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, [email protected]
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
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
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
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
y
n
− v
≤
x
n
− v
,
Q
n
{
v ∈ C :
x
n
− v, x
n
− x
0
≥ 0
}
,
T
n
x
n
,
C
n
v ∈ C :
y
n
− v
2
≤
x
n
− v
2
θ
n
,
Q
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
− v
2
θ
n
,
Q
n
{
v ∈ C :
x
n
− v, x
n
− x
0
≥ 0
}
,
x
n1
P
C
n
∩Q
n
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, 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
0
T
1
u
x
n
du
1
t
n2
t
n2
0
T
2
u
x
n
du
···
≤
x
n
− v
2
θ
n
,
Q
n
{
v ∈ C :
x
n
− v, x
n
− x
0
≥ 0
}
,
x
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
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
,
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:
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
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
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
/
− 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;
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
0
C,
y
n
α
n
x
n
1 − α
n
T
n
i
x
n
,
C
n1
v ∈ C
n
:
y
n
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
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
T
n
i
x
n
− p
x
n
− p
2
1 − α
n
t
2
ni
x
n
− p
2
−
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
. 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
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
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
0
− x
n
2
x
0
− x
n
x
0
− p
.
3.9
This implies that
x
0
− 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
0
,x
0
− x
n
x
n
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
− 2
x
0
− x
0
2
− 2
x
n
− x
0
2
x
0
− x
n1
2
−
x
n
− x
0
2
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
T
n
i
x
n
− x
n
,
1 − α
n
T
n
i
x
n
1 − α
n
x
n
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
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
i
x
n
− T
n1
i
x
n
T
n1
i
x
n
− T
n1
i
x
n1
x
n
T
n1
i
x
n1
− x
n1
1 t
∞
x
n
− x
n1
,
≤
T
n
x
n
− T
i
x
n
T
i
x
n
− x
n
≤
T
n
x
n
− T
n1
x
n
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 →∞
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
− 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
− 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
− 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
}
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
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
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
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
.
C
n1
v ∈ C
n
:
y
n
− v
≤
x
n
− v
,
x
n1
P
C
n1
x
∞
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
≤
x
n
− v
2
θ
n
,
x
n1
P
C
n1
x
,n 0, 1, 2 ,
4.1
where
θ
n
1 − α
n
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
1
t
ni
t
ni
0
T
i
u
x
n
du − p
2
≤ α
n
x
n
2
≤ α
n
x
n
− p
2
1 − α
n
1
t
ni
t
ni
0
T
i
t
ni
0
L
Ti
u
du
x
n
− p
2
α
n
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
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
n
1
t
ni
t
ni
0
T
i
u
x
n
du −
1 − α
n
x
n
and then
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
1
1 − α
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
ni
t
ni
0
T
i
u
x
n
du − x
n
≤
1
1 − a
2
x
n1
− x
n
T
i
s
x
n
− T
i
s
1
t
ni
t
ni
0
T
i
u
x
n
du
du
−
1
t
ni
t
ni
0
T
i
u
x
n
du
1
1
t
ni
t
ni
0
T
i
u
x
n
du − x
n
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.
. 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
C
n1
v ∈ C
n
:
y
n
− v
2
≤
x
n
− v
2
θ
n
,
x
n1
P
C
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
du,
C
n1
v ∈ C :
y
n
− v
≤
x
n
− v
,
x
n1
P
C
n1
x
0
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
96215, 11 pages, 2006.
9 K. Nakajo, K. 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, “Demiclosedness principle and asymptotic behavior for
asymptotically nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24,
no. 6, pp. 929–946, 1995.
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