Báo cáo hóa học: " Research Article Strong Convergence Theorems for an Inﬁnite Family of Equilibrium Problems and Fixed Point Problems for an Inﬁnite Family of Asymptotically Strict Pseudocontractions"

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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 859032, 15 pages doi:10.1155/2011/859032 Research Article Strong Convergence Theorems for an Inﬁnite Family of Equilibrium Problems and Fixed Point Problems for an Inﬁnite Family of Asymptotically Strict Pseudocontractions Shenghua Wang,1 Shin Min Kang,2 and Young Chel Kwun3 1 School of Applied Mathematics and Physics, North China Electric Power University, Baoding 071003, China 2 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea 3 Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea Correspondence should be addressed to Young Chel Kwun, yckwun@dau.ac.kr Received 12 October 2010; Accepted 29 January 2011 Academic Editor: Jong Kim Copyright q 2011 Shenghua Wang 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. We prove a strong convergence theorem for an inﬁnite family of asymptotically strict pseudo- contractions and an inﬁnite family of equilibrium problems in a Hilbert space. Our proof is simple and diﬀerent from those of others, and the main results extend and improve those of many others. 1. Introduction Let C be a closed convex subset of a Hilbert space H . Let S : C → H be a mapping and if there exists an element x ∈ C such that x Sx, then x is called a ﬁxed point of S. The set of ﬁxed points of S is denoted by F S . Recall that 1 S is called nonexpansive if Sx − Sy ≤ x − y , ∀x, y ∈ C, 1.1 2 S is called asymptotically nonexpansive 1 if there exists a sequence {kn } ⊂ 1, ∞ with kn → 1 such that
2 Fixed Point Theory and Applications Sn x − Sn y ≤ k n x − y , ∀x, y ∈ C, n ≥ 1, 1.2 3 S is called to be a κ-strict pseudo-contraction 2 if there exists a constant κ with 0 ≤ κ < 1 such that 2 2 2 Sx − Sy ≤ x−y κ x − y − Sx − Sy , ∀x, y ∈ C, 1.3 4 S is called an asymptotically κ-strict pseudo-contraction 3, 4 if there exists a constant κ with 0 ≤ κ < 1 and a sequence {γn } ⊂ 0, ∞ with limn → ∞ γn 0 such that Sn x − Sn y x − y − Sn x − Sn y 2 2 2 ≤1 γn x−y κ , ∀x, y ∈ C, n ≥ 1. 1.4 It is clear that every asymptotically nonexpansive mapping is an asymptotically 0- strict pseudo-contraction and every κ-strict pseudo-contraction is an asymptotically κ-strict pseudo-contraction with γn 0 for all n ≥ 1. Moreover, every asymptotically κ-strict pseudo-contraction with sequence {γn } is uniformly L-Lispchitzian, where L sup{ κ 1 γn 1 − κ / 1 − κ : n ≥ 1} and the ﬁxed point set of asymptotically κ-strict pseudo- contraction is closed and convex; see 3, Proposition 2.6 . Let Φ be a bifunction from C × C to Ê, where Ê is the set of real numbers. The equilibrium problem for Φ : C × C → Ê is to ﬁnd x ∈ C such that Φ x, y ≥ 0 for all y ∈ C. The set of such solutions is denoted by EP Φ . In 2007, S. Takahashi and W. Takahashi 5 ﬁrst introduced an iterative scheme by the viscosity approximation method for ﬁnding a common element of the set of solutions of the equilibrium problem and the set of ﬁxed points of a nonexpansive mapping in a Hilbert space H and proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result 6 and Wittmann’s result 7 . More precisely, they gave the following theorem. Theorem 1.1 see 5 . Let C be a nonempty closed convex subset of H . Let Φ be a bifunction from C × C to Ê satisfying the following assumptions: A1 Φ x, x 0 for all x ∈ C; A2 Φ is monotone, that is, Φ x, y Φ y, x ≤ 0 for all x, y ∈ C; A3 for all x, y, z ∈ C, lim Φ tz 1 − t x, y ≤ Φ x, y ; 1.5 t↓0 A4 for all x ∈ C, y → Φ x, y is convex and lower semicontinuous.
Fixed Point Theory and Applications 3 Let S : C → H be a nonexpansive mapping such that F S ∩ EP Φ / ∅, f : H → H be a contraction and {xn }, {un } be the sequences generated by x1 ∈ H, 1 Φ un , y y − un , un − xn ≥ 0, ∀y ∈ C, 1.6 rn xn αn f xn 1 − αn Sun , ∀n ≥ 1 , 1 where {αn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy the following conditions: ∞ ∞ lim αn 0, αn ∞, |αn − αn | < ∞, 1 n→∞ n1 n1 1.7 ∞ lim inf rn > 0, |rn − rn | < ∞. 1 n→∞ n1 Then, the sequences {xn } and {un } converge strongly to z ∈ F S ∩ EP Φ , where z PF S ∩EP Φ f z . In 8 , Tada and Takahashi proposed a hybrid algorithm to ﬁnd a common element of the set of ﬁxed points of a nonexpansive mapping and the set of solutions of an equilibrium problem and proved the following strong convergence theorem. Theorem 1.2 see 8 . Let C be a nonempty closed convex subset of a Hilbert space H . Let Φ be a bifunction from C × C → Ê satisfying A1 – A4 and let S be a nonexpansive mapping of C into H such that F S ∩ EP Φ / ∅. Let {xn } and {un } be sequences generated by x1 x ∈ H and 1 un ∈ C such that Φ un , y y − un , un − xn ≥ 0, ∀y ∈ C, rn wn 1 − αn xn αn Sun , 1.8 Cn {z ∈ H : wn − z ≤ xn − z }, Dn {z ∈ H : xn − z, x − xn ≥ 0}, xn PCn ∩Dn x, ∀n ≥ 1 , 1 where {αn } ⊂ a, 1 for some a ∈ 0, 1 and {rn } ⊂ 0, ∞ satisﬁes lim infn → ∞ rn > 0. Then {xn } converges strongly to PF S ∩EP Φ x. Many methods have been proposed to solve the equilibrium problems and ﬁxed point problems; see 9–13 . Recently, Kim and Xu 3 proposed a hybrid algorithm for ﬁnding a ﬁxed point of an asymptotically κ-strict pseudo-contraction and proved a strong convergence theorem in a Hilbert space.
4 Fixed Point Theory and Applications Theorem 1.3 see 3 . Let C be a closed convex subset of a Hilbert space H . Let T : C → C be an asymptotically κ-strict pseudo-contraction for some 0 ≤ κ < 1. Assume that F T is nonempty and bounded. Let {xn } be the sequence generated by the following algorithm: x0 ∈ C chosen arbitrarily, 1 − αn T n xn , yn αn xn xn − T n xn 2 2 Cn z ∈ H : yn − z ≤ xn − z κ − αn 1 − αn θn , 1.9 Dn {z ∈ H : xn − z, x0 − xn ≥ 0}, xn PCn ∩Dn x0 , ∀n ≥ 1 , 1 where θn Δ2 1 − αn γn −→ 0 n −→ ∞ , Δn sup{ xn − z : z ∈ F T } < ∞. 1.10 n Assume that the control sequence {αn } is chosen such that lim supn → ∞ αn < 1 − κ. Then {xn } converges strongly to PF T x0 . In this paper, motivated by 3, 8 , we propose a new algorithm for ﬁnding a common element of the set of ﬁxed points of an inﬁnite family of asymptotically strict pseudo- contractions and the set of solutions of an inﬁnite family of equilibrium problems and prove a strong convergence theorem. Our proof is simple and diﬀerent from those of others, and the main results extend and improve those Kim and Xu 3 , Tada and Takahashi 8 , and many others. 2. Preliminaries Let H be a Hilbert space, and let C be a nonempty closed convex subset of H . It is well known that, for all x, y ∈ C and t ∈ 0, 1 , 2 2 2 tx 1−t y tx 1−t y −t 1−t x−y , 2.1 and hence 2 2 2 tx 1−t y ≤t x 1−t y , 2.2 which implies that 2 n n 2 ti xi ≤ ti xi 2.3 i1 i1 n for all {xi } ⊂ H and {ti } ⊂ 0, 1 with i 1 ti 1.
Fixed Point Theory and Applications 5 For any x ∈ H , there exists a unique nearest point in C, denoted by PC x, such that z PC x ⇐⇒ x − z, z − y ≥ 0, ∀y ∈ C. 2.4 Let I denote the identity operator of H , and let {xn } be a sequence in a Hilbert space H and x ∈ H . Throughout the rest of the paper, xn → x denotes the strong convergence of {xn } to x. We need the following lemmas for our main results in this paper. Lemma 2.1 see 14 . Let C be a nonempty closed convex subset of a Hilbert space H . Let Φ be a bifunction from C × C to Ê satisfying A1 – A4 . Let r > 0 and x ∈ H . Then there exists z ∈ C such that 1 Φ z, y y − z, z − x ≥ 0, ∀y ∈ C. 2.5 r Lemma 2.2 see 6 . Let C be a nonempty closed convex subset of a Hilbert space H . Let Φ be a bifunction from C × C to Ê satisfying A1 – A4 . For any r > 0 and x ∈ H , deﬁne a mapping Tr : H → C as follows: 1 Tr x z ∈ C : Φ z, y y − z, z − x ≥ 0, ∀y ∈ C , ∀x ∈ H. 2.6 r Then the following hold: 1 Tr is single-valued, 2 Tr is ﬁrmly nonexpansive, that is, for any x, y ∈ H , 2 Tr x − Tr y ≤ Tr x − Tr y, x − y , 2.7 3 F Tr EP Φ , and 4 EP Φ is closed and convex. 3. Main Results Now, we are ready to give our main results. Lemma 3.1. Let C be a nonempty closed convex subset of a Hilbert space H . Let T : C → C be an asymptotically κ-strict pseudo-contraction with sequence {γn } ⊂ 0, ∞ such that F T / ∅. Assume 1 − βn T n for each n ≥ 1. Then the following that {βn } ⊂ κ, 1 and deﬁne a mapping Sn βn I hold: 2 2 Sn x − Sn y ≤1 γn x−y , ∀x, y ∈ C, 3.1 ≤ γn x − x∗ 2 x − Sn x, x − x∗ , ∀x ∈ C, x∗ ∈ F T . 2 2 Sn x − x
6 Fixed Point Theory and Applications Proof. For all x, y ∈ C, we have T nx − T ny 2 Sn x − Sn y βn x − y 1 − βn 2 T nx − T ny I − Tn x − I − Tn y βn x − y 1 − βn − βn 1 − βn 2 2 2 κ I − Tn x − I − Tn y ≤ βn x − y 1 − βn γn x−y 2 2 2 1 I − Tn x − I − Tn y − βn 1 − βn 2 βn x − y 1 − βn γn x−y 2 2 1 I − Tn x − I − Tn y 1 − βn κ − βn 2 ≤ βn x − y 1 − βn γn x−y 2 2 1 ≤1 γn x − y 2. 3.2 By this result, for all x ∈ C and x∗ ∈ F T , we have x − x∗ ≥ Sn x − Sn x ∗ x − x∗ 2 2 2 γn Sn x − x 1 3.3 x − x∗ 2 Sn x − x, x − x∗ , 2 2 Sn x − x and hence ≤ γn x − x∗ 2 x − Sn x, x − x∗ . 2 2 Sn x − x 3.4 This completes the proof. Lemma 3.2. Let C be a nonempty closed subset of a Hilbert space H . Let T : C → C be an asymptotically κ-strict pseudo-contraction with sequence {γn } ⊂ 0, ∞ satisfying γn → 0 as n → ∞. Let {zn } be a sequence in C such that zn − zn 1 → 0 and zn − T n zn → 0 as n → ∞. Then zn − Tzn → 0 as n → ∞. Proof. The proof method of this lemma is mainly from 15, Lemma 2.7 . Since T is an asymptotically κ-strict pseudo-contraction, we obtain from 3, Proposition 2.6 that T n 1 zn − T n 1 zn ≤ L zn − zn , 3.5 1 1 where L sup{ κ 1 γn 1 − κ / 1 − κ : n ≥ 1}. Note that zn − zn → 0, which implies 1 that T n 1 zn − T n 1 zn 1 → 0, and observe that − T n 1 zn T n 1 zn − T n 1 zn T n 1 zn − Tzn zn − Tzn ≤ zn − zn zn 1 1 1 1 3.6 n1 T n 1 zn − Tzn . ≤1 L zn − zn zn 1 −T zn 1 1
Fixed Point Theory and Applications 7 Since T is uniformly Lipschitzian, T is uniformly continuous. So we have T n 1 zn − Tzn −→ 0 as n −→ ∞. 3.7 It follows from zn − zn 1 → 0 and zn − T n zn → 0 as n → ∞ that limn → ∞ zn − Tzn 0. This completes the proof. Let H be a Hilbert space, and, let C be a nonempty closed and convex subset of H . Let {Φn } be a countable family of bifunctions from C × C to Ê satisfying A1 – A4 and let {rn } be a real number sequence in r, ∞ with r > 0. Deﬁne 1 Tri x z ∈ C : Φi z, y y − z, z − x ≥ 0, ∀y ∈ C , ∀x ∈ H. 3.8 ri Lemma 2.2 shows that every Tri i ≥ 1 is a ﬁrmly nonexpansive mapping and hence nonexpansive and F Tri EP Φ i . Theorem 3.3. Let C be a nonempty closed convex subset of a Hilbert space H . Let {Ti } : C → C be an inﬁnite family of asymptotically κi -strict pseudocontractions with the sequence {γi,n } ⊂ 0, ∞ satisfying γi,n → 0 as n → ∞ for each i ≥ 1 and γ1,n ≥ γi,n for each i ≥ 1 and n ≥ 1. Let {Φn } be a countable family of bifunctions from C × C to Ê satisfying A1 – A4 . Assume that Ω ∞ i 1 F Ti ∩ EP Φi is nonempty and bounded. Set α0 1 and θ0 1. Assume that {αi } is a strictly decreasing sequence in 0, a for some 0 < a < 1, {θn } is a strictly decreasing sequence in 0, 1 , {βi,n } is a sequence in κi , κ with 0 < κi < κ < 1 for each i ≥ 1, and {rn } is a sequence in r, ∞ with r > 0. The sequence {xn } is generated by x1 x ∈ C and n zn θn xn θi−1 − θi Tri xn , i1 n 1 − βi,n Tin zn , wn αn xn αi−1 − αi βi,n I i1 3.9 Cn {v ∈ C : wn − v ≤ xn − v λn }, n Dn Cj , j1 xn PDn x, ∀n ≥ 1 , 1 where {Tri } is deﬁned by 3.8 and λn 1 − αn γ1,n Δn −→ 0 n −→ ∞ , Δn sup{ xn − v : v ∈ Ω}. 3.10 Then {xn } converges strongly to PΩ x.
8 Fixed Point Theory and Applications Proof. We show ﬁrst that the sequence {xn } is well deﬁned. Obviously, Cn is closed for all n ≥ 1. Since wn − v ≤ xn − v λn 3.11 is equivalent to 2 wn − xn 2 wn − xn , xn − z ≤ λn , 3.12 n Cn is convex for all n ≥ 1. So Dn j 1 Cj is also closed and convex for all n ≥ 1. 1 − βi,n Tin . Let p ∈ Ω. Note that θ0 For each n ≥ 1 and i ≥ 1, put Si,n βi,n I 1, {θn } is strictly decreasing and each Tri is ﬁrmly nonexpansive. Hence we have n zn − p ≤ θn xn − p θi−1 − θi Tri xn − p i1 n ≤ θn xn − p θi−1 − θi xn − p 3.13 i1 ≤ θn xn − p 1 − θn xn − p xn − p , ∀n ≥ 1 . Since α0 1 and {αn } is strictly decreasing, by 3.13 and Lemma 3.1, we have n wn − p ≤ αn xn − p αi−1 − αi Si,n zn − p i1 n ≤ αn xn − p αi−1 − αi γi,n zn − p 1 i1 3.14 n ≤ αn xn − p αi−1 − αi 1 γ1,n xn − p i1 ≤ xn − p λn . n So we have p ∈ Cn and hence p ∈ Dn j 1 Cj for all n ≥ 1. This shows that Ω ⊂ Dn for all n ≥ 1. This implies that the sequence {xn } is well deﬁned. Since Ω is a nonempty closed convex subset of H , there exists a unique z∗ ∈ Ω such that z∗ PΩ x. 3.15 From xn PDn x, we have 1 xn −x ≤ z−x , ∀z ∈ Dn . 3.16 1
Fixed Point Theory and Applications 9 Since z∗ ∈ Ω ⊂ Dn , we have − x ≤ z∗ − x , xn ∀n ≥ 1 . 3.17 1 Therefore, {xn } is bounded. From 3.13 and 3.14 , {zn } and {wn } are also bounded. From xn 1 PDn x and Dn 1 ⊂ Dn , one sees that xn 2 PDn 1 x ∈ Dn 1 ⊂ Dn for all n ≥ 1. It follows that xn − x ≤ xn −x , ∀n ≥ 1 . 3.18 1 2 Since {xn } is bounded, the sequence { x − xn } is bounded and nondecreasing. So there exists c ∈ Ê such that c lim x − xn . 3.19 n→∞ Since xn PDn x ∈ Dn , xn PDn 1 x ∈ Dn ⊂ Dn and xn xn /2 ∈ Dn , we have 1 2 1 1 2 xn xn 2 1 2 2 x − xn ≤ x− 1 2 2 1 1 x − xn x − xn 3.20 1 2 2 2 1 1 1 2 2 2 x − xn x − xn − xn − xn . 1 2 1 2 2 2 4 So we get 1 1 1 2 2 2 xn − xn ≤ x − xn − x − xn . 3.21 1 2 2 1 4 2 2 Since limn → ∞ x − xn limn → ∞ x − xn c, we obtain 1 2 lim xn − xn 0, 3.22 1 2 n→∞ that is, lim xn − xn 0. 3.23 1 n→∞ Now, for each l ≥ 1, from 3.23 we get xn l − xn ≤ xn l − xn ··· xn − xn l−1 1 3.24 −→ 0 as n −→ ∞.
10 Fixed Point Theory and Applications This implies that there exists an element x ∈ C such that xn → x as n → ∞. Next we show that x ∈ ∞1 F Ti and x ∈ ∞1 EP Φi . i i From xn 1 ∈ Cn , we have xn − wn ≤ xn − xn xn − wn 1 1 3.25 ≤ 2 xn − xn λn . 1 By 3.10 and 3.23 , we obtain lim xn − wn 0. 3.26 n→∞ For p ∈ Ω, we have, from Lemma 2.2, 2 2 Tri xn − p Tri xn − Tri p ≤ Tri xn − Tri p, xn − p 3.27 Tri xn − p, xn − p 1 2 2 2 Tri xn − p xn − p − xn − Tri xn , 2 and hence 2 2 − xn − Tri xn 2 , Tri xn − p ≤ xn − p ∀i ≥ 1 . 3.28 Therefore n 2 2 2 zn − p ≤ θn xn − p θi−1 − θi Tri xn − p i1 n 2 2 2 ≤ θn xn − p θi−1 − θi xn − p − xn − Tri xn 3.29 i1 n 2 θi−1 − θi xn − Tri xn 2 . xn − p − i1
Fixed Point Theory and Applications 11 By 3.29 and Lemma 3.1, we have n 2 2 2 wn − p ≤ αn xn − p αi−1 − αi Si,n zn − p i1 n 2 2 2 ≤ αn xn − p αi−1 − αi 1 γ1,n zn − p i1 2 2 2 αn xn − p 1 − αn 1 γ1,n zn − p n 2 2 2 2 ≤ αn xn − p 1 − αn 1 γ1,n xn − p − θi−1 − θi xn − Tri xn i1 2 2 xn − p 1 − αn 2γ1,n γ1,n xn − p 2 n 2 θi−1 − θi xn − Tri xn 2 , − 1 − αn 1 γ1,n i1 3.30 and hence n 2 2 1 − αn 1 γ1,n θi−1 − θi xn − Tri xn i1 3.31 2 2 2 ≤ xn − p − wn − p 1 − αn 2γ1,n γ1,n xn − p 2 2 ≤ xn − wn xn − p wn − p 1 − αn 2γ1,n γ1,n xn − p . 2 This shows that 2 2 1 − αn 1 γ1,n θi−1 − θi xn − Tri xn ≤ xn − wn xn − p wn − p 3.32 2 1 − αn 2γ1,n γ1,n xn − p , ∀i ≥ 1 . 2 Since {αn } ⊂ 0, a with 0 < a < 1, γ1,n → 0, {θn } is strictly decreasing and xn − wn → 0, we get lim xn − Tri xn 0, ∀i ≥ 1 . 3.33 n→∞
12 Fixed Point Theory and Applications Let Mn supi≥1 { xn − Tri xn } for each n ≥ 1. Then Mn → 0 as n → ∞. Hence, from 3.33 , one has n xn − zn ≤ θi−1 − θi Tri xn − xn i1 n 3.34 ≤ θi−1 − θi Mn 1 − θn Mn i1 −→ 0. From 3.26 and 3.34 , we obtain zn − wn ≤ zn − xn xn − wn −→ 0. 3.35 Noting that n αi−1 − αi zn − Si,n zn αn xn 1 − αn zn − wn i1 3.36 αn xn − wn 1 − αn zn − wn , we have n αi−1 − αi zn − Si,n zn , zn − p i1 3.37 αn xn − wn , zn − p 1 − αn zn − wn , zn − p . By Lemma 3.1, we have 2 2 zn − Si,n zn ≤ γi,n zn − p 2 zn − Si,n zn , zn − p 3.38 2 ≤ γ1,n zn − p 2 zn − Si,n zn , zn − p . Therefore, combining this inequality with 3.37 , we get n 2 αi−1 − αi zn − Si,n zn i1 3.39 2 ≤ γ1,n 1 − αn zn − p 2αn xn − wn , zn − p 2 1 − αn zn − wn , zn − p ,
Fixed Point Theory and Applications 13 and hence noting that αi−1 > αi for each i ≥ 1 γ1,n 1 − αn 2αn 2 2 zn − Si,n zn ≤ zn − p xn − wn , zn − p αi−1 − αi αi−1 − αi 3.40 2 1 − αn zn − wn , zn − p . αi−1 − αi From 3.26 , 3.35 and limn → ∞ γ1,n 0, we have lim zn − Si,n zn 0, ∀i ≥ 1 . 3.41 n→∞ From the deﬁnition of Si,n and 3.41 , we have noting that {βi,n } ⊂ κi , κ ⊂ 0, 1 1 zn − Tin zn ≤ zn − Si,n zn −→ 0, ∀i ≥ 1 . 3.42 1 − βi,n We next show 3.42 implies that lim zn − Ti zn 0, ∀i ≥ 1 . 3.43 n→∞ As a matter of fact, from 3.23 and 3.34 we have zn − zn ≤ zn − xn xn − xn xn − zn 1 1 1 1 3.44 −→ 0. Now, 3.42 , 3.44 , and Lemma 3.2 imply 3.43 . Since each Ti is uniformly continuous and zn → x as n → ∞, one get x ∈ F Ti for each i ≥ 1 and hence x ∈ ∞1 F Ti . i Now we show x ∈ ∞1 EP Φi . i Since every Tri is nonexpansive, from 3.33 and xn → x, we have x ∈ F Tri and hence x ∈ ∞1 F Tri . Lemma 2.2 shows that x ∈ ∞1 EP Φi . i i Finally, we prove that x PΩ x. From xn 1 PDn x, one sees xn − z, x − xn ≥ 0, ∀z ∈ Dn . 3.45 1 1 Since Ω ⊂ Dn for all n ≥ 1, one arrives at xn − z, x − xn ≥ 0, ∀z ∈ Ω. 3.46 1 1 Taking the limit for above inequality, we get x − z, x − x ≥ 0, ∀z ∈ Ω. 3.47 Hence x PΩ x. This completes the proof.
14 Fixed Point Theory and Applications As direct consequences of Theorem 3.3, we can obtain the following corollaries. Corollary 3.4. Let C be a nonempty closed convex subset of a Hilbert space H . Let {Φn } be a countable family of bifunctions from: C × C to Ê satisfying A1 – A4 . Assume that Ω ∞ i 1 EP Φi is nonempty and bounded. Let {rn } be a sequence in r, ∞ with r > 0. Set θ0 1. The sequence {xn } is generated by x1 x ∈ C and n zn θn xn θi−1 − θi Tri xn , i1 Cn {v ∈ C : zn − v ≤ xn − v }, 3.48 n Dn Cj , j1 xn PDn x, ∀n ≥ 1 , 1 where {Tri } is deﬁned by 3.8 and {θn } is a strictly decreasing sequence in 0, 1 . Then {xn } converges strongly to PΩ x. Proof. Putting Ti I for all i ≥ 1 and αn 0 for all n ≥ 1 in Theorem 3.3, we obtain Corollary 3.4. Corollary 3.5. Let C be a nonempty closed subset of a Hilbert space H . Let T be an asymptotically κ-strict pseudo-contraction with sequence {γn } ⊂ 0, ∞ satisfying γn → 0 as n → ∞ and F T / ∅. Let {xn } and {un } be sequences generated by x1 x ∈ H and zn θn xn 1 − θn PC xn , 1 − βn T n zn , wn αn xn 1 − αn βn I Cn {v ∈ C : wn − v ≤ xn − v }, 3.49 n Dn Cj , j1 xn PDn x, ∀n ≥ 1 , 1 where {θn } ⊂ 0, 1 , {αn } ⊂ 0, a with 0 < a < 1, and {βn } ⊂ κ, κ with κ < κ < 1. Then {xn } converges strongly to PF T x. Proof. Put Φi x, y 0 for all x, y ∈ C and set rn 1 for all n ≥ 1 in Theorem 3.3. By Lemma 2.2, we have Tri xn PC xn for each i ≥ 1. Hence, by Theorem 3.3, we obtain Corollary 3.5. Remark 3.6. Our algorithms are of interest because the sequence {xn } in Theorem 3.3 is very diﬀerent from the known manner. The proof is simple and diﬀerent from those of others. The main results extend and improve those of Kim and Xu 3 , Tada and Takahashi 8 , and many others.
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