Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 681214, 7 pages
doi:10.1155/2011/681214
Research Article
Strong Convergence Theorems by Shrinking
Projection Methods for Class
T Mappings
Qiao-Li Dong,
1, 2
Songnian He,
1, 2
and Fang Su
3
1
College of Science, Civil Aviation University of China, Tianjin 300300, China
2
Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China,
Tianjin 300300, China
3
Department of Mathematics and Systems Science, National University of Defense Technology,
Changsha 410073, China
Correspondence should be addressed to Qiao-Li Dong,
Received 9 December 2010; Accepted 2 February 2011
Academic Editor: S. Al-Homidan
Copyright q 2011 Qiao-Li Dong et al. 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 a strong convergence theorem by a shrinking projection method for the class of
T
mappings. Using this theorem, we get a new result. We also describe a shrinking projection method
in 1 as follows: starting x
0
∈ H,
x
n1
P
Hx
0
,x
n
∩Hx
n
,T
n
x
n
x
0
. 1.2
Using Lemma 1.2 given below and the fact that a nonexpansive mapping is quasi-
nonexpansive, one can easily obtain hybrid methods introduced by Nakajo and Takahashi
3 for a nonexpansive mapping.
Recently, Takahashi et al. 4 proposed a shrinking projection method for nonexpan-
sive mappings T
n
: C → C.Letx
0
∈ H, C
1
− z≤x
n
− z
,
x
n1
P
C
n1
x
0
,n 1, 2, ,
1.3
where 0 ≤ α
n
≤ a<1, P
K
denotes the metric projection from H onto a closed convex subset
K of H.
Inspired by Bauschke and Combettes 1 and Takahashi et al. 4, we present a
shrinking projection method for the class of T mappings. Furthermore, we obtain a shrinking
projection method for a nonexpansive mapping on Hilbert spaces, which is the same as
presented by Takahashi et al. 4.
We will use the following notations:
1 for weak convergence and → for strong convergence;
2 ω
w
x
n
n0
v
n
− T
n
v
n
2
< ∞,
⇒ ω
w
v
n
⊂
∞
n0
Fix
T
n
. 1.4
Definition 1.4. T is called demiclosed at y ∈ H if Tx y whenever {x
n
x
n
⊂ K and satisfies the condition
x
n
− u≤u − q, ∀n, 1.5
then x
n
→ q.
Lemma 1.7 Goebel and Kirk 5. Let K be a closed convex subset of real Hilbert space H, and let
P
K
be the (metric or nearest point) projection from H onto K (i.e., for x ∈ H, P
K
x is the only point
in K such that x − P
K
x inf{x − z : z ∈ K}). Given x ∈ H and z ∈ K,thenz P
K
x if and
only if there holds the relation
x − z, y − z
≤ 0, ∀y ∈ K. 1.6
2. Main Results
In this section, we will introduce a shrinking projection method for the class of T mappings
and prove strong convergence theorem.
Theorem 2.1. Let T
n
C
n1
{
z ∈ C
n
:
z − T
n
x
n
,x
n
− T
n
x
n
≤ 0
}
.
2.1
Then, {x
n
} converges strongly to P
F
x
0
.
≤ 0, ∀z ∈F. 2.2
From the definition of C
k1
and F⊂C
k
,weobtainF⊂C
k1
. This implies that
F⊂C
n
, ∀n ∈ N. 2.3
It is obvious that C
1
H is closed and convex. So, from the definition, C
n
is closed and convex
for all n ∈ N.Sowegetthat{x
n
} is well defined.
Since x
n
is the projection of x
0
onto C
n
which contains F, we have
x
0
− x
C
n
x
0
and x
n1
P
C
n1
x
0
∈
C
n1
⊂ C
n
,usingLemma 1.7,weget
x
n1
− x
n
,x
0
− x
n
≤ 0. 2.6
It follows that
2
− 2
x
0
− x
n
,x
n1
− x
n
x
n1
− x
n
2
≥
x
0
− x
n
2
exists. From
2.7, it follows that
x
n1
− x
n
2
≤
x
0
− x
n1
2
−
x
0
− x
n
2
,
2.8
and
∞
≤ 0. 2.9
Hence,
x
n1
− x
n
2
x
n1
− T
n
x
n
− x
n
− T
n
x
n
2
x
2
≥
x
n1
− T
n
x
n
2
x
n
− T
n
x
n
2
.
2.10
We therefore get
∞
n1
x
n
∅. Suppose that the sequence {T
n
}
is coherent. Let x
0
∈ H. For C
1
H and x
1
x
0
, define a sequence {x
n
} as follows:
y
n
α
n
x
n
1 − α
n
T
n
x
n
,
n
} converges strongly to P
F
x
0
.
Proof. Set S
n
α
n
I 1 − α
n
T
n
.ByLemma 1.2 ii, we have that S
n
∈ T.Fromx
n
− S
n
x
n
1 − α
n
x
n
− T
n
x
n1
FixS
n
∞
n1
FixT
n
.UsingTheorem 2.1, we get the desired result.
3. Deduced Results
In this section, using Theorem 2.3, we obtain some new strong convergence results for the
class of T mappings, a quasi-nonexpansive mapping and a nonexpansive mapping in a
Hilbert space.
Theorem 3.1. Let T ∈ T such that FixT
/
∅ and satisfying that I −T is demiclosed at 0. Let x
0
∈ H.
For C
1
H and x
1
x
0
, define a sequence {x
n
} as follows:
y
n
C
n1
x
0
,n 1, 2, ,
3.1
where 0 ≤ α
n
≤ a<1. Then, {x
n
} converges strongly to P
FixT
x
0
.
Proof. Let T
n
T in 2.11 for all n ∈ N. Following the proof of Theorem 2.1, we can easily
get 2.5 and
∞
n1
x
n
− Tx
n
2
< ∞.By2.5,weobtainthat{x
n
− Tx
n
→0. Since I − T is
demiclosed at 0, we get x ∈ FixT.Thusω
w
x
n
⊂ FixT which together with Lemma 1.6
and 2.5 implies that x
n
→ P
FixT
x
0
.
Theorem 3.2. Let H be a Hilbert space. Let S be a quasi-nonexpansive mapping on H such that
FixS
/
∅ and satisfying that I − S is demiclosed at 0. Let x
0
∈ H. For C
1
H and x
1
x
0
, define a
sequence {x
n
} as follows:
C
n1
x
0
,n 1, 2, ,
3.2
where 0 ≤ α
n
≤ a<1. Then, {x
n
} converges strongly to P
FixS
x
0
.
6 Fixed Point Theory and Applications
Proof. By Lemma 1.2i, S I/2 ∈ T. Substitute T in 3.1 by S I/2. Then y
n
1
α
n
/2x
n
1 − α
n
/2Sx
n
.Setu
n
2y
n
− y
n
≤ 0
{
z ∈ C
n
:
2z −
x
n
u
n
,x
n
− u
n
≤ 0
}
{
z ∈ C
n
x
n
1 − α
n
Sx
n
,
C
n1
{
z ∈ C
n
: z − u
n
≤x
n
− z
}
,
x
n1
P
C
n1
x
0
nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–
379, 2003.
4 W. Takahashi, Y. 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.
Fixed Point Theory and Applications 7
5 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced
Mathematics, Cambridge University Press, Cambridge, UK, 1990.
6 C. Martinez-Yanes and H K. Xu, “Strong convergence of the CQ method for fixed point iteration
processes,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006.
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