On strong convergence by the hybrid method for families of mappings in Hilbert spaces

Nonlinear Analysis 71 (2009) 112–119

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

On strong convergence by the hybrid method for families of mappings in Hilbert spaces Kazuhide Nakajo a,∗ , Kazuya Shimoji b , Wataru Takahashi c a

Faculty of Engineering, Tamagawa University, Tamagawa-Gakuen, Machida-shi, Tokyo, 194-8610, Japan

b

Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara-cho, Okinawa, 903-0213, Japan

c

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152-8552, Japan

article

info

Article history: Received 25 March 2008 Accepted 10 October 2008 MSC: 49M05 47H09 47H20 Keywords: Strong convergence Hybrid method W -mapping Asymptotically nonexpansive Asymptotically nonexpansive semigroup

a b s t r a c t Let C be a nonempty closed convex subset of a real Hilbert space and let {Tn } be a family of mappings of C into itself such that the set of all common fixed points of {Tn } is nonempty. We consider a sequence {xn } generated by the hybrid method in mathematical programming. We give the new conditions of {Tn } under which {xn } converges strongly to a common fixed point of {Tn } and generalize the unified result for families of nonexpansive mappings [K. Nakajo, K. Shimoji, W. Takahashi, Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces, Taiwanese J. Math. 10 (2006) 339–360] and the results for asymptotically nonexpansive mappings and semigroups [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152; S. Plubtieng and K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 67 (2007) 2306–2315]. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Throughout this paper, let H be a real Hilbert space with inner product (· , ·) and norm k · k and let N and R be the set of all positive integers and the set of all real numbers, respectively. Haugazeau [5] introduced a sequence {xn } generated by the hybrid method, that is, let {Tn } be a family of mappings of H into itself with ∩∞ n=0 F (Tn ) 6= ∅, where F (Tn ) is the set of all fixed points of Tn and let {xn } be a sequence generated by

 x0 = x ∈ H ,   yn = Tn xn ,  Cn = {z ∈ H : (xn − yn , yn − z ) ≥ 0},    Qn = {z ∈ H : (xn − z , x0 − xn ) ≥ 0}, xn+1 = PCn ∩Qn (x0 )

(1)

for each n ∈ N ∪ {0}, where PCn ∩Qn is the metric projection onto Cn ∩ Qn . He proved a strong convergence theorem when Tn = Pn(mod m)+1 for every n ∈ N ∪{0}, where Pi is the metric projection onto a nonempty closed convex subset Ci of H for each i = 1, 2, . . . , m and ∩m i=1 Ci 6= ∅. Later, many authors [2,3,7–9,12–14,20] studied strong convergence by the hybrid method. Recently, Nakajo, Shimoji and Takahashi [15] proved the following unified result: Let C be a nonempty closed convex subset

∗

Corresponding author. E-mail addresses: [email protected] (K. Nakajo), [email protected] (K. Shimoji), [email protected] (W. Takahashi).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.10.034

K. Nakajo et al. / Nonlinear Analysis 71 (2009) 112–119

113

of H and let {Tn } be a family of mappings of C into itself with F := ∩∞ n=0 F (Tn ) 6= ∅ which satisfies the following condition: (I-i) There exists {an } ⊂ (−1, ∞) such that kTn x − z k2 ≤ kx − z k2 − an k(I − Tn )xk2 for every n ∈ N ∪ {0}, x ∈ C and z ∈ F (Tn ). Let {xn } be a sequence generated by

 x0 = x ∈ C ,    yn = Tn PC (xn + εn ), (2) Cn = {z ∈ C : kyn − z k2 ≤ kxn + εn − z k2 − an kPC (xn + εn ) − yn k2 },   Q = { z ∈ C : ( x − z , x − x ) ≥ 0 },  n n 0 n  xn+1 = PCn ∩Qn (x0 ) P∞ for each n ∈ N ∪ {0}, where {εn } ⊂ H and lim infn→∞ an > −1. If (II-i) n=0 kεn k2 < ∞ and for every bounded sequence P∞ P 2 {zn } in C , n=0 kzn+1 − zn k2 < ∞ and ∞ n=0 kzn − Tn zn k < ∞ imply ωw (zn ) ⊂ F , where ωw (zn ) is the set of all weak cluster points of {zn } or if (III-i) limn→∞ kεn k = 0 and for every bounded sequence {zn } in C , limn→∞ kzn − Tn zn k = 0 implies ωw (zn ) ⊂ F , then, {xn } converges strongly to z0 = PF (x0 ). This result generalizes the results of [2,3,5,7–9,12–14,20]. On the other hand, Kim and Xu [10] and Plubtieng and Ungchittrakool [17] proved the strong convergence for asymptotically nonexpansive mappings and semigroups. Motivated by [10,15,17], in this paper, we consider the unification of these results. And we get a strong convergence theorem and new results for W -mapping generated by asymptotically nonexpansive mappings, convex combination of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. 2. Preliminaries and a lemma We denote max{a, 0} by (a)+ for a real number a. We write xn * x to indicate that a sequence {xn } converges weakly to x. Similarly, xn → x will symbolize strong convergence. We know that H satisfies Opial’s condition [16], that is, for any sequence {xn } ⊂ H with xn * x, the inequality lim infn→∞ kxn − xk < lim infn→∞ kxn − yk holds for every y ∈ H with y 6= x. It is known that kλx + (1 − λ)yk2 = λkxk2 + (1 − λ)kyk2 − λ(1 − λ)kx − yk2 for each x, y ∈ H and λ ∈ R. We also know that the norm is weakly lower semicontinuous, that is, for any sequence {xn } ⊂ H with xn * x, kxk ≤ lim infn→∞ kxn k holds. Further, it is known that for any sequence {xn } ⊂ H with xn * x and kxn k → kxk, xn → x holds. Let C be a nonempty closed convex subset of H and let PC be the metric projection of H onto C . kPC x − PC yk2 ≤ kx − yk2 − k(I − PC )x − (I − PC )yk2 holds for every x, y ∈ H, where I is the identity mapping. Let C be a nonempty closed convex subset of H and let T be a mapping of C into itself. T is called nonexpansive if kTx − Tyk ≤ kx − yk for every x, y ∈ C . T is said to be asymptotically nonexpansive [4] if there exist Lipschitz constants {kn } ⊂ (0, ∞) with lim supn→∞ kn ≤ 1 such that kT n x − T n yk ≤ kn kx − yk for each n ∈ N and x, y ∈ C . We can always assume that Lipschitz constants {kn } satisfy kn ≥ 1 for all n ∈ N and limn→∞ kn = 1 for asymptotically nonexpansive mappings. Let C be a nonempty closed convex subset of H. Let {Sn } be a family of asymptotically nonexpansive mappings of C into itself and let {βn,k : n, k ∈ N, 1 ≤ k ≤ n} be a sequence of real numbers such that 0 ≤ βi,j ≤ 1 for every i, j ∈ N with i ≥ j. Then, motivated by the W -mapping which Takahashi [18,22,24] introduced, for any n ∈ N, we consider the following mapping of C into itself: Un,n = βn,n Snn + (1 − βn,n )I , Un,n−1 = βn,n−1 Snn−1 Un,n + (1 − βn,n−1 )I ,

.. . Un,k = βn,k Skn Un,k+1 + (1 − βn,k )I ,

.. . Un,2 = βn,2 S2n Un,3 + (1 − βn,2 )I , Wn = Un,1 = βn,1 S1n Un,2 + (1 − βn,1 )I . Such a mapping Wn is called the modified W -mapping generated by Sn , . . . , S2 , S1 and βn,n , . . . , βn,2 , βn,1 . Let S be a semigroup and let B(S ) be the Banach space of all bounded real valued functions on S with supremum norm. Then, for every s ∈ S and f ∈ B(S ), we can define ls f ∈ B(S ) by (ls f )(t ) = f (st ) for each t ∈ S. We also denote by l∗s the conjugate operators of ls . Let X be a subspace of B(S ) containing constants and let X ∗ be its dual. Let µ be an element of X ∗ . µ is called a mean on X if kµk = µ(1) = 1. Let C be a nonempty closed convex subset of H. A family S = {T (s) : s ∈ S } of mappings of C into itself is called a uniformly Lipschitzian semigroup on C with Lipschitz constants {kt : t ∈ S } if it satisfies the following conditions: (i) kt is a nonnegative real number for each t ∈ S and supt ∈S kt < ∞; (ii) kT (s)x − T (s)yk ≤ ks kx − yk for every s ∈ S and x, y ∈ C ; (iii) T (st ) = T (s)T (t ) for all s, t ∈ S.

114

K. Nakajo et al. / Nonlinear Analysis 71 (2009) 112–119

We denote by F (S ) the set of all common fixed points of S , that is, F (S ) = ∩s∈S F (T (s)). A uniformly Lipschitzian semigroup on C with Lipschitz constants {kt : t ∈ S } is said to be asymptotically nonexpansive if infs∈S supt ∈S kst ≤ 1 and it is said to be nonexpansive if kt = 1 for all t ∈ S. We have the following result from [21]; see also [6]. Lemma 2.1. Let C be a nonempty closed convex subset of H and let S be a semigroup. Let S = {T (s) : s ∈ S } be a uniformly Lipschitzian semigroup on C such that {T (s)u : s ∈ S } is bounded for some u ∈ C and let X be a subspace of B(S ) such that X contains constants and the mapping t 7→ kT (t )x − yk2 is an element of X for every x ∈ C and y ∈ H. For each mean µ on X and x ∈ C , there exists a unique element Tµ x of C such that µt (T (t )x, y) = (Tµ x, y) for every y ∈ H. Further, Tµ u = u for all u ∈ F (S ). 3. Results for families of mappings Let C be a nonempty closed convex subset of H and let {Tn } be a family of mappings of C into itself such that F := ∩∞ n=0 F (Tn ) is nonempty, closed and convex. Assume that the following condition (I-ii): There exist {an } ⊂ (−1, ∞) and {bn } ⊂ [0, ∞) such that

kTn x − z k2 ≤ kx − z k2 − an k(I − Tn )xk2 + bn for every n ∈ N ∪ {0}, x ∈ C and z ∈ F . We consider a sequence {xn } as follows:

 x0 = x ∈ C ,    yn = Tn PC (xn + εn ), Cn = {z ∈ C : kyn − z k2 ≤ kxn + εn − z k2 − an kPC (xn + εn ) − yn k2 + bn },    Qn = {z ∈ C : (xn − z , x0 − xn ) ≥ 0}, xn+1 = PCn ∩Qn (x0 ) for each n ∈ N ∪ {0}, where {εn } ⊂ H and lim infn→∞ an > −1. Now, we get the following result by using the method of [15]. Theorem 3.1. The followings hold: (i) If (II–ii) n=0 (kεn k2 + bn ) < ∞ and for every bounded sequence {zn } in C , n=0 kzn+1 − zn k2 < ∞ and n=0 kzn − Tn zn k2 < ∞ imply ωw (zn ) ⊂ F , then, {xn } converges strongly to z0 = PF (x0 ); (ii) if (II–iii) limn→∞ (kεn k+ bn ) = 0 and for every bounded sequence {zn } in C , limn→∞ kzn+1 − zn k = limn→∞ kzn − Tn zn k = 0 imply ωw (zn ) ⊂ F , then, {xn } converges strongly to z0 = PF (x0 ); (iii) if (III–ii) limn→∞ (kεn k + bn ) = 0 and for every bounded sequence {zn } in C , limn→∞ kzn − Tn zn k = 0 implies ωw (zn ) ⊂ F , then, {xn } converges strongly to z0 = PF (x0 ).

P∞

P∞

P∞

Proof. As in the proof of [[14], Theorem 4.2], we have that {xn } ⊂ C is P well defined, F ⊂ Cn ∩ Qn for every n ∈ N ∪ {0}, 2 kxn+1 − x0 k ≤ kz0 − x0 k for all n ∈ N ∪ {0}, limn→∞ kxn − x0 k exists and ∞ n=0 kxn+1 − xn k < ∞, where z0 = PF (x0 ). From a lim infn→∞ an > −1, there exists a ∈ (0, 1) such that an ≥ −a for all n ∈ N ∪ {0}. Let α ∈ (0, 1− ) and β = 1−a(a1+α) (> 0). a Further, we also have

2 2 + α + α1 1+α (kxn − xn+1 k2 + kεn k2 ) + bn kPC (xn + εn ) − yn k ≤ aβ aβ




2

(3)

for every n ∈ N ∪ {P 0}. ∞ (i) Assume that n=0 (kεn k2 + bn ) < ∞. If zn = PC (xn + εn ), we have that {zn } is bounded and ∞ X

kzn − Tn zn k2 =

n =0

∞ X

kPC (xn + εn ) − yn k2 < ∞

n =0

from obtain kzn − zn+1 k2 ≤ 3kxn − xn+1 k2 + 3kεn k2 + 3kεn+1 k2 for all n ∈ N ∪ {0} which implies P∞ (3). Further we 2 k z − z k < ∞. Therefore, we have ωw (zn ) ⊂ F which implies ωw (xn ) ⊂ F by limn→∞ kxn − zn k = 0. So, n n + 1 n =0 assume that a subsequence {xni } of {xn } converges weakly to w1 ∈ F . We have

kx0 − z0 k ≤ kx0 − w1 k ≤ lim kx0 − xni k ≤ kx0 − z0 k i→∞

by the weakly lower semicontinuity of the norm. Thus, we obtain limi→∞ kxni − x0 k = kx0 − w1 k = kx0 − z0 k. This implies xni → w1 = z0 . Therefore, we have xn → z0 . So, the proof of (i) is complete. Similarly, (ii) holds. (iii) Assume limn→∞ (kεn k + bn ) = 0. If zn = PC (xn + εn ), we get that {zn } is bounded, limn→∞ kxn − zn k = 0 and limn→∞ kzn − Tn zn k = 0 by (3). Therefore, we obtain ωw (xn ) ⊂ F . As in the proof of (i), we have xn → z0 . So, the proof of (iii) is complete. 

K. Nakajo et al. / Nonlinear Analysis 71 (2009) 112–119

115

We get the following result proved by Nakajo, Shimoji and Takahashi [15] by Theorem 3.1. Theorem 3.2. Let C be a nonempty closed convex subset of H and let {Tn } be a family of mappings of C into itself with 2 F := ∩∞ n=0 F (Tn ) 6= ∅ which satisfies the following condition: (I–i) There exists {an } ⊂ (−1, ∞) such that kTn x − z k ≤ kx − z k2 − an k(I − Tn )xk2 for every n ∈ N ∪ {0}, x ∈ C and z ∈ F (Tn ). Let {xn } be a sequence generated by

 x0 = x ∈ C ,   yn = Tn PC (xn + εn ),  Cn = {z ∈ C : kyn − z k2 ≤ kxn + εn − z k2 − an kPC (xn + εn ) − yn k2 },    Qn = {z ∈ C : (xn − z , x0 − xn ) ≥ 0}, xn+1 = PCn ∩Qn (x0 ) P∞ for each n ∈ N ∪ {0}, where {εn } ⊂ H and lim infn→∞ an > −1. If (II–i) n=0 kεn k2 < ∞ and for every bounded sequence P∞ P 2 {zn } in C , n=0 kzn+1 − zn k2 < ∞ and ∞ n=0 kzn − Tn zn k < ∞ imply ωw (zn ) ⊂ F or if (III–i) limn→∞ kεn k = 0 and for every bounded sequence {zn } in C , limn→∞ kzn − Tn zn k = 0 implies ωw (zn ) ⊂ F , then, {xn } converges strongly to z0 = PF (x0 ). Proof. We know that F is closed and convex, see [15]. Since bn = 0 (∀n ∈ N) in (I-ii), Theorem 3.2 holds by Theorem 3.1.  4. Strong convergence theorems for asymptotically nonexpansive mappings First, we prove the following lemma for the modified W -mappings. Lemma 4.1. Let C be a nonempty closed convex subset of H. Let {Sm } be a family of asymptotically nonexpansive mappings of C n n into itself with Lipschitz constants {tm,n }, i.e. kSm x − Sm yk ≤ tm,n kx − yk (∀m, n ∈ N, ∀x, y ∈ C ) such that F := ∩∞ i=1 F (Si ) 6= ∅ and let {βn,k : n, k ∈ N, 1 ≤ k ≤ n} be a sequence of real numbers with 0 < a ≤ βn,1 ≤ 1 for all n ∈ N and 0 < b ≤ βn,i ≤ c < 1 for every n ∈ N and i = 2, . . . , n for some a, b, c ∈ (0, 1). Let Wn be the modified W mapping generated by Sn , Sn−1 , . . . , S1 and βn,n , βn,n−1 , . . . , βn,1 . Let γn,k = {βn,k (tk2,n − 1) + βn,k βn,k+1 tk2,n (tk2+1,n − 1) +

· · · + βn,k βn,k+1 · · · βn,n−1 tk2,n tk2+1,n · · · tn2−2,n (tn2−1,n − 1) + βn,k βn,k+1 · · · βn,n tk2,n tk2+1,n · · · tn2−1,n (tn2,n − 1)} for every n ∈ N and k = 1, 2, . . . , n. Then, the followings hold: (i) kWn x − z k2 ≤ (1 + γn,1 )kx − z k2 for all n ∈ N, x ∈ C and z ∈ ∩ni=1 F (Si ); (ii) if C is bounded and limn→∞ γn,1 = 0, for every sequence {zn } in C , lim kzn+1 − zn k = 0 and

n→∞

lim kzn − Wn zn k = 0 imply ωw (zn ) ⊂ F ;

n→∞

(iii) if limn→∞ γn,1 = 0, F = ∩∞ n=1 F (Wn ) and F is closed convex. Proof. (i) Let n ∈ N, x ∈ C and z ∈ ∩ni=1 F (Si ). We have

kUn,n x − z k2 ≤ βn,n kSnn x − z k2 + (1 − βn,n )kx − z k2 ≤ βn,n tn2,n kx − z k2 + (1 − βn,n )kx − z k2 ≤ kx − z k2 + βn,n (tn2,n − 1)kx − z k2 which implies

kUn,n−1 x − z k2 ≤ βn,n−1 tn2−1,n kUn,n x − z k2 + (1 − βn,n−1 )kx − z k2 ≤ βn,n−1 tn2−1,n {kx − z k2 + βn,n (tn2,n − 1)kx − z k2 } + (1 − βn,n−1 )kx − z k2 = kx − z k2 + {βn,n−1 (tn2−1,n − 1) + βn,n−1 βn,n tn2−1,n (tn2,n − 1)}kx − z k2 . By repeating this, we get

kUn,k x − z k2 ≤ kx − z k2 + {βn,k (tk2,n − 1) + βn,k βn,k+1 tk2,n (tk2+1,n − 1) + · · · + βn,k βn,k+1 · · · βn,n−1 tk2,n tk2+1,n · · · tn2−2,n (tn2−1,n − 1) + βn,k βn,k+1 · · · βn,n tk2,n tk2+1,n · · · tn2−1,n (tn2,n − 1)}kx − z k2 = (1 + γn,k )kx − z k2 for every k = 1, 2, . . . , n. So, we obtain

kWn x − z k2 ≤ (1 + γn,1 )kx − z k2 .

116

K. Nakajo et al. / Nonlinear Analysis 71 (2009) 112–119

(ii) Let C be bounded and let {zn } ⊂ C be a sequence such that limn→∞ kzn+1 − zn k = limn→∞ kzn − Wn zn k = 0. We have limn→∞ kzn − S1n Un,2 zn k = 0 by 0 < a ≤ βn,1 . Let z ∈ F . We get

kzn − z k2 ≤ (kzn − S1n Un,2 zn k + kS1n Un,2 zn − z k)2 = kzn − S1n Un,2 zn k(kzn − S1n Un,2 zn k + 2kS1n Un,2 zn − z k) + kS1n Un,2 zn − z k2 ≤ M kzn − S1n Un,2 zn k + t12,n kUn,2 zn − z k2 = M kzn − S1n Un,2 zn k + t12,n {βn,2 kS2n Un,3 zn − z k2 + (1 − βn,2 )kzn − z k2 − βn,2 (1 − βn,2 )kS2n Un,3 zn − zn k2 } ≤ M kzn − S1n Un,2 zn k + t12,n {βn,2 t22,n kUn,3 zn − z k2 + (1 − βn,2 )kzn − z k2 − βn,2 (1 − βn,2 )kS2n Un,3 zn − zn k2 } ≤ M kzn − S1n Un,2 zn k + t12,n {βn,2 t22,n (1 + γn,3 )kzn − z k2 + (1 − βn,2 )kzn − z k2 − βn,2 (1 − βn,2 )kS2n Un,3 zn − zn k2 } which implies M kzn − S1n Un,2 zn k ≥ {1 − t12,n βn,2 t22,n (1 + γn,3 ) − t12,n (1 − βn,2 )}kzn − z k2 + βn,2 (1 − βn,2 )kS2n Un,3 zn − zn k2 for each n ∈ N, where M = supn∈N {kzn − S1n Un,2 zn k + 2kS1n Un,2 zn − z k}. Since limn→∞ kzn − S1n Un,2 zn k = limn→∞ γn,3 = 0, limn→∞ t1,n = limn→∞ t2,n = 1 and b ≤ βn,2 ≤ c, we obtain limn→∞ kzn − S2n Un,3 zn k = 0. By induction, we have n lim kSm Un,m+1 zn − zn k = 0

n→∞

for all m ∈ N. Since

kzn − Smn zn k ≤ ≤ = ≤

kzn − Smn Un,m+1 zn k + kSmn Un,m+1 zn − Smn zn k kzn − Smn Un,m+1 zn k + tm,n kUn,m+1 zn − zn k kzn − Smn Un,m+1 zn k + tm,n βn,m+1 kSmn +1 Un,m+2 zn − zn k kzn − Smn Un,m+1 zn k + ctm,n kSmn +1 Un,m+2 zn − zn k

n for every n ∈ N, we get limn→∞ kzn − Sm zn k = 0 for all m ∈ N. So, we obtain limn→∞ kzn − Sm zn k = 0 for all m ∈ N because

kzn − Sm zn k ≤ kSm zn − Smn+1 zn k + kSmn+1 zn − Smn+1 zn+1 k + kSmn+1 zn+1 − zn+1 k + kzn+1 − zn k ≤ tm,1 kzn − Smn zn k + (tm,n+1 + 1)kzn+1 − zn k + kSmn+1 zn+1 − zn+1 k. By [[11], Theorem 3.1], we get ωw (zn ) ⊂ F . ∞ (iii) F ⊂ ∩∞ n=1 F (Wn ) is trivial. Let u ∈ ∩n=1 F (Wn ) and z ∈ F . As in the proof of (ii), we have ku − Sm uk = 0, i.e. u = Sm u for every m ∈ N. So, we get F = ∩∞  F ( W n ). And we know that F is closed and convex from [[4], Theorem 2]. n =1 We have the following result for the modified W -mappings from Lemma 4.1 and Theorem 3.1(ii). Theorem 4.2. Let C be a nonempty bounded closed convex subset of H. Let {Sm } be a family of asymptotically nonexpansive n n mappings of C into itself with Lipschitz constants {tm,n }, i.e. kSm x − Sm yk ≤ tm,n kx − yk (∀m, n ∈ N, ∀x, y ∈ C ) such that ∞ F := ∩i=1 F (Si ) 6= ∅ and let {βn,k : n, k ∈ N, 1 ≤ k ≤ n} be a sequence of real numbers with 0 < a ≤ βn,1 ≤ 1 for all n ∈ N and 0 < b ≤ βn,i ≤ c < 1 for every n ∈ N and i = 2, . . . , n for some a, b, c ∈ (0, 1). Let Wn be the modified W -mapping generated by Sn , Sn−1 , . . . , S1 and βn,n , βn,n−1 , . . . , βn,1 . Let γn,k = {βn,k (tk2,n − 1) + βn,k βn,k+1 tk2,n (tk2+1,n − 1) +

· · · + βn,k βn,k+1 · · · βn,n−1 tk2,n tk2+1,n · · · tn2−2,n (tn2−1,n − 1) + βn,k βn,k+1 · · · βn,n tk2,n tk2+1,n · · · tn2−1,n (tn2,n − 1)} for every n ∈ N and k = 1, 2, . . . , n. Let {xn } be a sequence generated by  x1 = x ∈ C ,    yn = Wn PC (xn + εn ), Cn = {z ∈ C : kyn − z k2 ≤ kxn + εn − z k2 + γn,1 · {diam(C)}2 },    Qn = {z ∈ C : (xn − z , x1 − xn ) ≥ 0}, xn+1 = PCn ∩Qn (x1 )

for each n ∈ N, where {εn } ⊂ H with limn→∞ kεn k = 0 and diam(C) is the diameter of C . If limn→∞ γn,1 = 0, {xn } converges strongly to z0 = PF (x1 ). By Theorem 4.2, we get the following proved by Plubtieng and Ungchittrakool [17]. Theorem 4.3. Let C be a bounded closed convex subset of H and let S and T be two asymptotically nonexpansive mappings of C into itself with Lipschitz constants {sn } and {tn }, respectively such that F := F (S )∩ F (T ) 6= ∅. Let {xn } be a sequence generated by

K. Nakajo et al. / Nonlinear Analysis 71 (2009) 112–119

117

 x1 = x ∈ C ,     y = αn xn + (1 − αn )T n zn ,   n zn = βn xn + (1 − βn )S n xn , Cn = {z ∈ C : kyn − z k2 ≤ kxn − z k2 + δn },     Qn = {z ∈ C : (xn − z , x1 − xn ) ≥ 0}, xn+1 = PCn ∩Qn (x1 ) for each n ∈ N, where 0 ≤ αn ≤ a < 1 and 0 < b ≤ βn ≤ c < 1 for all n ∈ N for some a, b, c and δn = (1 − αn ){(tn2 − 1) + (1 − βn )tn2 (s2n − 1)} · {diam(C)}2 (∀n ∈ N). Then, {xn } converges strongly to z0 = PF (x1 ). Proof. Since Sm = I for all m = 3, 4, . . . in Theorem 4.2, Theorem 4.3 holds.



Aoyama, Pn Kimura, Takahashi and Toyoda [1] considered the following convex combination of nonexpansive mappings: Tn = N) for a family of nonexpansive mappings {Sn }, where 0 ≤ βn,k ≤ 1 for every n = 1, 2, 3, . . . k=1 βn,k Sk (∀n ∈ P n and k = 1, 2, . . . , n with k=1 βn,k = 1 for each n ∈ N and limn→∞ βn,k > 0 for every k ∈ N. Motivated by this convex combination, we have the following result for the convex combination of asymptotically nonexpansive mappings. Lemma 4.4. Let C be a nonempty closed convex subset of H. Let {Sm } be a family of asymptotically nonexpansive mappings of C n n into itself with Lipschitz constants {tm,n }, i.e. kSm x − Sm yk ≤ tm,n kx − yk (∀m, n ∈ N, ∀x, y ∈ C ) such that F := ∩∞ Pn Pi=n1 F (Si ) 6= ∅. n Let Tn = n = 1, 2, 3, . . . and k = 1, 2, . . . , n with k=1 βn,k = 1 k=1 βn,k Sk for every n ∈ N, where 0 ≤ βn,k ≤ 1 for every P n 2 for each n ∈ N and limn→∞ βn,k > 0 for every k ∈ N and let γn = k=1 βn,k (tk,n − 1) for every n ∈ N. Then, the followings hold: (i) kTn x − z k2 ≤ (1 + γn )kx − z k2 for all n ∈ N, x ∈ C and z ∈ ∩ni=1 F (Si ); (ii) if C is bounded and limn→∞ γn = 0, for every sequence {zn } in C , lim kzn+1 − zn k = 0 and

n→∞

lim kzn − Tn zn k = 0 imply ωw (zn ) ⊂ F ;

n→∞

(iii) if limn→∞ γn = 0, F = ∩∞ n=1 F (Tn ) and F is closed convex. Proof. (i) Let n ∈ N and z ∈ ∩ni=1 F (Si ). We have

2

n n

X X

βn,k (Skn x − z ) ≤ βn,k tk2,n kx − z k2 = (1 + γn )kx − z k2 kT n x − z k2 =

k =1 k=1 for every x ∈ C . (ii) Let C be bounded and let limn→∞ γn = 0. Let {zn } be a sequence in C such that limn→∞ kzn+1 − zn k = 0 and limn→∞ kzn − Tn zn k = 0 and let z ∈ F . By parallelogram law, we have

kzn − z k2 ≤ kzn − Tn zn k(kzn − Tn zn k + 2kTn zn − z k) + kTn zn − z k2

2 n

X

n βn,k (Sk zn − z ) ≤ kzn − Tn zn k · M +

k=1

X ≤ kzn − Tn zn k · M + (1 + γn )kzn − z k2 − βn,i βn,j kSin zn − Sjn zn k2 , i
where M = supn∈N {kzn − Tn zn k + 2kTn zn − z k} and

kzn − Tn zn k2 =

n X

βn,k kzn − Skn zn k2 −

k=1

X

βn,i βn,j kSin zn − Sjn zn k2

i
for every n ∈ N. So we get n X

βn,k kzn − Skn zn k2 ≤ kzn − Tn zn k2 + kzn − Tn zn k · M + γn kzn − z k2

k=1

for all n ∈ N. From γn → 0 and βn,k → βk > 0 for each k ∈ N, we obtain lim kzn − Skn zn k = 0

n→∞

for all k ∈ N. As in the proof of Lemma 4.1(ii), we have ωw (zn ) ⊂ F . (iii) Similarly in Lemma 4.1(iii), we get conclusion.  We have the following result for the convex combination of asymptotically nonexpansive mappings by Lemma 4.4 and Theorem 3.1(ii).

118

K. Nakajo et al. / Nonlinear Analysis 71 (2009) 112–119

Theorem 4.5. Let C be a nonempty bounded closed convex subset of H. Let {Sm } be a family of asymptotically nonexpansive n n mappings of C into itself with P Lipschitz constants {tm,n }, i.e. kSm x − Sm yk ≤ tm,n kx − yk (∀m, n ∈ N, ∀x, y ∈ C ) such that n n ∞ β F := ∩i=1 F (Si ) 6= ∅. Let Tn = n,k Sk for every n ∈ N, where 0 ≤ βn,k ≤ 1 for every n = 1, 2, 3, . . . and k = 1, 2, . . . , n k = 1 Pn Pn 2 with k=1 βn,k = 1 for each n ∈ N and limn→∞ βn,k > 0 for every k ∈ N and let γn = k=1 βn,k (tk,n − 1) for every n ∈ N. Let {xn } be a sequence generated by

 x1 = x ∈ C ,    yn = Tn PC (xn + εn ), Cn = {z ∈ C : kyn − z k2 ≤ kxn + εn − z k2 + γn · {diam(C)}2 },   Qn = {z ∈ C : (xn − z , x1 − xn ) ≥ 0},  xn+1 = PCn ∩Qn (x1 ) for each n ∈ N, where {εn } ⊂ H with limn→∞ kεn k = 0. If limn→∞ γn = 0, {xn } converges strongly to z0 = PF (x1 ). 5. Strong convergence theorems for asymptotically nonexpansive semigroups We have the following results for asymptotically nonexpansive semigroups by Shioji and Takahashi [19]. Lemma 5.1. Let C be a nonempty closed convex subset of H and let S be a semigroup. Let S = {T (s) : s ∈ S } be an asymptotically nonexpansive semigroup on C with Lipschitz constants {kt } such that F (S ) is nonempty. Let X be a subspace of B(S ) containing constants and invariant under ls for all s ∈ S. Suppose that the mappings t 7→ kt and t 7→ kT (t )x − z k2 are elements of X for every x ∈ C and z ∈ H. Let {µn } be a sequence of means on X such that limn→∞ kµn − l∗s µn k = 0 for each s ∈ S. Then, the following hold: (i) kTµn x − Tµn yk ≤ (µn )t (kt )kx − yk for every n ∈ N and x, y ∈ C ; (ii) for every bounded sequence {zn } ⊂ C , limn→∞ kzn − Tµn zn k = 0 implies ωw (zn ) ⊂ F (S ); (iii) F (S ) = ∩∞ n=1 F (Tµn ) and if S is commutative, F (S ) is closed and convex. Proof. (i), (ii) By [[19], Lemmas 1, 2 and 3], we get conclusions. (iii) From (ii) and the definition of Tµn , we have F (S ) = ∩∞ n=1 F (Tµn ). And if S is commutative, we obtain that F (S ) is closed and convex by [[23], Theorem 3.6.5].  We have the following result for asymptotically nonexpansive semigroups from Lemmas 2.1 and 5.1 and Theorem 3.1(iii). Theorem 5.2. Let C be a nonempty bounded closed convex subset of H and let S be a commutative semigroup. Let S = {T (s) : s ∈ S } be an asymptotically nonexpansive semigroup on C with Lipschitz constants {kt } such that F (S ) is nonempty. Let X be a subspace of B(S ) containing constants and invariant under ls for all s ∈ S. Suppose that the mappings t 7→ kt and t 7→ kT (t )x − z k2 are elements of X for every x ∈ C and z ∈ H. Let {µn } be a sequence of means on X such that limn→∞ kµn − l∗s µn k = 0 for each s ∈ S. Let {xn } be a sequence generated by

 x0 = x ∈ C ,    yn = (αn I + (1 − αn )Tµn )PC (xn + εn ), Cn = {z ∈ C : kyn − z k2 ≤ kxn + εn − z k2 + δn },    Qn = {z ∈ C : (xn − z , x0 − xn ) ≥ 0}, xn+1 = PCn ∩Qn (x0 ) for each n ∈ N ∪ {0}, where 0 ≤ αn ≤ a < 1 for all n ∈ N for some a, {εn } ⊂ H with limn→∞ kεn k = 0 and δn = (1 − αn )({(µn )t (kt )}2 − 1)+ · {diam(C)}2 (∀n ∈ N). Then, {xn } converges strongly to z0 = PF (S ) (x0 ). Proof. Let Tn = αn I +(1 −αn )Tµn for every n ∈ N. By Lemma 5.1, kTn x − z k2 ≤ kx − z k2 +(1 −αn )({(µn )t (kt )}2 − 1)+ kx − z k2 for each n ∈ N, x ∈ C and z ∈ F (S ). It follows from infs∈S supt ∈S kst ≤ 1 and limn→∞ kµn − l∗s µn k = 0 for each s ∈ S that limn→∞ δn = 0. By Lemma 5.1 and Theorem 3.1(iii), Theorem 5.2 holds.  We get the following proved by Kim and Xu [10] from Theorem 5.2. Theorem 5.3. Let C be a nonempty bounded closed convex subset of H. Let S = {T (s) : 0 ≤ s < ∞} be an asymptotically nonexpansive semigroup on C with Lipschitz constants {kt } such that lim supt →∞ kt ≤ 1, where T (0) = I and let the mapping t 7→ kt be measurable and the mapping t 7→ T (t )x be continuous for every x ∈ C . Let {xn } be a sequence generated by

 x0 = x ∈ C ,  Z   1   yn = αn xn + (1 − αn ) tn

tn 0

T (s)xn ds,

 Cn = {z ∈ C : kyn − z k ≤ kxn − z k2 + δn },     Qn = {z ∈ C : (xn − z , x0 − xn ) ≥ 0}, xn+1 = PCn ∩Qn (x0 ) 2

K. Nakajo et al. / Nonlinear Analysis 71 (2009) 112–119

119

for every n ∈ N ∪ {0}, where  {αn } ⊂ [0, a] for some a ∈ [0, 1), {tn } is a positive real divergent sequence and δn =  R 2 tn 1 (1 − αn ) kt dt − 1 · {diam(C)}2 for all n ∈ N. Then, {xn } converges strongly to z0 = PF (S ) (x0 ). 0 t n

+

Proof. We have F (S ) 6= ∅ by theR boundedness of C , see [25]. Let S = (0, ∞) and X be the set of all bounded measurable s functions on S. Define µs (f ) = 1s 0 f (t ) dt for every s > 0 and f ∈ X . We know that µ is a mean on X with kµs − l∗t µs k → 0 as s → ∞ for each t > 0 and Tµn x =

1 tn

R tn 0

T (s)x ds for all x ∈ C , see [6,23]. By Theorem 5.2, we get Theorem 5.3.



References [1] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350–2360. [2] S. Atsushiba, W. Takahashi, Strong convergence theorems for nonexpansive semigroups by a hybrid method, J. Nonlinear Convex Anal. 3 (2002) 231–242. [3] H.H. Bauschke, P.L. Combettes, A weak-to-strong convergence principle for fejér-monotone methods in Hilbert spaces, Math. Oper. Res. 26 (2001) 248–264. [4] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972) 171–174. [5] Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, Thèse, Université de Paris, Paris, France, 1968. [6] N. Hirano, K. Kido, W. Takahashi, Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Anal. 12 (1988) 1269–1281. [7] H. Iiduka, W. Takahashi, M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, Panamerican Math. J. 14 (2004) 49–61. [8] H. Iiduka, W. Takahashi, Strong convergence theorems by a hybrid method for nonexpansive mappings and inverse-strongly-monotone mappings, in: J.G. Falset, E.L. Fuster, B. Sims (Eds.), Fixed Point Theory and Applications, Yokohama Publishers, Yokohama, 2004, pp. 81–94. [9] M. Kikkawa, W. Takahashi, Approximating fixed points of infinite nonexpansive mappings by the hybrid method, J. Optim. Theory Appl. 117 (2003) 93–101. [10] T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152. [11] P.K. Lin, K.K. Tan, H.K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. 24 (1995) 929–946. [12] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379. [13] K. Nakajo, W. Takahashi, Strong and weak convergence theorems by an improved splitting method, Comm. Appl. Nonlinear Anal. 9 (2002) 99–107. [14] K. Nakajo, K. Shimoji, W. Takahashi, Weak and strong convergence theorems by Mann’s type iteration and the hybrid method in Hilbert spaces, J. Nonlinear Convex Anal. 4 (2003) 463–478. [15] K. Nakajo, K. Shimoji, W. Takahashi, Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces, Taiwanese J. Math. 10 (2006) 339–360. [16] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597. [17] S. Plubtieng, K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 67 (2007) 2306–2315. [18] K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5 (2001) 387–404. [19] N. Shioji, W. Takahashi, Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces, Nonlinear Anal. 34 (1998) 87–99. [20] M.V. Solodov, B.F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Programming Ser. A 87 (2000) 189–202. [21] W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981) 253–256. [22] W. Takahashi, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Ann. Univ. Mariae CurieSklodowska Sect. A 51 (1997) 277–292. [23] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000. [24] W. Takahashi, Convex Analysis and Approximation of Fixed Points, Yokohama Publishers, Yokohama, 2000 (in Japanese). [25] K.K. Tan, H.K. Xu, Fixed point theorems for Lipschitzian semigroups in Banach spaces, Nonlinear Anal. 20 (1993) 395–404.