Báo cáo hóa học: "Research Article Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces" docx

Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2007, Article ID 21972, 18 pages
doi:10.1155/2007/21972
Research Article
Block Iterative Methods for a Finite Family of Relatively
Nonexpansive Mappings in Banach Spaces
Fumiaki Kohsaka and Wataru Takahashi
Received 7 November 2006; Accepted 12 November 2006
Recommended by Ravi P. Agarwal
Using t he convex combination based on Bregman distances due to Censor and Reich, we
define an operator from a given family of relatively nonexpansive mappings in a Banach
space. We first prove that the fixed-point set of this operator is identical to the set of all
common fixed points of the mappings. Next, using this operator, we construct an iterative
sequence to approximate common fixed points of the family. We finally apply our results
to a convex feasibility problem in Banach spaces.
Copyright © 2007 F. Kohsaka and W. Takahashi. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, dis-
tribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let H be a Hilbert space and let
{C
i
}
m
i
=1
be a family of closed convex subsets of H such
that F
=


m
i
=1
C
i
is nonempty. Then the problem of image recovery is to find an element
of F using the metric projection P
i
from H onto C
i
(i = 1,2, ,m), where
P
i
(x) = argmin
y∈C
i
y − x (1.1)
for all x
∈ H. This problem is connected with the convex feasibility problem. In fact,
if
{g
i
}
m
i
=1
is a family of continuous convex functions from H into R, then the convex
feasibility problem is to find an element of the feasible set
m


i=1

x ∈ H : g
i
(x) ≤ 0

. (1.2)
We know that each P
i
is a nonexpansive retraction from H onto C
i
, that is,


P
i
x − P
i
y


≤
x − y (1.3)
2 Fixed Point Theory and Applications
for all x, y
∈ H and P
2
i
= P
i

. Further, it holds that F =

m
i
=1
F(P
i
), where F(P
i
) denotes
the set of all fixed points of P
i
(i = 1,2, ,m). Thus the problem of image recovery in the
setting of Hilbert spaces is a common fixed point problem for a family of nonexpansive
mappings.
A well-known method for finding a solution to the problem of image recovery is the
block-iterative projection algorithm which was proposed by Aharoni and Censor [1]in
finite-dimensional spaces; see also [2–5] and the references therein. This is an iterative
procedure, which generates a sequence
{x
n
} by the rule x
1
= x ∈ H and
x
n+1
=
m

i=1

ω
n
(i)

α
i
x
n
+

1 − α
i

P
i
x
n

(n = 1,2, ), (1.4)
where
{ω
n
(i)}
m
i
=1
⊂ [0,1] (n ∈ N)with

m
i

=1
ω
n
(i) = 1(n ∈ N)and{α
i
}
m
i
=1
⊂ (−1,1). In
particular, Butnariu and Censor [3] studied the strong convergence of
{x
n
} to an element
of F.
Recently, Kikkawa and Takahashi [6] applied this method to the problem of finding a
common fixed point of a finite family of nonexpansive mappings in Banach spaces. Let C
beanonemptyclosedconvexsubsetofaBanachspaceE and let
{T
i
}
m
i
=1
be a finite family
of nonexpansive mappings from C into itself. T hen the iterative scheme they dealt with is
stated as follows: x
1
= x ∈ C and
x

n+1
=
m

i=1
ω
n
(i)

α
n,i
x
n
+

1 − α
n,i

T
i
x
n

(n = 1,2, ), (1.5)
where
{ω
n
(i)}
m
i

=1
⊂ [0,1] with

m
i
=1
ω
n
(i) = 1(n ∈ N)and{α
i
}
m
i
=1
⊂ [0,1]. The y proved
that the generated sequence
{x
n
} converges weakly to a common fix ed point of {T
i
}
m
i
=1
under some conditions on E, {α
n,i
},and{ω
n
(i)}. Then they applied their result to the
problem of finding a common point of a family of nonexpansive retracts of E;seealso

[7–10] for the previous results on this subject.
Our purpose in the present paper is to obtain an analogous result for a finite family
of relatively nonexpansive mappings in Banach spaces. This notion was originally intro-
duced by Butnariu et al. [11]. Recently, Matsushita and Takahashi [12–14]reformulated
the definition of the notion and obtained weak and strong convergence theorems to ap-
proximate a fixed point of a single relatively nonexpansive mapping. It is known that if
C is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach
space E, then the generalized projection Π
C
(see, Alber [15] or Kamimura and Takahashi
[16]) from E onto C is relatively nonexpansive, whereas the metric projection P
C
from E
onto C is not generally nonexpansive.
Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive
Banach space E,letJ be the duality mapping from E into E
∗
,andlet{T
i
}
m
i
=1
be a finite
family of relatively nonexpansive mappings from C into itself such that the set of all com-
mon fixed points of
{T
i
}
m

i
=1
is nonempty. Motivated by the convex combination based
on Bregman distances [17] due to Censor and Reich [18], the iterative methods intro-
duced by Matsushita and Takahashi [12–14], and the proximal-typ e algorithm due to the
F. Kohsaka and W. Takahashi 3
authors [19], we define an operator U
n
(n ∈ N)by
U
n
x = Π
C
J
−1

m

i=1
ω
n
(i)

α
n,i
Jx+

1 − α
n,i


JT
i
x


(1.6)
for all x
∈ C,where{ω
n
(i)}⊂[0,1] and {α
n,i
}⊂[0,1] with

m
i
=1
ω
n
(i) = 1(n ∈ N). Such
amappingU
n
is cal led a block mapping defined by T
1
,T
2
, ,T
m
, {α
n,i
} and {ω

n
(i)}.In
Section 4, we show that the set of all fixed points of U
n
is identical to the set of all common
fixed points of
{T
i
}
m
i
=1
(Theorem 4.2). In Section 5, under some additional assumptions,
we show that the sequence
{x
n
} generated by x
1
= x ∈ C and
x
n+1
= U
n
x
n
(n = 1,2, ) (1.7)
converges weakly to a common fixed point of
{T
i
}

m
i
=1
(Theorem 5.3). This result gener-
alizes the result of Matsushita and Takahashi [12]. If E is a Hilbert space and each T
i
is a
nonexpansive mapping from C into itself, then J is the identity operator on E, and hence
(1.5)and(1.7) are coincident with each other. In Section 6, we deduce some results from
Theorems 4.2 and 5.3.
2. Preliminaries
Let E be a (real) Banach space with norm
·and let E
∗
denote the topological dual of E.
We denote the strong convergence and the weak convergence of a sequence
{x
n
} to x in E
by x
n
→ x and x
n
 x, respectively. We also denote the weak
∗
convergence of a sequence
{x
∗
n
} to x

∗
in E
∗
by x
∗
n
∗
x
∗
.Forallx ∈ E and x
∗
∈ E
∗
,wedenotethevalueofx
∗
at x
by
x, x
∗
. We also denote by R and N the set of al l real numbers and the set of all positive
integers, respectively. The duality mapping J from E into 2
E
∗
is defined by
J(x)
=

x
∗
∈ E

∗
:

x, x
∗

=
x
2
=


x
∗


2

(2.1)
for all x
∈ E.
ABanachspaceE is said to be strictly convex if
x=y=1andx = y imply
(x + y)/2 < 1. It is also said to be uniformly convex if for each ε ∈ (0,2], there exists
δ>0suchthat
x=y=1, x − y≥ε (2.2)
imply
(x + y)/2≤1 − δ. The space E is also said to be smooth if the limit
lim
t→0

x + ty−x
t
(2.3)
exists for all x, y
∈ S(E) ={z ∈ E : z=1}.Itisalsosaidtobeuniformly smooth if the
limit (2.3) exists uniformly in x, y
∈ S(E). It is well known that 
p
and L
p
(1 <p<∞)are
uniformly convex and uniformly smooth; s ee Cioranescu [20] or Diestel [21]. We know
that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single-
valued, one-to-one, and onto. The duality mapping from a smooth Banach space E into
4 Fixed Point Theory and Applications
E
∗
is said to be weakly sequentially continuous if Jx
n
∗
Jx whenever {x
n
} isasequence
in E converging weakly to x in E; see, for instance, [20, 22].
Let E be a smooth, strictly convex, and reflexive Banach space, let J be the duality
mapping from E into E
∗
,andletC be a nonempty closed convex subset of E. Throughout
the present paper, we denote by φ the mapping defined by
φ(y,x)

=y
2
− 2y,Jx + x
2
(2.4)
for all y,x
∈ E. Following Alber [15], the generalized projection from E onto C is defined
by
Π
C
(x) = argmin
y∈C
φ(y,x) (2.5)
for all x
∈ E; see also Kamimura and Takahashi [16]. If E is a Hilbert space, then φ(y,x) =

y − x
2
for all y,x ∈ E, and hence Π
C
is reduced to the metric projection P
C
. It should
be noted that the mapping φ is known to be the Bregman distance [17] corresponding
to the Bregman function
·
2
, and hence the projection Π
C
is the Bregman project ion

corresponding to φ. We know the following lemmas concerning generalized projections.
Lemma 2.1 (see [15]; see also [16]). Let C be a nonempty closed c onvex subset of a smooth,
strictly convex, and reflexive Banach space E. Then
φ

x, Π
C
y

+ φ

Π
C
y, y

≤
φ(x, y) (2.6)
for all x
∈ C and y ∈ E.
Lemma 2.2 (see [15]; see also [16]). Let C be a nonempty closed c onvex subset of a smooth,
strictly convex, and reflexive Banach space E,letx
∈ E,andletz ∈ C. Then z = Π
C
x is
equivalent to
y − z, Jx− Jz≤0 (2.7)
for all y
∈ C.
Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive
Banach space E,letT be a mapping from C into itself, and let F(T) be the set of al l fixed

points of T. Then a point z
∈ C is said to be an asymptotic fixed point of T (see Reich
[23]) if there exists a sequence
{z
n
} in C converging weakly to z and lim
n
z
n
− Tz
n
=0.
We denote the set of all asymptotic fixed points of T by

F(T). Following Matsushita and
Takahashi [12–14], we say that T is a relatively nonexpansive mapping if the following
conditions are satisfied:
(R1) F(T)isnonempty;
(R2) φ(u,Tx)
≤ φ(u, x)forallu ∈ F(T)andx ∈ C;
(R3)

F(T) = F(T).
F. Kohsaka and W. Takahashi 5
Some examples of relatively nonexpansive mapping s are listed below; see Reich [23]and
Matsushita and Takahashi [12] for more details.
(a) If C is a nonempty closed convex subset of a Hilbert space E and T isanon-
expansive mapping from C into itself such that F(T) is nonempty, then T is a
relatively nonexpansive mapping from C into itself.
(b) If E is a uniformly smooth and strictly convex Banach space and A

⊂ E × E
∗
is
a maximal monotone operator such that A
−1
0 is nonempty, then the resolvent
J
r
= (J + rA)
−1
J (r>0) is a relatively nonexpansive mapping from E onto D(A)
(the domain of A)andF(J
r
) = A
−1
0.
(c) If Π
C
is the generalized projection from a smooth, strictly convex, and reflex-
ive Banach space E onto a nonempty closed convex subset C of E,thenΠ
C
is a
relatively nonexpansive mapping from E onto C and F(Π
C
) = C.
(d) If
{C
i
}
m

i
=1
is a finite family of closed convex subset of a uniformly convex and
uniformly smooth Banach space E such that

m
i
=1
C
i
is nonempty and T =
Π
1
Π
2
···Π
m
is the composition of the generalized projections Π
i
from E onto
C
i
(i = 1,2, ,m), then T is a relatively nonexpansive mapping from E into itself
and F(T)
=

m
i
=1
C

i
.
The following lemma is due to Matsushita and Takahashi [14].
Lemma 2.3 (see [14]). Let C be a nonempty closed convex subset of a smooth, strictly convex,
and reflexive Banach space E and let T be a relatively nonexpansive mapping from C into
itself. Then F(T) is clos ed and convex.
We also know the following lemmas.
Lemma 2.4 (see [16]). Let E be a smooth and uniformly convex Banach space and let
{x
n
}
and {y
n
} be sequences in E such that eithe r {x
n
} or {y
n
} is bounded. If lim
n
φ(x
n
, y
n
) = 0,
then lim
n
x
n
− y
n

=0.
Lemma 2.5 (see [16]). Let E be a smooth and uniformly convex Banach space and let r>0.
Then there exists a strictly increasing, continuous, and convex function g :[0,2r]
→ R such
that g(0)
= 0 and
g


x − y

≤ φ(x, y) (2.8)
for all x, y
∈ B
r
={z ∈ E : z≤r}.
Lemma 2.6 (see [24]; see also [25, 26]). Let E be a uniformly convex Banach space and let
r>0. Then there exists a strictly increasing, continuous, and convex function g :[0,2r]
→ R
such that g(0) = 0 and


tx +(1− t)y


2
≤ tx
2
+(1− t)y
2

− t(1 − t)g


x − y

(2.9)
for all x, y
∈ B
r
and t ∈ [0,1].
6 Fixed Point Theory and Applications
3. Lemmas
The following lemma is well known. For the sake of completeness, we give the proof.
Lemma 3.1. Let E be a strictly convex Banach space and let
{t
i
}
m
i
=1
⊂ (0,1) with

m
i
=1
t
i
= 1.
If
{x

i
}
m
i
=1
is a finite sequence in E such that





m

i=1
t
i
x
i





2
=
m

i=1
t
i



x
i


2
, (3.1)
then x
1
= x
2
=···=x
m
.
Proof. If x
k
= x
l
for some k,l ∈{1,2, ,m}, then the strict convexity of E implies that





t
k
t
k
+ t

l
x
k
+
t
l
t
k
+ t
l
x
l





2
<
t
k
t
k
+ t
l


x
k



2
+
t
l
t
k
+ t
l


x
l


2
. (3.2)
Using this inequality, we have





m

i=1
t
i
x
i






2
=






t
k
+ t
l


t
k
t
k
+ t
l
x
k
+
t
l

t
k
+ t
l
x
l

+

i=k,l
t
i
x
i





2
≤

t
k
+ t
l







t
k
t
k
+ t
l
x
k
+
t
l
t
k
+ t
l
x
l





2
+

i=k,l
t
i



x
i


2
<

t
k
+ t
l


t
k
t
k
+ t
l
x
2
+
t
l
t
k
+ t
l

y
2

+

i=k,l
t
i


x
i


2
=
m

i=1
t
i


x
i


2
.
(3.3)

This is a contradiction.

We also need the following lemmas.
Lemma 3.2. Let E be a smooth, strictly convex and reflexive Banach space, let z
∈ E and let
{t
i
}⊂(0,1) with

m
i
=1
t
i
= 1.If{x
i
}
m
i
=1
is a finite sequence in E such that
φ

z, J
−1

m

j=1
t

j
Jx
j

=
φ

z, x
i

(3.4)
for all i
∈{1,2, ,m}, then x
1
= x
2
=···=x
m
.
Proof. By assumption, we have
φ

z, J
−1

m

j=1
t
j

Jx
j

=
m

i=1
t
i
φ

z, x
i

. (3.5)
F. Kohsaka and W. Takahashi 7
This is equivalent to
z
2
− 2

z,
m

i=1
t
i
Jx
i


+





m

i=1
t
i
Jx
i





2
=
m

i=1
t
i


z
2
− 2


z, Jx
i

+


x
i


2

, (3.6)
which is also equivalent to





m

i=1
t
i
Jx
i






2
=
m

i=1
t
i


Jx
i


2
. (3.7)
Since E is smooth and reflexive, E
∗
is strictly convex. Thus, Lemma 3.1 implies that Jx
1
=
Jx
2
=···= Jx
m
. By the str ict convexity of E, J is one-to-one. Hence we have the desired
result.

Lemma 3.3. Let E be a smooth, strictly convex, and reflexive Banach space, let {x

i
}
m
i
=1
be a
finite sequence in E and let
{t
i
}
m
i
=1
⊂ [0,1] with

m
i
=1
t
i
= 1. Then
φ

z, J
−1

m

i=1
t

i
Jx
i

≤
m

i=1
t
i
φ

z, x
i

(3.8)
for all z
∈ E.
Proof. Let V : E
× E
∗
→ R be the function defined by
V

x, x
∗

=
x
2

− 2

x, x
∗

+


x
∗


2
(3.9)
for all x
∈ E and x
∗
∈ E
∗
. In other words,
V

x, x
∗

=
φ

x, J
−1

x
∗

(3.10)
for all x
∈ E and x
∗
∈ E
∗
.Wealsohaveφ(x, y) = V (x,Jy)forallx, y ∈ E.Thenwehave
from the convexity of V in its second variable that
φ

z, J
−1

m

i=1
t
i
Jx
i

=
V

z,
m


i=1
t
i
Jx
i

≤
m

i=1
t
i
V

z, Jx
i

=
m

i=1
t
i
φ

z, x
i

. (3.11)
This completes the proof.


4. Block mappings by relatively nonexpansive mappings
Let E be a smooth, strictly convex, and reflexive Banach space and let J be the duality
mapping from E into E
∗
.LetC be a nonempty closed convex subset of E and let {T
i
}
m
i
=1
be a finite family of relatively nonexpansive mappings from C into itself. In this section,
we study some properties of the mapping U defined by
Ux
= Π
C
J
−1

m

i=1
ω
i

α
i
Jx+

1 − α

i

JT
i
x


(4.1)
8 Fixed Point Theory and Applications
for all x
∈ C,where{α
i
}
m
i
=1
⊂ [0,1] and {ω
i
}
m
i
=1
⊂ [0,1] with

m
i
=1
ω
i
= 1. Recall that such

amappingU is called a block mapping defined by T
1
,T
2
, ,T
m
, {α
n,i
} and {ω
n
(i)}.
Lemma 4.1. Let E be a smooth, strictly convex, and reflexive Banach space and let C be a
nonempty closed convex subset of E.Let
{T
i
}
m
i
=1
be a finite family of relatively nonexpansive
mappings from C into itself such that

m
i
=1
F(T
i
) is nonempty and let U be the block mapping
defined by (4.1), where
{α

i
}⊂[0,1] and {ω
i
}⊂[0,1] with

m
i
=1
ω
i
= 1. Then
φ(u,Ux)
≤ φ(u, x) (4.2)
for all u
∈

m
i
=1
F(T
i
) and x ∈ C.
Proof. Let u
∈

m
i
=1
F(T
i

)andx ∈ C. Then i t holds from Lemmas 2.1 and 3.3 that
φ(u,Ux)
= φ

u,Π
C
J
−1

m

i=1
ω
i

α
i
Jx+

1 − α
i

JT
i
x


≤
φ


u,J
−1

m

i=1
ω
i

α
i
Jx+

1 − α
i

JT
i
x


≤
m

i=1
ω
i

α
i

φ(u,x)+

1 − α
i

φ

u,T
i
x

≤
φ(u,x).
(4.3)
This completes the proof.

Theorem 4.2. Let E be a smooth, strictly convex and reflexive Banach space and let C be a
nonempty closed convex subset of E.Let
{T
i
}
m
i
=1
be a finite family of relatively nonexpansive
mappings from C into itself such that

m
i
=1

F(T
i
) is nonempty and let U be the block mapping
defined by (4.1), where
{α
i
}⊂[0,1) and {ω
i
}⊂(0,1] with

m
i
=1
ω
i
= 1. Then
F(U)
=
m

i=1
F

T
i

. (4.4)
Proof. Since the inclusion F(U)
⊃


m
i
=1
F(T
i
)isobvious,itsuffices to show the inverse
inclusion F(U)
⊂

m
i
=1
F(T
i
). Let z ∈ F(U)begivenandfixu ∈

m
i
=1
F(T
i
). Let V : E ×
E
∗
→ R be the function defined by (3.9). Then, as in the proof of Lemma 4.1,wehave
φ(u,z)
= φ(u, Uz) ≤ φ

u,J
−1


m

i=1
ω
i

α
i
Jz+

1 − α
i

JT
i
z


≤
m

i=1
ω
i

α
i
φ(u,z)+


1 − α
i

φ

u,T
i
z

≤
φ(u,z).
(4.5)
If k
∈{1,2, ,m},thenwehave
φ(u,z)
=
m

i=1
ω
i

α
i
φ(u,z)+

1 − α
i

φ


u,T
i
z

≤

i=k
ω
i
φ(u,z)+ω
k

α
k
φ(u,z)+

1 − α
k

φ

u,T
k
z

.
(4.6)
F. Kohsaka and W. Takahashi 9
Using (4.6), we have

ω
k
φ(u,z) =

1 −

i=k
ω
i

φ(u,z) ≤ ω
k

α
k
φ(u,z)+

1 − α
k

φ

u,T
k
z

. (4.7)
Hence we have
ω
k


1 − α
k

φ(u,z) ≤ ω
k

1 − α
k

φ

u,T
k
z

. (4.8)
Since ω
k
> 0, α
k
< 1, and u ∈ F(T
k
), we have
φ(u,z)
≤ φ

u,T
k
z


≤
φ(u,z). (4.9)
Thus
φ

u,J
−1

m

i=1
ω
i

α
i
Jz+

1 − α
i

JT
i
z


=
φ


u,T
j
z

=
φ(u,z) (4.10)
for all j
∈{1,2, ,m}.
If m
= 1, then ω
1
= 1. In this case,
Ux
= Π
C
J
−1

α
1
Jx+

1 − α
1

JT
1
x

(4.11)

for all x
∈ C.Ifα
1
= 0, then U = T
1
, and hence the conclusion obviously holds. If α
1
> 0,
then we have from (4.10)that
φ

u,J
−1

α
1
Jz+

1 − α
1

JT
1
z

=
φ

u,T
1

z

=
φ(u,z). (4.12)
Then, using Lemma 3.2,wehavez
= T
1
z.
We next consider the case where m
≥ 2. In this case, it holds that 0 <ω
i
< 1forall
i
∈{1,2, ,m}.LetI ={i ∈{1,2, ,m} : α
i
= 0}.IfI is empty, then we have from (4.10)
that
φ

u,J
−1

m

i=1
ω
i
JT
i
z


=
φ

u,T
i
z

(4.13)
for all i
∈{1,2, ,m}. Using Lemma 3.2,wehaveT
1
z = T
2
z =···=T
m
z.Hencewehave
z
= Uz = Π
C
J
−1

m

i=1
ω
i
JT
i

z

=
Π
C
J
−1

m

i=1
ω
i
JT
j
z

=
Π
C
T
j
z = T
j
z (4.14)
for all j
∈{1,2, ,m}.Thusz ∈

m
i

=1
F(T
i
).
On the other hand, if I is nonempty, then we have from (4.10)that
φ

u,J
−1


i∈I
ω
i
α
i
Jz+
m

i=1
ω
i

1 − α
i

JT
i
z


=
φ

u,T
i
z

=
φ(u,z) (4.15)
for all i
∈{1,2, ,m}.Then,fromLemma 3.2,wehavez = T
1
z = T
2
z =···=T
m
z.Thus
z
∈

m
i
=1
F(T
i
). This completes the proof. 
10 Fixed Point Theory and Applications
5. Weak and strong convergence theorems
Let E be a smooth, strictly convex, and reflexive Banach space and let C be a nonempty
closed con vex subset of E.Let

{T
i
}
m
i
=1
be a finite family of relatively nonexpansive map-
pings from C into itself such that

m
i
=1
F(T
i
)isnonemptyandletU
n
be a block mapping
from C into itself defined by
U
n
x = Π
C
J
−1

m

i=1
ω
n

(i)

α
n,i
Jx+

1 − α
n,i

JT
i
x


(5.1)
for all x
∈ C,where{ω
n
(i)}⊂[0,1] and {α
n,i
}⊂[0,1] with

m
i
=1
ω
n
(i) = 1foralln ∈ N.
In this section, we study the asymptotic behavior of
{x

n
} generated by x
1
= x ∈ C and
x
n+1
= U
n
x
n
(n = 1,2, ). (5.2)
Lemma 5.1. Let E be a smooth and uniformly convex Banach space and let C be a none mpty
closed convex subset of E.Let
{T
i
}
m
i
=1
be a finite family of relatively nonexpansive mappings
from C into itself such that F
=

m
i
=1
F(T
i
) is nonempty and let {α
n,i

: n,i ∈ N,1 ≤ i ≤ m}
and {ω
n
(i):n,i ∈ N,1 ≤ i ≤ m} be sequences in [0,1] such that

m
i
=1
ω
n
(i) = 1 for all n ∈
N
.Let{U
n
} be a sequence of block mappings defined by (5.1)andlet{x
n
} beasequence
generated by (5.2). Then
{Π
F
x
n
} converges strongly to the unique element z of F such that
lim
n→∞
φ

z, x
n


=
min

lim
n→∞
φ

y,x
n

: y ∈ F

. (5.3)
Proof. If u
∈ F,thenwehavefromLemma 4.1 that
φ

u,x
n+1

≤
φ

u,x
n

(5.4)
for all n
∈ N. T hus the limit of φ(u,x
n

) exists. Since φ(u,x
n
) ≥ (u−x
n
)
2
for all u ∈ F
and n
∈ N, the sequence {x
n
} is bounded. By Lemma 2.1,wehaveφ(u,Π
F
x
n
) ≤ φ(u,x
n
).
So, the sequence
{Π
F
x
n
} is also bounded. By the definition of Π
F
and (5.4), we have
φ

Π
F
x

n+1
,x
n+1

≤
φ

Π
F
x
n
,x
n+1

≤
φ

Π
F
x
n
,x
n

. (5.5)
Thus lim
n
φ(Π
F
x

n
,x
n
) exists. We next show that {Π
F
x
n
} is a Cauchy sequence. Take r>0
such that
{Π
F
x
n
}⊂B
r
.Then,byLemma 2.5 , we have a strictly increasing, continuous
and convex function g :[0,2r]
→ R such that g(0) = 0and
g



Π
F
x
m
− Π
F
x
n




≤
φ

Π
F
x
m
,Π
F
x
n

(5.6)
F. Kohsaka and W. Takahashi 11
for all m,n
∈ N.Ifm>n,thenitfollowsfromLemma 2.1 that
φ

Π
F
x
n
,Π
F
x
m


≤
φ

Π
F
x
n
,x
m

−
φ

Π
F
x
m
,x
m

≤
φ

Π
F
x
n
,x
n


−
φ

Π
F
x
m
,x
m

. (5.7)
Thus, for all ε>0, there exists N
∈ N such that m>n≥ N implies that
g



Π
F
x
m
− Π
F
x
n



≤
φ


Π
F
x
n
,x
n

−
φ

Π
F
x
m
,x
m

≤
ε. (5.8)
Therefore,
{Π
F
x
n
} is a Cauchy sequence in F, and hence it converges strongly to an ele-
ment z of F.
We next show that z is the unique element of F such that
lim
n→∞

φ

z, x
n

=
min

lim
n→∞
φ

y,x
n

: y ∈ F

. (5.9)
We define a function h : F
→ R by
h(y)
= lim
n→∞
φ

y,x
n

(5.10)
for all y

∈ F. Then we can show that h is a continuous convex function. In fact, if y
1
, y
2
∈
F and t ∈ (0,1), then
φ

ty
1
+(1− t)y
2
,x
n

≤
tφ

y
1
,x
n

+(1− t)φ

y
2
,x
n


(5.11)
for all n
∈ N.Tendingn →∞, we have the convexity of h. We next show the continuity of
h.Lety
1
, y
2
∈ F and take M>0suchthat{x
n
},{y
1
, y
2
}⊂B
M
.Thenwehave
φ

y
1
,x
n

−
φ

y
2
,x
n


=


y
1


2
−


y
2


2
+2

y
2
− y
1
,Jx
n

≤




y
1


+


y
2





y
1


−


y
2



+2


x

n




y
1
− y
2


≤ 4M


y
1
− y
2


(5.12)
for all n
∈ N.Tendingn →∞,wehaveh(y
1
) − h(y
2
) ≤ 4My
1
− y
2

. Similarly, we have
h(y
2
) − h(y
1
) ≤ 4My
1
− y
2
.Thush is continuous. We can also show that z
n
→∞
implies that h(z
n
) →∞.SinceE is reflexive and F is closed and convex by Lemma 2.3,
the set
A
=

p ∈ F : h(p) = inf
y∈F
h(y)

(5.13)
is nonempty ; see Takahashi [27, 28] for more details.
12 Fixed Point Theory and Applications
On the other hand, if y
∈ F,thenwehave
h


Π
F
x
n

=
lim
m→∞
φ

Π
F
x
n
,x
m

≤
φ

Π
F
x
n
,x
n

≤
φ


y,x
n

(5.14)
for all n
∈ N.Tendingn →∞,wehaveh(z) ≤ h(y), and hence z ∈ A. We finally show that
A is singleton. Suppose that there exist z
1
,z
2
∈ A such that z
1
= z
2
.Takes>0suchthat
{z
1
,z
2
}⊂B
s
.Then,byLemma 2.6, we have a strictly increasing, continuous, and convex
function
g :[0,2s] → R such that g(0) = 0and





z

1
+ z
2
2





2
≤
1
2


z
1


2
+
1
2


z
2


2

−
1
4
g



z
1
− z
2



. (5.15)
Using this inequality, we have
h

z
1
+ z
2
2

=
lim
n→∞






z
1
+ z
2
2




2
− 2

z
1
+ z
2
2
,Jx
n

+


x
n


2


≤
lim
n→∞

φ

z
1
,x
n

2
+
φ

z
2
,x
n

2
−

g



z
1

− z
2



4

=
h

z
1

2
+
h

z
2

2
−

g



z
1
− z

2



4
<
h

z
1

2
+
h

z
2

2
= min
y∈F
h(y).
(5.16)
This is a contradiction.

Following an idea due to Matsushita and Takahashi [12], we prove the following
lemma.
Lemma 5.2. Let E be a uniformly smooth and uniformly convex Banach space and let C
be a nonempty closed convex subset of E.Let
{T

i
}
m
i
=1
be a finite family of relatively non-
expansive mapping s from C into itself such that F
=

m
i
=1
F(T
i
) is nonempty and let {α
n,i
:
n,i
∈ N,1 ≤ i ≤ m}⊂[0,1] and {ω
n
(i):n,i ∈ N,1 ≤ i ≤ m}⊂[0, 1] besequencessuchthat
liminf
n
α
n,i
(1 − α
n,i
) > 0 and liminf
n
ω

n
(i) > 0 for all i ∈{1,2, ,m} and

m
i
=1
ω
n
(i) = 1
for all n
∈ N.Let{U
n
} be a sequence of block mappings defined by (5.1)andlet{z
n
} be
aboundedsequenceinC such that lim
n
{φ(u,z
n
) − φ(u,U
n
z
n
)}=0 for some u ∈ F and
z
n
k
 z. The n z ∈ F.
Proof. Since
{z

n
} is bounded and φ(u,T
i
z
n
) ≤ φ(u,z
n
)foralln ∈ N, {T
i
z
n
} is also
bounded. It follows from the uniform smoothness of E that E
∗
is uniformly convex; see
Takahashi [27, 28]. Take r>0suchthat
{z
n
},{T
i
z
n
}⊂B
r
(i = 1,2, ,m). Then,
Lemma 2.6 ensures the existence of a strictly increasing, continuous and convex function
g :[0,2r]
→ R such that g(0) = 0and



tJz
n
+(1− t)JT
i
z
n


2
≤ t


z
n


2
+(1− t)


T
i
z
n


2
− t(1 − t)g




Jz
n
− JT
i
z
n



(5.17)
F. Kohsaka and W. Takahashi 13
for all t
∈ [0,1], n ∈ N,andi ∈{1,2, ,m}.Sinceu is an element of F, we can show from
Lemma 2.1 that
φ

u,U
n
z
n

≤
φ

u,J
−1

m


i=1
ω
n
(i)

α
n,i
Jz
n
+

1 − α
n,i

JT
i
z
n


=
V

u,
m

i=1
ω
n
(i)


α
n,i
Jz
n
+

1 − α
n,i

JT
i
z
n


≤
m

i=1
ω
n
(i)V

u,α
n,i
Jz
n
+


1 − α
n,i

JT
i
z
n

=
m

i=1
ω
n
(i)


u
2
− 2

u,α
n,i
Jz
n
+

1 − α
n,i


JT
i
z
n

+


α
n,i
Jz
n
+

1 − α
n,i

JT
i
z
n


2

.
(5.18)
Using (5.17)andφ(u,T
i
z

n
) ≤ φ(u,z
n
), we have
φ

u,U
n
z
n

≤
m

i=1
ω
n
(i)


u
2
− 2

u,α
n,i
Jz
n
+


1 − α
n,i

JT
i
z
n

+


α
n,i
Jz
n
+

1 − α
n,i

JT
i
z
n


2

≤
m


i=1
ω
n
(i)


u
2
− 2

u,α
n,i
Jz
n
+

1− α
n,i

JT
i
z
n

+ α
n,i


z

n


2
+

1 − α
n,i



T
i
z
n


2
− α
n,i
(1 − α
n,i
)g(Jz
n
− JT
i
z
n
)


=
m

i=1
ω
n
(i)

α
n,i
φ

u,z
n

+

1 − α
n,i

φ

u,T
i
z
n

−
α
n,i


1 − α
n,i

g



Jz
n
− JT
i
z
n



≤
φ

u,z
n

−
m

i=1
ω
n
(i)α

n,i

1 − α
n,i

g



Jz
n
− JT
i
z
n



.
(5.19)
Thus we have
m

i=1
ω
n
(i)α
n,i

1 − α

n,i

g



Jz
n
− JT
i
z
n



≤
φ

u,z
n

−
φ

u,U
n
z
n

(5.20)

for all n
∈ N. Then it follows from lim
n
{φ(u,z
n
) − φ(u,U
n
z
n
)}=0that
lim
n→∞
m

i=1
ω
n
(i)α
n,i

1 − α
n,i

g



Jz
n
− JT

i
z
n



=
0. (5.21)
Since liminf
n
ω
n
(i) > 0 and liminf
n
α
n,i
(1 − α
n,i
) > 0foralli ∈{1,2, ,m},wehave
lim
n→∞
g



Jz
n
− JT
i
z

n



=
0 (5.22)
14 Fixed Point Theory and Applications
for all i
∈{1,2, ,m}. Then, the properties of g yield
lim
n→∞


Jz
n
− JT
i
z
n


=
0 (5.23)
for all i
∈{1,2, ,m}.SinceE is uniformly convex, the duality mapping J
−1
from E
∗
into
E is uniformly norm-to-norm continuous on every bounded subset of E

∗
; see Takahashi
[27, 28]. Hence, we have
lim
n→∞


z
n
− T
i
z
n


=
lim
n→∞


J
−1

Jz
n

−
J
−1


JT
i
z
n



=
0 (5.24)
for all i
∈{1,2, ,m}.Thusz ∈

F(T
i
)foralli ∈{1,2, ,m}.SinceeachT
i
is relatively
nonexpansive, we have

F(T
i
) = F(T
i
)foralli ∈{1, 2, ,m}, and hence z ∈ F. This com-
pletes the proof.

Using Lemmas 5.1 and 5.2,westudytheasymptoticbehaviorof{x
n
} generated by
(5.2).

Theorem 5.3. Let E be a uniformly smooth and uniformly convex Banach space and let
C be a nonempty closed c onvex subs et of E.Let
{T
i
}
m
i
=1
be a finite family of relatively non-
expansive mapping s from C into itself such that F
=

m
i
=1
F(T
i
) is nonempty and let {α
n,i
:
n,i
∈ N,1 ≤ i ≤ m}⊂[0,1] and {ω
n
(i):n,i ∈ N,1 ≤ i ≤ m}⊂[0, 1] besequencessuchthat
liminf
n
α
n,i
(1 − α
n,i

) > 0 and liminf
n
ω
n
(i) > 0 for all i ∈{1,2, ,m} and

m
i
=1
ω
n
(i) = 1
for all n
∈ N.Let{U
n
} be a sequence of block mappings defined by (5.1)andlet{x
n
} be a
sequence generated by (5.2). Then the following hold:
(a) the sequence
{x
n
} is bounded and each weak subsequential limit of {x
n
} belongs to

m
i
=1
F(T

i
);
(b) if the duality mapping J from E into E
∗
is weakly sequentially continuous, then {x
n
}
converges weakly to the strong limit of {Π
F
x
n
}.
Proof. We first prove part (a). Let u
∈ F.AsintheproofofLemma 5.1, we can show that
{φ(u,x
n
)} is nonincreasing a nd {x
n
}, {T
i
x
n
} are bounded. It also holds that
φ

u,x
n

−
φ


u,U
n
x
n

=
φ

u,x
n

−
φ

u,x
n+1

−→
0 (5.25)
as n
→ 0. Using Lemma 5.2, we know that every weak subsequential limit of {x
n
} belongs
to F.
We next prove part (b). Suppose that J is weakly sequentially continuous. If x
n
k
 z,
then z

∈ F by part (a). It follows from Lemma 2.2 that

z − Π
F
x
n
,Jx
n
− JΠ
F
x
n

≤
0 (5.26)
for all n
∈ N.ByLemma 5.1, Π
F
x
n
→ w ∈ F.Tendingn
k
→∞,wehave
z − w,Jz− Jw≤0. (5.27)
Since J is a monotone operator, we have
z − w,Jz− Jw=0.Thenthestrictconvexityof
E implies that z
= w; see Takahashi [27, 28]. This completes the proof. 
F. Kohsaka and W. Takahashi 15
6. Deduced results

As direct consequences of Theorem 4.2, we have the following two corollaries.
Corollar y 6.1. Let E be a smooth, strictly convex, and reflexive Banach space and let C be
a nonempty c losed convex s ubset of E.LetT be a relatively nonexpansive mapping from C
into itself and let U be the mapping defined by
Ux
= Π
C
J
−1

αJx +

1 − α

JTx

(6.1)
for all x
∈ C,whereα ∈ [0,1). Then
F(U)
= F(T). (6.2)
Corollar y 6.2. Let H beaHilbertspaceandletC be a nonempty closed convex subset
of H.Let
{T
i
}
m
i
=1
be a finite family of nonexpansive mappings from C into itself such that


m
i
=1
F(T
i
) is nonempty and let U be the mapping defined by
Ux
=
m

i=1
ω
i

α
i
x +

1 − α
i

T
i
x

(6.3)
for all x
∈ C,where{α
i

}⊂[0,1), {ω
i
}⊂(0,1] and

m
i
=1
ω
i
= 1. Then
F(U)
=
m

i=1
F

T
i

. (6.4)
As a direct consequence of Theorem 5.3, we obtain the weak convergence theorem
according to Matsushita and Takahashi [12].
Corollar y 6.3 (see [12]). Let E be a uniformly smooth and uniformly convex Banach
space and let C beanonemptyclosedconvexsubsetofE.LetT be a relatively nonexpansive
mapping from C into itself and let
{x
n
} be a sequence gene rated by x
1

= x ∈ C and
x
n+1
= Π
C
J
−1

α
n
Jx
n
+

1 − α
n

JTx
n

(n = 1,2, ), (6.5)
where
{α
n
}⊂[0,1] satisfies liminf
n
α
n
(1 − α
n

) > 0.Thenthefollowinghold:
(a) the sequence
{x
n
} is bounded and each weak subsequential limit of {x
n
} belongs to
F(T);
(b) if the duality mapping J from E into E
∗
is weakly sequentially continuous, then {x
n
}
converges weakly to the strong limit of {Π
F
x
n
}.
If E is a Hilbert space and each T
i
is a nonexpansive mapping from C into itself, then
Theorem 5.3 is reduced to the following.
Corollar y 6.4. Let H beaHilbertspaceandletC be a nonempty closed convex subset
of H.Let
{T
i
}
m
i
=1

be a finite family of nonexpansive mappings from C into itself such that
F
=

m
i
=1
F(T
i
) is nonempty and let {x
n
} beasequencegeneratedbyx
1
= x ∈ C and
x
n+1
=
m

i=1
ω
n
(i)

α
n,i
x
n
+


1 − α
n,i

T
i
x
n

(n = 1,2, ), (6.6)
16 Fixed Point Theory and Applications
where
{α
n,i
: n,i ∈ N,1 ≤ i ≤ m}⊂[0,1] and {ω
n
(i):n,i ∈ N,1 ≤ i ≤ m}⊂[0,1] satisfy
liminf
n
α
n,i
(1 − α
n,i
) > 0 and liminf
n
ω
n
(i) > 0 for all i ∈{1,2, ,m} and

m
i

=1
ω
n
(i) = 1
for all n
∈ N. Then {x
n
} converges weakly to the strong limit of {P
F
x
n
},whereP
F
is the
metric projection from H onto F.
Using Theorems 4.2 and 5.3, we can deal with the image recovery problem in Banach
spaces as follows.
Corollar y 6.5. Let E be a smooth, strictly convex, and reflexive Banach space, let
{C
i
}
m
i
=1
be a finite family of closed convex subsets of E such that

m
i
=1
C

i
is nonempty, and let Π
i
be the
generalized projection from E onto C
i
for all i ∈{1,2, ,m}.LetU be the mapping defined
by
Ux
= J
−1

m

i=1
ω
i

α
i
Jx+

1 − α
i

JΠ
i
x



, (6.7)
where
{α
i
}⊂[0,1) and {ω
i
}⊂(0,1] with

m
i
=1
ω
i
= 1. Then
F(U)
=
m

i=1
C
i
. (6.8)
Corollar y 6.6. Let E be a uniformly smooth and uniformly convex Banach space, let
{C
i
}
m
i
=1
be a finite family of closed convex subsets of E such that


m
i
=1
C
i
is nonempty, and
let Π
i
be the generalized projection from E onto C
i
for all i ∈{1,2, ,m}.Let{x
n
} be a
sequence generated by x
1
= x ∈ E and
x
n+1
= J
−1

m

i=1
ω
n
(i)

α

n,i
Jx
n
+

1 − α
n,i

JΠ
i
x
n


(n = 1,2, ), (6.9)
where
{α
n,i
: n,i ∈ N,1 ≤ i ≤ m}⊂[0,1] and {ω
n
(i):n,i ∈ N,1 ≤ i ≤ m}⊂[0,1] satisfy
liminf
n
α
n,i
(1 − α
n,i
) > 0 and liminf
n
ω

n
(i) > 0 for all i ∈{1,2, ,m} and

m
i
=1
ω
n
(i) = 1
for all n
∈ N. Then the following hold:
(a) the sequence
{x
n
} is bounded and each weak subsequential limit of {x
n
} belongs to

m
i
=1
C
i
;
(b) if the duality mapping J from E into E
∗
is weakly sequentially continuous, then {x
n
}
converges weakly to the strong limit of {Π


m
i
=1
C
i
x
n
}.
References
[1] R. Aharoni and Y. Censor, “Block-iterative projection methods for parallel computation of s olu-
tions to convex feasibility problems,” Linear Algebra and Its Applications, vol. 120, pp. 165–175,
1989.
[2] D. Butnariu and Y. Censor, “On the behavior of a block-iterative projection method for solving
convex feasibility problems,” International Journal of Computer Mathematics,vol.34,no.1-2,pp.
79–94, 1990.
[3] D. Butnariu and Y. Censor, “Strong convergence of almost simultaneous block-iterative pro-
jection methods in Hilbert spaces,” Journal of Computational and Applied Mathematic s, vol. 53,
no. 1, pp. 33–42, 1994.
F. Kohsaka and W. Takahashi 17
[4] N. Cohen and T. Kutscher, “On spherical convergence, convexity, and block iterative projection
algorithms in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 226, no. 2,
pp. 271–291, 1998.
[5] S.D.Fl
˚
am and J. Zowe, “Relaxed outer projections, weighted averages and convex feasibility,”
BIT, vol. 30, no. 2, pp. 289–300, 1990.
[6] M. Kikkawa and W. Takahashi, “Approximating fixed points of nonexpansive mappings by the
block iterative method in Banach spaces,” International Journal of Computational and Numerical
Analysis and Applications, vol. 5, no. 1, pp. 59–66, 2004.

[7] G. Crombez, “Image recovery by convex combinations of projections,” Journal of Mathematical
Analysis and Applications, vol. 155, no. 2, pp. 413–419, 1991.
[8] S. Kitahara and W. Takahashi, “Image recovery by convex combinations of sunny nonexpansive
retractions,” Topological Methods in Nonlinear Analysis, vol. 2, no. 2, pp. 333–342, 1993.
[9] W. Takahashi, “Iterative methods for approximation of fixed points and their applications,” Jour-
nal of the Operations Research Society of Japan, vol. 43, no. 1, pp. 87–108, 2000.
[10] W. Takahashi and T. Tamura, “Limit theorems of operators by convex combinations of non-
expansive retractions in Banach spaces,” Journal of Approximation Theory,vol.91,no.3,pp.
386–397, 1997.
[11] D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive op-
erators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001.
[12] S. Matsushita and W. Takahashi, “Weak and strong convergence theorems for relatively nonex-
pansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2004, no. 1, pp.
37–47, 2004.
[13] S. Matsushita and W. Takahashi, “An iterative algorithm for relatively nonexpansive mappings
by a hybrid method and applications,” in Nonlinear Analysis and Convex Analysis, W. Takahashi
and T. Tanaka, Eds., pp. 305–313, Yokohama Publishers, Yokohama, Japan, 2004.
[14] S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive
mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266,
2005.
[15] Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and appli-
cations,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type,A.
G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 15–50, Markel Dekker,
New Yor k, NY, USA, 1996.
[16] S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach
space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
[17] L. M. Bregman, “A relaxation method of finding a common point of convex sets and its appli-
cation to the solution of problems in convex programming,” USSR Computational Mathematics
and Mathematical Physics, vol. 7, pp. 200–217, 1967.
[18] Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with

applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996.
[19] F. Kohsaka and W. Takahashi, “Strong convergence of an iterative sequence for maximal mono-
tone operators in a Banach space,” Abstract and Applied Analysis, vol. 2004, no. 3, pp. 239–249,
2004.
[20] I. Cioranescu, Geometry of B anach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of
MathematicsandItsApplications, Kluwer Academic, Dordrecht, The Netherlands, 1990.
[21] J. Diestel, Geometry of Banach Spaces—Selected Topics, vol. 485 of Lecture Notes in Mathematics,
Springer, Berlin, Germany, 1975.
[22] J P. Gossez and E. Lami Dozo, “Some geometric properties related to the fixed point theory for
nonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, pp. 565–573, 1972.
18 Fixed Point Theory and Applications
[23] S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in
Theory and Applications of Nonlinear Operators of Accretive and Monotone Type,A.G.Kartsatos,
Ed., vol. 178 of Lecture Notes in Pure and Appl. Math., pp. 313–318, Markel Dekker, New York,
NY, USA, 1996.
[24] H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis, vol. 16, no. 12,
pp. 1127–1138, 1991.
[25] C. Z
˘
alinescu, “On uniformly convex functions,” Journal of Mathematical Analysis and Applica-
tions, vol. 95, no. 2, pp. 344–374, 1983.
[26] C. Z
˘
alinescu, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, USA,
2002.
[27] W. Takahashi, Convex Analysis and Approximation of Fixed Points, vol. 2 of Mathematical Analysis
Series, Yokohama Publishers, Yokohama, Japan, 2000.
[28] W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama
Publishers, Yokohama, Japan, 2000.
Fumiaki Kohsaka: Department of Information Environment, Tokyo Denki University,

Muzai Gakuendai, Inzai 270-1382, Chiba, Japan
Email address:
Wataru Takahashi: Department of Mathematical and Computing Sciences,
Tokyo Institute of Technology, Oh-Okayama, Meguro-Ku 152-8552, Tokyo, Japan
Email address: