Volume 2010, Article ID 301868, 15 pagesdoi:10.1155/2010/301868 Research Article Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings
Trang 1Volume 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 Wattanawitoon1, 2 and Poom Kumam2, 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
Copyrightq 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 Takahashi2003 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 Let C be
a nonempty closed convex subset of H, and let T : C → C be a mapping Recall that T is
FT, that is, FT {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
Mann’s iterative algorithm was introduced by Mann1 in 1953 This iteration process is now known as Mann’s iteration process, which is defined as
x n1 α n x n 1 − α n Tx n , n ≥ 0, 1.2
Trang 2where the initial guess x0 is taken in C arbitrarily and the sequence {α n}∞n0is in the interval
0, 1.
In 1967, Halpern2 first introduced the following iteration scheme:
x n1 α n u 1 − α n Tx n 1.3
for all n ∈ N, where x1 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 {Tt : 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 T0x x, x ∈ C;
b Tt sx TtTsx, for all t, s ≥ 0, x ∈ C;
c for each x ∈ C, the map t → Ttx is continuous on 0, ∞;
d there exists a bounded measurable function L : 0, ∞ → 0, ∞ such that, for each
t > 0, Ttx − Tty ≤ 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 supt → ∞ L t ≤ 1 We denote by FT the set of fixed points
of the semigroupT, that is, FT {x ∈ C : Tsx x, ∀s > 0}.
In 2003, Nakajo and Takahashi3 proposed the following modification of the Mann
iteration method for a nonexpansive mapping T in a Hilbert space H:
x0 ∈ C, chosen arbitrarily,
y n α n x n 1 − α n Tx n ,
C nv ∈ C :y n − v ≤ x n − v,
Q n {v ∈ C : x n − v, x n − x0 ≥ 0},
x n1 P C n ∩Q n x0,
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 semigroupT {Tt : 0 ≤
t < ∞} in a Hilbert space H:
x0 ∈ C, chosen arbitrarily,
y n α n x n 1 − α n1
t n
t n 0
T ux n du,
C nv ∈ C :y n − v ≤ x n − v,
Q n {v ∈ C : x n − v, x n − x0 ≥ 0},
x n1 P C ∩Q x0.
1.5
Trang 3In 2006, Kim and Xu4 adapted iteration 1.4 to an asymptotically nonexpansive
mapping in a Hilbert space H:
x0 ∈ C, chosen arbitrarily,
y n α n x n 1 − α n T n x n ,
C n v ∈ C :y n − v2
≤ x n − v2 θ n
,
Q n {v ∈ C : x n − v, x n − x0 ≥ 0},
x n1 P C n ∩Q n x0,
1.6
where θ n 1 − α n k2
n − 1diam C2 → 0 as n → ∞ 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 semigroupT {Tt : 0 ≤ t < ∞} in a Hilbert space H:
x0 ∈ C, chosen arbitrarily,
y n α n x n 1 − α n1
t n
t n 0
T ux n du,
C n v ∈ C :y n − v2≤ x n − v2 θ n
,
Q n {v ∈ C : x n − v, x n − x0 ≥ 0},
x n1 P C n ∩Q n x0,
1.7
where θ n 1 − α n 1/t nt n
0 L u du2− 1diam C2 → 0 as n → ∞.
In 2007, Zegeye and Shahzad5 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:
x0 ∈ C, chosen arbitrarily,
y n α n0 x n α n1 T1n x n α n2 T2n x n · · · α nr T r n x n ,
C n v ∈ C :y n − v2
≤ x n − v2 θ n
,
Q n {v ∈ C : x n − v, x n − x0 ≥ 0},
x n1 P C ∩Q x0,
1.8
Trang 4where θ n k2
n1 − 1α n1 k2
n2 − 1α n2 · · · k2
nr − 1α nr diam C2 → 0 as n → ∞ and
x0 ∈ C, chosen arbitrarily,
y n α n0 x n α n1
1
t n1
t n1 0
T1uxn du
1
t n2
t n2 0
T2uxn du · · ·
1
t nr
t nr 0
T r ux n du ,
C n v ∈ C :y n − v2
≤ x n − v2 θ n
,
Q n {v ∈ C : x n − v, x n − x0 ≥ 0},
x n1 P C n ∩Q n x0,
1.9
where θ n L2
u1 − 1α n1 L2
u2 − 1α n2 · · · L2
ur − 1α nr diam C2 → 0 as n → ∞, with
L ui 1/t nit 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 n1v ∈ C n:y n − v ≤ u n − v,
x n1 P C n1 x0, 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 su n ds,
C n1v ∈ C n:y n − v ≤ u n − v,
x n1 P C n1 x0, 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
in3,5 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
Trang 5Lemma 2.1 Here holds the identity in a Hilbert space H:
λx 1 − λy2
λx2 1 − λy2
− λ1 − λx − y2 2.1
for all x, y ∈ H and λ ∈ 0, 1.
Using thisLemma 2.1, we can prove that the set FT 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 We know that for x ∈ H and z ∈ C, z P C x is
equivalent tox − 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 − x2 x2− 2x n , x x2 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 ∅ / FT ⊂ ∞n1 FT n , where FT N is the set of all fixed
points of T n and FT 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 limn → ∞ z n − Tz n 0 for all T ∈ T.
ii NST-condition II For each bounded sequence {z n } ⊂ C, lim n → ∞ z n1 − T n z n 0 implies that limn → ∞ z n − T m z n 0 for all m ∈ N.
iii NST-condition III There exists {a n } ⊂ 0, ∞ with ∞n1 a n < ∞ such that for every
bounded subset B of C, there exists M B > 0 such that T n x − T n1 x ≤ a n M Bholds
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 FT / ∅ Then, the following hold:
i {T n } with T n T∀n ∈ N and T {T} satisfy the condition (I) with ∞
n1 FT n
FT FT.
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
x n − Tx n → 0, then z Tz.
Trang 6Lemma 2.4 Lin et al 11 Let T be an asymptotically nonexpansive mapping defined on a bounded
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 − v2≤ x − v2 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 {Tt : 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 supt → ∞lim supn → ∞ Ttx n − x n 0,
then z ∈ FT.
Lemma 2.7 Kim and Xu 4 Let C be a nonempty bounded closed convex subset of H and T {Tt : 0 ≤ t < ∞} be an asymtotically nonexpansive semigroup on C Then it holds that
lim sup
s → ∞
lim sup
t → ∞
sup
x∈C
1t
t 0
T uxdu − Ts
1
t
t 0
T uxdu 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 → ∞ Let FT ∞
i1 FT i / ∅ Further, suppose that {T i } satisfies NST-condition
x0 x ∈ C, C0 C,
y n α n x n 1 − α n T n
i x n ,
C n1v ∈ C n:y n − v2≤ x n − v2 θ n
,
x n1 P C n1 x, n 0, 1, 2 ,
3.1
where θ n 1 − α n t2
ni − 1diam C2 → 0 as n → ∞ Then {x n } converges in norm to P FT x0.
is observed that C n1 is closed and convex for each n ∈ N ∪ {0}.
Trang 7Next, we show that FT ⊂ C n for all n ≥ 0 Indeed, let p ∈ FT, we have
y n − p2α n x n 1 − α n T n
i x n − p2
α n x n − p 1 − α n T n
i x n − p2
≤ α nx n − p2 1 − α nT n
i x n − p2
≤ α nx n − p2
1 − α n t2
nix n − p2
x n − p2
1 − α nt2nix n − p2−x n − p2
x n − p2
1 − α nt2ni− 1x
n − p2
x n − p2 θ n −→ 0 as n −→ ∞.
3.2
Thus p ∈ C n1 and hence FT ⊂ C n1 for all n ≥ 0 Thus {x n} is well defined
From x n P C n x0and x n1 P C n1 x0∈ C n1 ⊂ C n, we have
x0 − x n , x n − x n1 ≥ 0 ∀x0 ∈ FT, n ∈ N ∪ {0}. 3.3
So, for x n1 ∈ C n, we have
0≤ x0 − x n , x n − x n1 ,
x0 − x n , x n − x0 x0 − x n1 ,
−x n − x0 , x n − x0 x0 − x n , x0− x n1 ,
≤ −x n − x02 x0 − x n x0 − x n1
3.4
for all n ∈ N This implies that
x0 − x n2≤ x0 − x n x0 − x n1 3.5 hence
x0 − x n ≤ x0 − x n1 3.6
for all n ∈ N ∪ {0} Therefore {x0 − x n} is nondecreasing
From x n P C n x0, we have
x0− x n , x n − y≥ 0 ∀y ∈ C n 3.7
Using FT ⊂ C n, we also have
x0− x n , x n − p≥ 0 ∀p ∈ FT, n ∈ N ∪ {0}. 3.8
Trang 8So, for p ∈ FT, we have
0≥ x0 − x n , x n − p,
x0 − x n , x n − x0 x0 − p,
−x0 − x n2 x0 − x nx0− p. 3.9 This implies that
x0 − x n ≤x0− p ∀p ∈ FT, n ∈ N ∪ {0}. 3.10
Thus,{x0 − x n} is bounded So, limn → ∞ x n − x0 exists.
Next, we show thatx n1 − x n → 0 From 3.3, we have
x n − x n12 x n − x0 x0 − x n12
x n − x02 2x n − x0 , x0− x n1 x0 − x n12
x n − x02 2x n − x0 , x0− x n x n − x n1 x0 − x n12
x n − x02− 2x0 − x n , x0− x n − 2x0 − x n , x n − x n1 x0 − x n12
≤ x n − x02− 2x n − x02 x0 − x n12
−x n − x02 x0 − x n12.
3.11
Since limn → ∞ x n − x0 exists, we conclude that lim n → ∞ x n − x n1 0
Since x n1 ∈ C n1 ⊂ C n, we havey n − x n12 ≤ x n − x n12 θ n which implies that
y n − x n1 ≤ x n − x n1 θ n Now we claim thatT i x n − x n → 0 as n → ∞ for all i ∈ N.
We first show thatT n
i x n − x n → 0 as n → ∞ 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 nT n
i x n − x n,
1 − α nT 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 n1 x n1 − x n,
≤ 1
1− α n
x n − x n1 θ n x n1 − x n.
3.13
Trang 9Sincex n − x n1 → 0 as n → ∞, 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 n1
i x n T n1
i x n − T n1
i x n1 T n1
i x n1 − x n1 x
n1 − x n ,
≤ t∞x n − T n
i x n T n1
i x n1 − x n1 1 t∞x
n − x n1 ,
3.15 from3.14 and x n − x n1 → 0 as n → ∞, yields
lim
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 n1 x n T n1 x n − T n2 x n · · · T i−1 x n − T i x n T i x n − x n
≤ M B i−1
kn
a k T i x n − x n .
3.17
By3.16 and i−1
kn a k < ∞, we get
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 −→ 0 as n −→ ∞. 3.19
Put z0 P FT x0 Sincex n − x0 ≤ z0 − x0 for all n ∈ N ∪ {0}, {x n } is bounded Let {x n i} be a subsequence of{x n } such that x n w Since C is closed and convex, C is weakly closed and
Trang 10hence 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
x0 − z0 ≤ x0 − w ≤ lim inf
i → ∞ x0 − x n i ≤ lim sup
i → ∞ x0 − x n i ≤ z0 − x0, 3.21
and hencex0 − z0 x0 − w From z0 P F x0, we have z0 w This implies that {x n}
converges weakly to z0, and we have
x0 − z0 ≤ lim inf
n → ∞ x0 − x n ≤ lim sup
n → ∞
x0 − x n ≤ z0 − x0, 3.22
and hence limn → ∞ x0 − x n z0 − x0 From x n z0, we also have x0− x n x0− z0 Since
x n − z0 x n − x0 − z0 − x0 −→ 0 3.23
and hence x n → z0 P F x0 This completes the proof.
Corollary 3.2 Let C be a nonempty bounded closed convex subset of a Hilbert space H and let T :
the following algorithm:
x0 x ∈ C, C0 C,
y n α n x n 1 − α n T n x n ,
C n1v ∈ C n:y n − v2
≤ x n − v2 θ n
,
x n1 P C n1 x, n 0, 1, 2 ,
3.24
where θ n 1 − α n t2
n − 1diam C2 → 0 as n → ∞ Then {x n } converges in norm to P FT x0.
obtain the corollary
... set of all fixedpoints of T n and FT is the set of all common fixed points of T We consider the following
conditions of< i>{T n} and T...
3 Strong Convergence for a Family of Asymptotically< /b>
Nonexpansive Mappings< /b>
Theorem 3.1 Let C be a nonempty bounded closed convex subset of a Hilbert... 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