xn − yn → 0. Therefore, we conclude that yn → w. We next show that (2) implies (1). Let (x, u) ∈ C × C be given. For each n ∈ N, let fn be a mapping defined by fn(z) = u for z ∈ C. Then, obviously, each fn is a 0-contraction and {fn} is stable on F. Thus it follows from (2) that {xn} converges strongly to w = PF ( f1(w) ) = PF(u). Remark 3.2. It is easy to check that Theorem 3.1 holds even if H is a Banach space under appropriate conditions. We know the following result; see [2,7] and see also [3,11]. 14 Koji Aoyama & Yasunori Kimura CUBO 16, 1 (2014) Theorem 3.3. Let H be a Hilbert space, C a nonempty closed convex subset of H, {Tn} a sequence of nonexpansive self-mappings of C with a common fixed point, F the set of common fixed points of {Tn}, and {λn} a sequence in [0, 1] such that λn → 0, ∞∑ n=1 λn = ∞, and ∞∑ n=1 |λn+1 − λn| < ∞. (3.3) Suppose that {Tn} satisfies the condition (Z) and ∞∑ n=1 sup{‖Tn+1y − Tny‖ : y ∈ D} < ∞ for every nonempty bounded subset D of C. Let x and u be points in C and {xn} a sequence defined by x1 = x and (3.1) for n ∈ N. Then {xn} converges strongly to PF(u). Using Theorems 3.1 and 3.3, we obtain the following: Theorem 3.4. Let H, C, {Tn}, F, and {λn} be the same as in Theorem 3.3. Let θ be a nonnegative real number with θ < 1 and {fn} a sequence of θ-contractions on C which is stable on F. Let y be a point in C and {yn} a sequence defined by y1 = y and (3.2) for n ∈ N. Then {yn} converges strongly to w, where w is the unique fixed point of PF ◦ f1. We also know the following result; see [1,5]. Theorem 3.5. Let H be a Hilbert space, C a nonempty closed convex subset of H, {Sn} a sequence of nonexpansive self-mappings of C with a common fixed point, F the set of common fixed points of {Sn}. Let {λn} and {βn} be sequences in [0, 1] such that λn → 0, ∞∑ n=1 λn = ∞, and 0 < lim inf n→∞ βn ≤ lim sup n→∞ βn < 1. Suppose that {Sn} satisfies the conditions (Z) and (R). Let x and u be points in C and {xn} a sequence defined by x1 = x and xn+1 = λnu + (1 − λn) ( (1 − βn)xn + βnSnxn ) for n ∈ N. Then {xn} converges strongly to PF(u). Using Theorems 3.1 and 3.5, we also obtain the following: Theorem 3.6. Let H, C, {Sn}, F, {λn}, and {βn} be the same as in Theorem 3.5. Let θ be a nonnegative real number with θ < 1 and {fn} a sequence of θ-contractions on C which is stable on F. Let y be a point in C and {yn} a sequence defined by y1 = y and yn+1 = λnfn(yn) + (1 − λn) ( (1 − βn)yn + βnSnyn ) for n ∈ N. Then {yn} converges strongly to w, where w is the unique fixed point of PF ◦ f1. CUBO 16, 1 (2014) Viscosity approximation methods with a sequence . . . 15 Proof. Set Tn = (1 − βn)I + βnSn for n ∈ N. Then it is clear that each Tn is nonexpansive and yn+1 = λnfn(yn) + (1 − λn)Tnyn for n ∈ N. Let x and u be points in C and {xn} a sequence defined by x1 = x and xn+1 = λnu + (1 − λn)Tnxn for n ∈ N. Then it follows from Theorem 3.5 that xn → PF(u). Therefore, Theorem 3.1 implies the conclusion. 4 Convergence theorems by the hybrid steepest descent method In this section, we deal with the variational inequality problem over the set of common fixed points of a sequence of nonexpansive mappings; see Problem 4.1 below. Then we prove some strong convergence theorems by the hybrid steepest descent method introduced by Yamada [30]. We know many results by using the hybrid steepest descent method; see [2,14,17,29,31]. Problem 4.1. Let H be a Hilbert space, {Tn} a sequence of nonexpansive self-mappings of H with a common fixed point, F the set of common fixed points of {Tn}, and A : H → H a κ-strongly monotone and η-lipschitzian mapping, where κ and η are positive real numbers such that η2 < 2κ. Then find z ∈ F such that 〈y − z, Az〉 ≥ 0 for all y ∈ F. Remark 4.2. The assumption that η2 < 2κ in Problem 4.1 is not restrictive. Indeed, suppose that a κ-strongly monotone and η-lipschitzian mapping A is given. Let us choose a positive constant µ such that µ < 2κ/η2, and define κ′ = µκ and η′ = µη. Then it is easy to verify that (η′)2 < 2κ′, µA is κ′-strongly monotone and η′-lipschitzian, and moreover, 〈y − z, Az〉 ≥ 0 is equivalent to 〈y − z, µAz〉 ≥ 0 for every y, z ∈ H. As a consequence of Theorem 3.1, we can obtain the following theorem, which shows that every sequence generated by the hybrid steepest descent method for Problem 4.1 is convergent whenever so is every sequence generated by the Halpern type iterative method for the sequence of nonexpansive mappings. Theorem 4.3. Let H, {Tn}, F, κ, η, and A be the same as in Problem 4.1. Let {λn} be a sequence in [0, 1] such that ∑ ∞ n=1 λn = ∞. Suppose that for any (x, u) ∈ H × H, the sequence {xn} defined by x1 = x and xn+1 = λnu + (1 − λn)Tnxn (4.1) for n ∈ N converges strongly to PF(u). Let y be a point in H and {yn} a sequence defined by y1 = y and yn+1 = (I − λnA)Tnyn (4.2) for n ∈ N. Then {yn} converges strongly to the unique solution of Problem 4.1. 16 Koji Aoyama & Yasunori Kimura CUBO 16, 1 (2014) Proof. Set fn = (I − A)Tn for n ∈ N. Since Tn is nonexpansive, fn is a θ-contraction by Lemma 2.4, where θ = √ 1 − 2κ + η2. By the definition of {yn}, it is clear that yn+1 = λn(I − A)Tnxn + (1 − λn)Tnyn = λnfn(yn) + (1 − λn)Tnyn for every n ∈ N. It is also clear that {fn} is stable on F. Thus Theorem 3.1 implies that {yn} converges strongly to w = PF ( (I − A)T1w ) = PF(I − A)w, which is the unique solution of Problem 4.1 by Lemma 2.4. Using Theorem 3.6 and other known results, we also obtain the following: Theorem 4.4 (Iemoto and Takahashi [17, Theorem 3.1]). Let H, {Tn}, F, κ, η, and A be the same as in Problem 4.1. Let {λn} be a sequence in [0, 1] such that λn → 0 and ∞∑ n=1 λn = ∞ and {γn} a sequence in [a, b], where 0 < a ≤ b < 1. For each n ∈ N and k ∈ {1, 2, . . . , n + 1}, let Un,k be a mapping defined by Un,k = { I if k = n + 1; Un,k = (1 − γk)I + γkTkUn,k+1 if k ∈ {1, 2, . . . , n}. Let y be a point in H and {yn} a sequence defined by y1 = y and yn+1 = (I − λnA)Un,1yn (4.3) for n ∈ N. Then {yn} converges strongly to the unique solution of Problem 4.1. Proof. Set fn = (I − A)Un,1 and Sn = T1Un,2 for n ∈ N. Then it is obvious from (4.3) that yn+1 = λn(I − A)Un,1yn + (1 − λn)Un,1yn = λnfn(yn) + (1 − λn) ( (1 − γ1)yn + γ1Snyn ) for every n ∈ N. It is known that Fix(Sn) = Fix(Un,1) = n ⋂ k=1 Fix(Tk) by [23, Lemma 3.2]; see also [9, Lemma 4.2]. Hence we have ∞ ⋂ n=1 Fix(Sn) = ∞ ⋂ n=1 Fix(Un,1) = ∞ ⋂ n=1 n ⋂ k=1 Fix(Tk) = F and thus fn(z) = (I − A)Un,1z = (I − A)z for all z ∈ F. Therefore, {fn} is stable on F. Since Un,1 is nonexpansive, fn is a θ-contraction by Lemma 2.4, where θ = √ 1 − 2κ + η2. It is also known that {Sn} satisfies the conditions (Z) and (R); see [3, 5, 9, 11]. Therefore, Theorem 3.6 implies that {yn} converges strongly to w = PF ( f1(w) ) = PF(I − A)w, which is the unique solution of Problem 4.1 by Lemma 2.4. CUBO 16, 1 (2014) Viscosity approximation methods with a sequence . . . 17 5 Zero point problems and fixed point problems Motivated by Ceng, Petruşel, and Yao [13], we consider the problem of finding a common solution of the zero point problem for a maximal monotone operator and the fixed point problems for nonexpansive mappings. Then, by using Theorem 3.6, we prove the following strong convergence theorem, which is a generalization of [13, Theorem 3.1]; see Remark 5.2 below. Theorem 5.1. Let H be a Hilbert space, C a nonempty closed convex subset of H, {Tn} a sequence of nonexpansive self-mappings of C, A ⊂ H × H a maximal monotone operator with dom(A) ⊂ C, θ a nonnegative real number with θ < 1, and f a θ-contraction on C. Let {αn}, {βn}, and {γn} be sequences in [0, 1) such that αn → 0, ∑ ∞ n=1 αn = ∞, 0 < lim infn→∞ βn ≤ supn βn < 1, αn + βn ≤ 1 for every n ∈ N, 0 < lim infn→∞ γn ≤ supn γn < 1, and γn+1 − γn → 0. Let {ρn} be a sequnece of positive real numbers such that infn ρn > 0 and ρn+1 − ρn → 0. Suppose that F = ⋂ ∞ n=1 Fix(Tn) ∩ A −10 is nonempty and lim n→∞ sup y∈D ‖Tny − TmTny‖ = 0 and lim n→∞ sup y∈D ‖Tn+1y − TmTny‖ = 0 (5.1) for any m ∈ N and nonempty bounded subset D of C. Let y be a point in C and {yn} a sequence defined by y1 = y and yn+1 = αnf(Vnxn) + (1 − αn − βn)xn + βnTnVnxn (5.2) for n ∈ N, where Vn = γnI + (1 − γn)TnJρn and Jρn is the resolvent of A. Then {yn} converges strongly to the unique fixed point of PF ◦ f. Proof. Since γn 6= 1 and Fix(Tn) ∩ Fix(Jρn) = Fix(Tn) ∩ A −10 is nonempty, it follows from [8, Corollary 3.9] and [9, Corollary 3.6] that Fix(Vn) = Fix(TnJρn) = Fix(Tn) ∩ Fix(Jρn) = Fix(Tn) ∩ A −10 and Fix(TnVn) = Fix(Tn) ∩ Fix(Vn) = Fix(Vn) for every n ∈ N. Therefore, we have ∞ ⋂ n=1 Fix(TnVn) = ∞ ⋂ n=1 Fix(Vn) = ∞ ⋂ n=1 Fix(Tn) ∩ A −10 = F 6= ∅. (5.3) It is clear that each Vn is nonexpansive and thus f ◦ Vn is a θ-contraction for every n ∈ N. Since f(Vnz) = f(z) for all z ∈ F by (5.3), we see that {f ◦ Vn} is stable on F. We next show that {TnVn} satisfies the condition (R). Let D be a nonempty bounded subset of C. By (5.1), we have lim n→∞ sup y∈D ‖Tn+1y − Tny‖ ≤ lim n→∞ sup y∈D ‖Tn+1y − T1Tny‖ + lim n→∞ sup y∈D ‖T1Tny − Tny‖ = 0 18 Koji Aoyama & Yasunori Kimura CUBO 16, 1 (2014) and hence {Tn} satisfies the condition (R). Since {Jρn} satisfies the condition (R) by [5, Example 4.2], Lemma 2.2 shows that {TnJρn} satisfies the condition (R). Thus Lemma 2.3 implies that {Vn} satisfies the condition (R). Therefore, it follows from Lemma 2.2 that {TnVn} satisfies the condition (R). We next show that {TnVn} satisfies the condition (Z). Let {xn} be a bounded sequence in C such that xn − TnVnxn → 0 and {xni} a subsequence of {xn} such that xni ⇀ z. It is enough to show that z ∈ F. It follows from [8, Theorem 3.10] that xn − Tnxn → 0 and xn − Vnxn → 0. Let D be a nonempty bounded subset of C such that xn ∈ D for all n ∈ N. For fixed m ∈ N, it follows from (5.1) and xn − Tnxn → 0 that ‖xn − Tmxn‖ ≤ ‖xn − Tnxn‖ + ‖Tnxn − TmTnxn‖ + ‖TmTnxn − Tmxn‖ ≤ 2 ‖xn − Tnxn‖ + sup y∈D ‖Tny − TmTny‖ → 0 as n → 0. Thus, by the demiclosedness [15, p.109] of I − Tm, z ∈ Fix(Tm) and hence z ∈ ⋂ ∞ n=1 Fix(Tn). On the other hand, xn−Vnxn → 0 and [9, Corollary 3.2] imply that xn−TnJρnxn → 0 and hence xn − Jρnxn → 0 by [8, Theorem 3.10]. Thus z ∈ A −10 because {Jρn} satisfies the condition (Z); see [8, Lemma 5.1], [10, Lemma 2.1], and [12, Lemma 2.4]. Consequently, we conclude that z ∈ F. Finally, by assumption, it is obvious that yn+1 = αnf(Vnxn) + (1 − αn) ( ( 1 − βn 1 − αn ) xn + βn 1 − αn TnVnxn ) for every n ∈ N and 0 < lim inf n→∞ βn 1 − αn ≤ lim sup n→∞ βn 1 − αn < 1. Thus Theorem 3.6 implies the conclusion. Remark 5.2. Ceng, Petruşel, and Yao [13] considered an equilibrium problem for a real-valued function φ defined on C×C and they adopted the resolvent of φ in [13, Theorem 3.1]. 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