Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 301868, 15 pages doi:10.1155/2010/301868 Research Article Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups Kriengsak Wattanawitoon 1, 2 and Poom Kumam 2, 3 1 Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand 2 Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand 3 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 15 April 2010; Accepted 11 October 2010 Academic Editor: A. T. M. Lau Copyright q 2010 K. Wattanawitoon 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 prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi 2003 and of Zegeye and Shahzad 2008 from the class of nonexpansive mappings to asymptotically nonexpansive mappings. 1. Introduction Throughout this paper, Let H be a real Hilbert space with inner product ·, · and norm ·, and we write x n → x to indicate that the sequence {x n } converges strongly to x.LetC be a nonempty closed convex subset of H,andletT : C → C be a mapping. Recall that T is nonexpansive if Tx − Ty≤x − y, for all x, y ∈ C. We denote the set of fixed points of T by FT,thatis,FT{x ∈ C : x  Tx}. A mapping T is said to be asymptotically nonexpansive if there exists a sequence {k n } with k n ≥ 1 for all n, lim n →∞ k n  1, and   T n x − T n y   ≤ k n   x − y   ∀n ≥ 1,x,y∈ C. 1.1 Mann’s iterative algorithm was introduced by Mann 1 in 1953. This iteration process is now known as Mann’s iteration process, which is defined as x n1  α n x n   1 − α n  Tx n ,n≥ 0, 1.2 2 Fixed Point Theory and Applications where the initial guess x 0 is taken in C arbitrarily and the sequence {α n } ∞ n0 is in the interval 0, 1. In 1967, Halpern 2 first introduced the following iteration scheme: x n1  α n u   1 − α n  Tx n 1.3 for all n ∈ N, where x 1  x ∈ C and {α n } is a sequence in 0, 1. This iteration process is called a Halpern-type iteration. Recall also that a one-parameter family T  {Tt :0≤ t<∞} of self-mappings of a nonempty closed convex subset C of a Hilbert space H is said to be a continuous Lipschitzian semigroup on C if the following conditions are satisfied: a T0x  x, x ∈ C; b Tt  sx  TtTsx, for all t, s ≥ 0, x ∈ C; c for each x ∈ C, the map t → Ttx is continuous on 0, ∞; d there exists a bounded measurable function L : 0, ∞ → 0, ∞ such that, for each t>0, Ttx − Tty≤L t x − y, for all x, y ∈ C. A Lipschitzian semigroup T is called nonexpansive if L t  1 for all t>0, and asymptotically nonexpansive if lim sup t →∞ L t ≤ 1. We denote by FT the set of fixed points of the semigroup T,thatis,FT{x ∈ C : Tsx  x, ∀s>0}. In 2003, Nakajo and Takahashi 3 proposed the following modification of the Mann iteration method for a nonexpansive mapping T in a Hilbert space H: x 0 ∈ C, chosen arbitrarily, y n  α n x n   1 − α n  Tx n , C n   v ∈ C :   y n − v   ≤  x n − v   , Q n  { v ∈ C :  x n − v, x n − x 0  ≥ 0 } , x n1  P C n ∩Q n  x 0  , 1.4 where P C denotes the metric projection from H onto a closed convex subset C of H. They proved that the sequence {x n } converges weakly to a fixed point of T. Moreover, they introduced and studied an iteration process of a nonexpansive semigroup T  {Tt :0≤ t<∞} in a Hilbert space H: x 0 ∈ C, chosen arbitrarily, y n  α n x n   1 − α n  1 t n  t n 0 T  u  x n du, C n   v ∈ C :   y n − v   ≤  x n − v   , Q n  { v ∈ C :  x n − v, x n − x 0  ≥ 0 } , x n1  P C n ∩Q n  x 0  . 1.5 Fixed Point Theory and Applications 3 In 2006, Kim and Xu 4 adapted iteration 1.4 to an asymptotically nonexpansive mapping in a Hilbert space H: x 0 ∈ C, chosen arbitrarily, y n  α n x n   1 − α n  T n x n , C n   v ∈ C :   y n − v   2 ≤  x n − v  2  θ n  , Q n  { v ∈ C :  x n − v, x n − x 0  ≥ 0 } , x n1  P C n ∩Q n  x 0  , 1.6 where θ n 1 − α n k 2 n − 1diam C 2 → 0asn →∞. They also proved that if α n ≤ a for all n and for some 0 <a<1, then the sequence {x n } converges weakly to a fixed point of T. Moreover, they modified an iterative method 1.5 to the case of an asymptotically nonexpansive semigroup T  {Tt :0≤ t<∞} in a Hilbert space H: x 0 ∈ C, chosen arbitrarily, y n  α n x n   1 − α n  1 t n  t n 0 T  u  x n du, C n   v ∈ C :   y n − v   2 ≤  x n − v  2  θ n  , Q n  { v ∈ C :  x n − v, x n − x 0  ≥ 0 } , x n1  P C n ∩Q n  x 0  , 1.7 where θ n 1 − α n 1/t n   t n 0 L u du 2 − 1diam C 2 → 0asn →∞. In 2007, Zegeye and Shahzad 5 developed the iteration process for a finite family of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups with C a closed convex bounded subset of a Hilbert space H: x 0 ∈ C, chosen arbitrarily, y n  α n0 x n  α n1 T n 1 x n  α n2 T n 2 x n  ··· α nr T n r x n , C n   v ∈ C :   y n − v   2 ≤  x n − v  2  θ n  , Q n  { v ∈ C :  x n − v, x n − x 0  ≥ 0 } , x n1  P C n ∩Q n  x 0  , 1.8 4 Fixed Point Theory and Applications where θ n k 2 n1 − 1α n1 k 2 n2 − 1α n2  ···k 2 nr − 1α nr diam C 2 → 0asn →∞and x 0 ∈ C, chosen arbitrarily, y n  α n0 x n  α n1  1 t n1  t n1 0 T 1  u  x n du    1 t n2  t n2 0 T 2  u  x n du   ···  1 t nr  t nr 0 T r  u  x n du  , C n   v ∈ C :   y n − v   2 ≤  x n − v  2   θ n  , Q n  { v ∈ C :  x n − v, x n − x 0  ≥ 0 } , x n1  P C n ∩Q n  x 0  , 1.9 where  θ n L 2 u1 − 1α n1 L 2 u2 − 1α n2  ···L 2 ur − 1α nr diam C 2 → 0asn →∞,with L ui 1/t ni   t ni 0 L Ti u du, for each i  1, 2, 3, ,r. Recently, Su and Qin 6 modified the hybrid iteration method of Nakajo and Takahashi through the monotone hybrid method, and to prove strong convergence theorems. In 2008, Takahashi et al. 7 proved strong convergence theorems by the new hybrid methods for a family of nonexpansive mappings and nonexpansive semigroups in Hilbert spaces: y n  α n u n   1 − α n  T n x n , C n1   v ∈ C n :   y n − v   ≤  u n − v   , x n1  P C n1  x 0  ,n∈ N, 1.10 where 0 ≤ α n ≤ a<1, and y n  α n u n   1 − α n  1 λ n  λ n 0 T  s  u n ds, C n1   v ∈ C n :   y n − v   ≤  u n − v   , x n1  P C n1  x 0  ,n∈ N, 1.11 where 0 ≤ α n ≤ a<1, 0 <λ n < ∞ and λ n →∞. In this paper, motivated and inspired by the above results, we modify iteration process 1.4–1.11 by the new hybrid methods for countable families of asymptotically nonexpansive mappings and semigroups in a Hilbert space, and to prove strong convergence theorems. Our results presented are improvement and extension of the corresponding results in 3, 5–8 and many authors. 2. Preliminaries This section collects some lemmas which will be used in the proofs for the main results in the next section. Fixed Point Theory and Applications 5 Lemma 2.1. Here holds the identity in a Hilbert space H:   λx 1 − λy   2  λ  x  2   1 − λ    y   2 − λ  1 − λ    x − y   2 2.1 for all x,y ∈ H and λ ∈ 0, 1. Using this Lemma 2.1, we can prove that the set FT of fixed points of T is closed and convex. 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 P C x, such that x−P C x≤x−y for all y ∈ C, where P C is called the metric projection of H onto C.Weknowthatforx ∈ H and z ∈ C, z  P C x is equivalent to x − z, z − u≥0 for all u ∈ C. We know that a Hilbert space H satisfies Opial’s condition, that is, for any sequence {x n }⊂H with x n x, the inequality lim inf n →∞  x n − x  < lim inf n →∞   x n − y   2.2 hold for every y ∈ H with y /  H. We also know that H has the Kadec-Klee property, that is, x n xand x n →x imply x n → x. In fact, from  x n − x  2   x  2 − 2x n ,x   x  2 2.3 we get that a Hilbert space has the Kadec-Klee property. Let C be a nonempty closed convex subset of a Hilbert space H. Motivated by Nakajo et al. 9, we give the following definitions: Let {T n } and T be families of nonexpansive mappings of C into itself such that ∅ /  FT ⊂  ∞ n1 FT n , where FT N  is the set of all fixed points of T n and FT is the set of all common fixed points of T. We consider the following conditions of {T n } and T see 9: i NST-condition I. For each bounded sequence {z n }⊂C, lim n →∞ z n − T n z n   0 implies that lim n →∞ z n − Tz n   0 for all T ∈T. ii NST-condition II. For each bounded sequence {z n }⊂C, lim n →∞ z n1 − T n z n   0 implies that lim n →∞ z n − T m z n   0 for all m ∈ N. iii NST-condition III. There exists {a n }⊂0, ∞ with  ∞ n1 a n < ∞ such that for every bounded subset B of C, there exists M B > 0 such that T n x − T n1 x≤a n M B holds for all n ∈ N and x ∈ B. Lemma 2.2. Let C be a nonempty closed convex subset of E and let T be a nonexpansive mapping of C into itself with FT /  ∅. Then, the following hold: i {T n } with T n  T∀n ∈ N and T  {T} satisfy the condition (I) with  ∞ n1 FT n  FTFT. ii {T n } with T n  T∀n ∈ N and T  {T} satisfy the condition (I) with α n  0 ∀n ∈ N. Lemma 2.3 Opial 10. Let C be a closed convex subset of a real Hilbert space H and let T : C → C be a nonexpansive mapping such that FT /  ∅.If{x n } is a sequence in C such that x n zand x n − Tx n → 0,thenz  Tz. 6 Fixed Point Theory and Applications Lemma 2.4 Lin et al. 11. Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset of a bounded closed convex subset C of a Hilbert space H.If{x n } is a sequence in C such that x n zand Tx n − x n → 0,thenz ∈ FT. Lemma 2.5 Nakajo and Takahashi 3. Let H be a real Hilbert space. Given a closed convex subset C ⊂ H and points x,y, z ∈ H. Given also a real number a ∈ R. The set D : {v ∈ C : y − v 2 ≤x − v 2  z, v  a} is convex and closed. Lemma 2.6 Kim and Xu 4. Let C be a nonempty bounded closed convex subset of H and T  {Tt :0≤ t<∞} be an asymptotically nonexpansive semigroup on C.If{x n } is a sequence in C satisfying the properties a x n z; b lim sup t →∞ lim sup n →∞ Ttx n − x n   0, then z ∈ FT. Lemma 2.7 Kim and Xu 4. Let C be a nonempty bounded closed convex subset of H and T  {Tt :0≤ t<∞} be an asymtotically nonexpansive semigroup on C. Then it holds that lim sup s →∞ lim sup t →∞ sup x∈C      1 t  t 0 T  u  xdu − T  s   1 t  t 0 T  u  xdu        0. 2.4 3. Strong Convergence for a Family of Asymptotically Nonexpansive Mappings Theorem 3.1. Let C be a nonempty bounded closed convex subset of a Hilbert space H and let T i : C → C for i  1, 2, 3, be a countable family of asymptotically nonexpansive mapping with sequence {t ni } n≥0 for i  1, 2, 3, , respectively. Assume {α n } n≥0 ⊂ 0, 1 such that α n ≤ a<1 for all n and α n → 0 as n →∞.LetFT  ∞ i1 FT i  /  ∅. Further, suppose that {T i } satisfies NST-condition (I) and (III) with T. Define a sequence {x n } in C by the following algorithm: x 0  x ∈ C, C 0  C, y n  α n x n   1 − α n  T n i x n , C n1   v ∈ C n :   y n − v   2 ≤  x n − v  2  θ n  , x n1  P C n1  x  ,n 0, 1, 2 , 3.1 where θ n 1 − α n t 2 ni − 1diam C 2 → 0 as n →∞.Then{x n } converges in norm to P FT x 0 . Proof. We first show that C n1 is closed and convex for all n ∈ N ∪{0}.FromtheLemma 2.5,it is observed that C n1 is closed and convex for each n ∈ N ∪{0}. Fixed Point Theory and Applications 7 Next, we show that FT ⊂ C n for all n ≥ 0. Indeed, let p ∈ FT, we h ave   y n − p   2    α n x n 1 − α n T n i x n − p   2    α n x n − p1 − α n T n i x n − p   2 ≤ α n   x n − p   2   1 − α n    T n i x n − p   2 ≤ α n   x n − p   2   1 − α n  t 2 ni   x n − p   2    x n − p   2   1 − α n   t 2 ni   x n − p   2 −   x n − p   2     x n − p   2   1 − α n   t 2 ni − 1    x n − p   2    x n − p   2  θ n −→ 0asn −→ ∞ . 3.2 Thus p ∈ C n1 and hence FT ⊂ C n1 for all n ≥ 0. Thus {x n } is well defined. From x n  P C n x 0 and x n1  P C n1 x 0 ∈ C n1 ⊂ C n , we have  x 0 − x n ,x n − x n1  ≥ 0 ∀x 0 ∈ F  T  ,n∈ N ∪ { 0 } . 3.3 So, for x n1 ∈ C n , we have 0 ≤  x 0 − x n ,x n − x n1  ,   x 0 − x n ,x n − x 0  x 0 − x n1  ,  −x n − x 0 ,x n − x 0    x 0 − x n ,x 0 − x n1  , ≤−  x n − x 0  2   x 0 − x n  x 0 − x n1  3.4 for all n ∈ N. This implies that  x 0 − x n  2 ≤  x 0 − x n  x 0 − x n1  3.5 hence  x 0 − x n  ≤  x 0 − x n1  3.6 for all n ∈ N ∪{0}. Therefore {x 0 − x n } is nondecreasing. From x n  P C n x 0 , we have  x 0 − x n ,x n − y  ≥ 0 ∀y ∈ C n . 3.7 Using FT ⊂ C n , we also have  x 0 − x n ,x n − p  ≥ 0 ∀p ∈ F  T  ,n∈ N ∪ { 0 } . 3.8 8 Fixed Point Theory and Applications So, for p ∈ FT, we have 0 ≥x 0 − x n ,x n − p,  x 0 − x n ,x n − x 0  x 0 − p,  −  x 0 − x n  2   x 0 − x n    x 0 − p   . 3.9 This implies that  x 0 − x n  ≤   x 0 − p   ∀p ∈ F  T  ,n∈ N ∪ { 0 } . 3.10 Thus, {x 0 − x n } is bounded. So, lim n →∞ x n − x 0  exists. Next, we show that x n1 − x n →0. From 3.3, we have  x n − x n1  2   x n − x 0  x 0 − x n1  2   x n − x 0  2  2  x n − x 0 ,x 0 − x n1    x 0 − x n1  2   x n − x 0  2  2  x n − x 0 ,x 0 − x n  x n − x n1    x 0 − x n1  2   x n − x 0  2 − 2  x 0 − x n ,x 0 − x n  − 2  x 0 − x n ,x n − x n1    x 0 − x n1  2 ≤  x n − x 0  2 − 2  x n − x 0  2   x 0 − x n1  2  −  x n − x 0  2   x 0 − x n1  2 . 3.11 Since lim n →∞ x n − x 0  exists, we conclude that lim n →∞ x n − x n1   0. Since x n1 ∈ C n1 ⊂ C n , we have y n − x n1  2 ≤x n − x n1  2  θ n which implies that y n − x n1 ≤x n − x n1    θ n . Now we claim that T i x n − x n →0asn →∞for all i ∈ N. We first show that T n i x n − x n →0asn →∞. Indeed, by the definition of y n , we have   y n − x n      α n x n   1 − α n  T n i x n − x n   ,     1 − α n  T n i x n   1 − α n  x n   ,     1 − α n   T n i x n − x n    ,   1 − α n    T n i x n − x n   3.12 for all i ∈ N and it follows that   T n i x n − x n    1 1 − α n   y n − x n   , ≤ 1 1 − α n    y n − x n1     x n1 − x n   , ≤ 1 1 − α n   x n − x n1    θ n   x n1 − x n   . 3.13 Fixed Point Theory and Applications 9 Since x n − x n1 →0asn →∞, we obtain lim n →∞   T n i x n − x n    0 3.14 for all i ∈ N. Let t ∞  sup{t n : n ≥ 1} < ∞. Now, for i  1, 2, 3, ,we get  T i x n − x n  ≤    T i x n − T n1 i x n        T n1 i x n − T n1 i x n1        T n1 i x n1 − x n1      x n1 − x n  , ≤ t ∞   x n − T n i x n       T n1 i x n1 − x n1      1  t ∞   x n − x n1  , 3.15 from 3.14 and x n − x n1 →0asn →∞, yields lim n →∞  x n − T i x n   0 3.16 for each i  1, 2, 3, Let m ∈ N and take n ∈ N with i>n. By NST-condition III, there exists M B > 0 such that  T n x n − x n  ≤  T n x n − T i x n    T i x n − x n  ≤  T n x n − T n1 x n    T n1 x n − T n2 x n   ···  T i−1 x n − T i x n    T i x n − x n  ≤ M B i−1  kn a k   T i x n − x n  . 3.17 By 3.16 and  i−1 kn a k < ∞,weget lim sup n →∞  x n − T n x n   0. 3.18 By the assumption of {T n } and NST-condition I, we have  Tx n − x n  −→ 0asn −→ ∞ . 3.19 Put z 0  P FT x 0 . Since x n − x 0 ≤z 0 − x 0  for all n ∈ N ∪{0}, {x n } is bounded. Let {x n i } be a subsequence of {x n } such that x n i w. Since C is closed and convex, C is weakly closed and 10 Fixed Point Theory and Applications hence w ∈ C.From3.19, we have that w  Tw. If not, since H satisfies Opial’s condition, we have lim inf n →∞  x n i − w  ≤ lim inf n →∞  x n i − Tw  , ≤ lim inf n →∞   x n i − Tx n i    Tx n i − Tw   , ≤ lim inf n →∞   x n i − Tx n i    x n i − w   ,  lim inf n →∞  x n i − w  . 3.20 This is a contradiction. So, we have that w  Tw. Then, we have  x 0 − z 0  ≤  x 0 − w  ≤ lim inf i →∞  x 0 − x n i  ≤ lim sup i →∞  x 0 − x n i  ≤  z 0 − x 0  , 3.21 and hence x 0 − z 0   x 0 − w.Fromz 0  P F x 0 , we have z 0  w. This implies that {x n } converges weakly to z 0 , and we have  x 0 − z 0  ≤ lim inf n →∞  x 0 − x n  ≤ lim sup n →∞  x 0 − x n  ≤  z 0 − x 0  , 3.22 and hence lim n →∞ x 0 − x n   z 0 − x 0 .Fromx n z 0 , we also have x 0 − x n x 0 − z 0 . Since H satisfies the Kadec-Klee property, it follows that x 0 − x n → x 0 − z 0 . So, we have  x n − z 0    x n − x 0 −  z 0 − x 0   −→ 0 3.23 and hence x n → z 0  P F x 0 . This completes the proof. Corollary 3.2. Let C be a nonempty bounded closed convex subset of a Hilbert space H and let T : C → C be an asymptotically nonexpansive mapping with sequence {t n } n≥0 . Assume {α n } n≥0 ⊂ 0, 1 such that α n ≤ a<1 for all n and α n → 0 as n →∞.LetFT /  ∅. Define a sequence {x n } in C by the following algorithm: x 0  x ∈ C, C 0  C, y n  α n x n   1 − α n  T n x n , C n1   v ∈ C n :   y n − v   2 ≤  x n − v  2  θ n  , x n1  P C n1  x  ,n 0, 1, 2 , 3.24 where θ n 1 − α n t 2 n − 1diam C 2 → 0 as n →∞.Then{x n } converges in norm to P FT x 0 . Proof. Setting T n i ≡ T n for all i ∈ N ∪{0} from Lemma 2.2i and Theorem 3.1, we immediately obtain the corollary. [...]... Takeuchi, and R Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 341, no 1, pp 276–286, 2008 8 Y Su and X Qin, Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups,” Fixed Point Theory and Applications, vol 2006, Article ID... Shimoji, and W Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol 8, no 1, pp 11–34, 2007 10 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 11 P.-K Lin, K.-K Tan, and H K Xu,... Mathematical Society, vol 73, pp 957–961, 1967 3 K Nakajo and W Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol 279, no 2, pp 372– 379, 2003 4 T.-H Kim and H.-K Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods... vol 64, no 5, pp 1140–1152, 2006 5 H Zegeye and N Shahzad, Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications, vol 69, no 12, pp 4496–4503, 2008 6 Y Su and X Qin, Strong convergence of monotone hybrid method for fixed point iteration processes,” Journal of Systems Science & Complexity, vol 21, no.. .Fixed Point Theory and Applications 11 Since every family’s nonexpansive mapping is family’s asymptotically nonexpansive mapping we obtain the following result Corollary 3.3 Let C be a nonempty bounded closed convex subset of a Hilbert space H and let {Ti } : C → C be a family of nonexpansive mappings with sequence {ti }i≥0 Assume {αn }n≥0 ⊂ 0, 1 ∞ such that αn ≤ a < 1 for all n and αn →... C 2 → 0 as Proof By Theorem 4.1, if the semigroup T {T t : 0 ≤ t < ∞} I : {I t : 0 ≤ t < ∞}, then t T t xn xn for all n and for all t > 0 Hence 1/tn 0n T u xn du xn for all n and zn xn then, 4.1 reduces to 4.11 Fixed Point Theory and Applications 15 Corollary 4.3 Takahashi et al 7, Theorem 4.4 Let C be a nonempty closed convex subset of a Hilbert space H and T {T t : 0 ≤ t < ∞} be a nonexpansive semigroup... discussions and the referees for helpful suggestions that improved the contents of the paper This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand References 1 W R Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol 4, pp 506–510, 1953 2 B Halpern, Fixed points of nonexpanding maps,” Bulletin of the... ≤ a < 1 for all n Then the sequence {xn } generated by x0 x ∈ C, yn αn xn C0 C, 1 − αn T xn , 3.26 Cn xn converges in norm to PF T v ∈ Cn : yn − v ≤ xn − v 1 1 PCn 1 x , n , 0, 1, 2 , x0 4 Strong Convergence for a Family of Asymptotically Nonexpansive Semigroups Theorem 4.1 Let C be a nonempty bounded closed convex subset of a Hilbert space H and let {Ti t : t ∈ R , i 1, 2, 3, } be a countable. .. Furthermore, from 4.9 and Lemma 2.6 and the boundedness of {xn } we obtain that PF x0 ∅ / ωw xn ⊂ F By the fact that xn − x0 ≤ p − x0 for any n ≥ 0, where p and the weak lower semi-continuity of the norm, we have ω − x0 ≤ p − x0 for all p for all w ∈ ωw xn w ∈ ωw xn However, since ωw xn ⊂ F, we must have w {p} and then xn converges weakly to p Moreover, following the method Thus ωw xn of Theorem 3.1, xn... convex subset of a Hilbert space H and let {Ti t : t ∈ R , i 1, 2, 3, } be a countable family of asymptotically nonexpansive Ti semigroups Assume {αn }n≥0 ⊂ 0, 1 such that αn ≤ a < 1 for all n and αn → 0 as n → ∞ Let 12 Fixed Point Theory and Applications ∞ {tni }, i 1, 2, 3, be a countable positive and divergent real sequence Let F i 1 F Ti / ∅ Further, suppose that {Ti } satisfies NST-condition . Corporation Fixed Point Theory and Applications Volume 2010, Article ID 301868, 15 pages doi:10.1155/2010/301868 Research Article Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive. 2008. 8 Y. Su and X. Qin, Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups,” Fixed Point Theory and Applications, vol. 2006, Article. results extend and improve the recent results of Nakajo and Takahashi 2003 and of Zegeye and Shahzad 2008 from the class of nonexpansive mappings to asymptotically nonexpansive mappings. 1.