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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,
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
FT,thatis,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
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
n1
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
}
∞
n0
is in the interval
0, 1.
In 1967, Halpern 2 first introduced the following iteration scheme:
x
n1
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 {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 sup
t →∞
L
t
≤ 1. We denote by FT the set of fixed points
of the semigroup T,thatis,FT{x ∈ C : Tsx 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
n1
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 {Tt :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
n1
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
n1
P
C
n
∩Q
n
x
0
,
1.6
where θ
n
1 − α
n
k
2
n
− 1diam 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 {Tt :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
n1
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
− 1diam 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
n1
P
C
n
∩Q
n
x
0
,
1.8
4 Fixed Point Theory and Applications
where θ
n
k
2
n1
− 1α
n1
k
2
n2
− 1α
n2
···k
2
nr
− 1α
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
n1
P
C
n
∩Q
n
x
0
,
1.9
where
θ
n
L
2
u1
− 1α
n1
L
2
u2
− 1α
n2
···L
2
ur
− 1α
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
n1
v ∈ C
n
:
y
n
− v
≤
u
n
− v
,
x
n1
P
C
n1
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
n1
v ∈ C
n
:
y
n
− v
≤
u
n
− v
,
x
n1
P
C
n1
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 − λy
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 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.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
− 2x
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 ∅
/
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 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
n1
− 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
∞
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
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 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
FTFT.
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 FT
/
∅.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 ∈ FT.
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
{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 sup
t →∞
lim sup
n →∞
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
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 →∞.LetFT
∞
i1
FT
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
n1
v ∈ C
n
:
y
n
− v
2
≤
x
n
− v
2
θ
n
,
x
n1
P
C
n1
x
,n 0, 1, 2 ,
3.1
where θ
n
1 − α
n
t
2
ni
− 1diam C
2
→ 0 as n →∞.Then{x
n
} converges in norm to P
FT
x
0
.
Proof. We first show that C
n1
is closed and convex for all n ∈ N ∪{0}.FromtheLemma 2.5,it
is observed that C
n1
is closed and convex for each n ∈ N ∪{0}.
Fixed Point Theory and Applications 7
Next, we show that FT ⊂ C
n
for all n ≥ 0. Indeed, let p ∈ FT, 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
− p1 − 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
n1
and hence FT ⊂ C
n1
for all n ≥ 0. Thus {x
n
} is well defined.
From x
n
P
C
n
x
0
and x
n1
P
C
n1
x
0
∈ C
n1
⊂ C
n
, we have
x
0
− x
n
,x
n
− x
n1
≥ 0 ∀x
0
∈ F
T
,n∈ N ∪
{
0
}
. 3.3
So, for x
n1
∈ C
n
, we have
0 ≤
x
0
− x
n
,x
n
− x
n1
,
x
0
− x
n
,x
n
− x
0
x
0
− x
n1
,
−x
n
− x
0
,x
n
− x
0
x
0
− x
n
,x
0
− x
n1
,
≤−
x
n
− x
0
2
x
0
− x
n
x
0
− x
n1
3.4
for all n ∈ N. This implies that
x
0
− x
n
2
≤
x
0
− x
n
x
0
− x
n1
3.5
hence
x
0
− x
n
≤
x
0
− x
n1
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 FT ⊂ 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 ∈ FT, 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
n1
− x
n
→0. From 3.3, we have
x
n
− x
n1
2
x
n
− x
0
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n
x
n
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
− 2
x
0
− x
n
,x
0
− x
n
− 2
x
0
− x
n
,x
n
− x
n1
x
0
− x
n1
2
≤
x
n
− x
0
2
− 2
x
n
− x
0
2
x
0
− x
n1
2
−
x
n
− x
0
2
x
0
− x
n1
2
.
3.11
Since lim
n →∞
x
n
− x
0
exists, we conclude that lim
n →∞
x
n
− x
n1
0.
Since x
n1
∈ C
n1
⊂ C
n
, we have y
n
− x
n1
2
≤x
n
− x
n1
2
θ
n
which implies that
y
n
− x
n1
≤x
n
− x
n1
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
n1
x
n1
− x
n
,
≤
1
1 − α
n
x
n
− x
n1
θ
n
x
n1
− x
n
.
3.13
Fixed Point Theory and Applications 9
Since x
n
− x
n1
→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
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
from 3.14 and x
n
− x
n1
→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
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
By 3.16 and
i−1
kn
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
FT
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.From3.19, 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 →∞.LetFT
/
∅. 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
n1
v ∈ C
n
:
y
n
− v
2
≤
x
n
− v
2
θ
n
,
x
n1
P
C
n1
x
,n 0, 1, 2 ,
3.24
where θ
n
1 − α
n
t
2
n
− 1diam C
2
→ 0 as n →∞.Then{x
n
} converges in norm to P
FT
x
0
.
Proof. Setting T
n
i
≡ T
n
for all i ∈ N ∪{0} from Lemma 2.2i and Theorem 3.1, we immediately
obtain the corollary.
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
{T
i
} : C → C be a family of nonexpansive mappings with sequence {t
i
}
i≥0
. Assume {α
n
}
n≥0
⊂ 0, 1
such that α
n
≤ a<1 for all n and α
n
→ 0 as n →∞.LetFT
∞
i1
FT
i
/
∅. Further, suppose
that {T
i
} satisfies NST-condition (I) 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
i
x
n
,
C
n1
v ∈ C
n
:
y
n
− v
2
≤
x
n
− v
2
,
x
n1
P
C
n1
x
,n 0, 1, 2
3.25
Assume that if for each bounded sequence {z
n
}∈C, lim
n →∞
z
n
− T
i
z
n
0, for all i ∈ N implies
that lim
n →∞
z
n
− Tz
n
0.Then{x
n
} converges in norm to P
FT
x
0
.
We have the following corollary for nonexpansive mappings by Lemma 2.2i and
Theorem 3.1.
Corollary 3.4 Takahashi et al. 7, Theorem 4.1. Let C be a bounded closed convex subset of a
Hilbert space H and let T : C → C be a nonexpansive mapping such that FT
/
∅. Assume that
0 ≤ α
n
≤ a<1 for all n. Then the sequence {x
n
} generated by
x
0
x ∈ C, C
0
C,
y
n
α
n
x
n
1 − α
n
Tx
n
,
C
n1
v ∈ C
n
:
y
n
− v
≤
x
n
− v
,
x
n1
P
C
n1
x
,n 0, 1, 2 ,
3.26
converges in norm to P
FT
x
0
.
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
T
i
{T
i
t : t ∈ R
, i 1, 2, 3, } be a countable family of asymptotically nonexpansive
semigroups. Assume {α
n
}
n≥0
⊂ 0, 1 such that α
n
≤ a<1 for all n and α
n
→ 0 as n →∞.Let
12 Fixed Point Theory and Applications
{t
ni
},i 1, 2, 3, be a countable positive and divergent real sequence. Let F
∞
i1
FT
i
/
∅. Fur-
ther, suppose that {T
i
} satisfies NST-condition (I) 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
1
t
ni
t
ni
0
T
i
u
x
n
du,
C
n1
v ∈ C
n
:
y
n
− v
2
≤
x
n
− v
2
θ
n
,
x
n1
P
C
n1
x
,n 0, 1, 2 ,
4.1
where
θ
n
1 − α
n
t
2
ni
− 1diam C
2
→ 0 as n →∞with
t
ni
1/t
ni
t
ni
0
L
T
i
u
du.Then{x
n
}
converges in norm to P
F
x
0
.
Proof. First observe that F ⊂ C
n
for all n. Indeed, we have for all p ∈ F
y
n
− p
2
α
n
x
n
1 − α
n
1
t
ni
t
ni
0
T
i
ux
n
du − p
2
α
n
x
n
− p
1 − α
n
1
t
ni
t
ni
0
T
i
u
x
n
du − p
2
≤ α
n
x
n
− p
2
1 − α
n
1
t
ni
t
ni
0
T
i
ux
n
du − p
2
≤ α
n
x
n
− p
2
1 − α
n
1
t
ni
t
ni
0
T
i
u
x
n
− p
du
2
≤ α
n
x
n
− p
2
1 − α
n
1
t
ni
t
ni
0
L
Ti
u
du
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
− 1
x
n
− p
2
≤
x
n
− p
2
θ
n
.
4.2
Fixed Point Theory and Applications 13
So, p ∈ C
n1
. Hence F ⊂ C
n
for all n ∈ N. By the same argument as in the proof of Theorem 3.1,
C
n
is closed and convex, {x
n
} is well defined. Also, similar to the proof of Theorem 3.1
lim
n →∞
x
n
− x
n1
0.
4.3
We next claim that lim sup
s →∞
lim sup
n →∞
T
i
sx
n
−x
n
0. Indeed, by definition of y
n
and
x
n1
∈ C
n
we have
y
n
− x
n
α
n
x
n
1 − α
n
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
1 − α
n
1
t
ni
t
ni
0
T
i
u
x
n
du −
1 − α
n
x
n
1 − α
n
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
4.4
and then
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
1
1 − α
n
y
n
− x
n
≤
1
1 − α
n
y
n
− x
n1
x
n1
− x
n
.
4.5
Since x
n1
∈ C
n1
⊂ C
n
, we have
y
n
− x
n1
2
≤
x
n
− x
n1
2
θ
n
4.6
which in turn implies that
y
n
− x
n1
≤
x
n
− x
n1
θ
n
.
4.7
It follows from 4.5 that
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
≤
1
1 − a
2
x
n1
− x
n
θ
n
−→ 0asn −→ ∞ . 4.8
14 Fixed Point Theory and Applications
Let L
∞
: sup{k
ni
,i 1, 2, 3, } and for each i ∈{1, 2, 3, },wegetthat
T
i
s
x
n
− x
n
≤
T
i
s
x
n
− T
i
s
1
t
ni
t
ni
0
T
i
u
x
n
du
T
i
s
1
t
ni
t
ni
0
T
i
u
x
n
du
−
1
t
ni
t
ni
0
T
i
u
x
n
du
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
≤
L
∞
1
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
T
i
s
1
t
ni
t
ni
0
T
i
u
x
n
du
−
1
t
ni
t
ni
0
T
i
u
x
n
du
.
4.9
By 4.8 and Lemma 2.7,weobtainthat
lim sup
s →∞
lim sup
n →∞
T
i
s
x
n
− x
n
0.
4.10
Furthermore, from 4.9 and Lemma 2.6 and the boundedness of {x
n
} we obtain that
∅
/
ω
w
x
n
⊂ F. By the fact that x
n
− x
0
≤p − x
0
for any n ≥ 0, where p P
F
x
0
and the weak lower semi-continuity of the norm, we have ω − x
0
≤p − x
0
for all
w ∈ ω
w
x
n
. However, since ω
w
x
n
⊂ F, we must have w p for all w ∈ ω
w
x
n
.
Thus ω
w
x
n
{p} and then x
n
converges weakly to p. Moreover, following the method
of Theorem 3.1, x
n
→ p P
F
x
0
. This completes t he proof.
Corollary 4.2. Let C be a b ounded closed convex subset of a Hilbert space H and T {Tt :0≤
t<∞} be an asymptotically nonexpansive semigroup on C. Assume also that 0 <α
n
≤ a<1 for all
n ∈ N ∪{0} and {t
n
} is a positive real divergent sequence. Then, the sequence {x
n
} generated by
x
0
x ∈ C, chosen arbitrarily,
y
n
α
n
x
n
1 − α
n
1
t
n
t
n
0
T
u
x
n
du,
C
n1
v ∈ C
n
:
y
n
− v
2
≤
x
n
− v
2
θ
n
,
x
n1
P
C
n1
x
,n 0, 1, 2 ,
4.11
converges in norm to P
FT
x
0
,whereθ
n
1 − α
n
1/t
n
t
n
0
L
u
du
2
− 1diam C
2
→ 0 as
n →∞.
Proof. By Theorem 4.1, if the semigroup T {Tt :0≤ t<∞} I : {It :0≤ t<∞}, then
Ttx
n
x
n
for all n and for all t>0. Hence 1/t
n
t
n
0
Tux
n
du x
n
for all n and z
n
x
n
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 {Tt :0≤ t<∞} be a nonexpansive semigroup on C. Assume that
0 <α
n
≤ a<1 for all n ∈ N ∪{0} and {t
n
} is a positive real divergent sequence. If FT
/
∅, then the
sequence {x
n
} generated by
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
n1
v ∈ C :
y
n
− v
≤
x
n
− v
,
x
n1
P
C
n1
x
0
,n 0, 1, 2 ,
4.12
converges in norm to P
FT
x
0
.
Acknowledgments
The authors would like to thank professor Somyot Plubtieng for drawing my attention to
the subject and for many useful 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.
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