FOR RELATIVELY NONEXPANSIVE MAPPINGS IN BANACH SPACES
SHIN-YA MATSUSHITA AND WATARU TAKAHASHI Received 29 October 2003
We first introduce an iterative sequence for finding fixed points of relatively nonexpansive mappings in Banach spaces, and then prove weak and strong convergence theorems by using the notion of generalized projection. We apply these results to the convex feasibility problem and a proximal-type algorithm for monotone operators in Banach spaces.
1. Introduction
LetEbe a real Banach space and letAbe a maximal monotone operator fromEtoE∗, whereE∗is the dual space ofE. It is well known that many problems in nonlinear analysis and optimization can be formulated as follows: find
u∈E such that 0∈Au. (1.1)
A well-known method for solving (1.1) in a Hilbert spaceHis the proximal point algo- rithm:x0∈Hand
xn+1=Jrnxn, n=0, 1, 2,..., (1.2) where{rn} ⊂(0,∞) andJr=(I+rA)−1for allr >0. This algorithm was first introduced by Martinet [9]. In [16], Rockafellar proved that if lim infn→∞rn>0 andA−10= ∅, then the sequence{xn}defined by (1.2) converges weakly to an element of solutions of (1.1).
On the other hand, Kamimura and Takahashi [4] considered an algorithm to generate a strong convergent sequence in a Hilbert space. Further, Kamimura and Takahashi’s re- sult was extended to more general Banach spaces by Kohsaka and Takahashi [7]. They introduced and studied the following iteration sequence:x=x0∈Eand
xn+1=J−1αnJx+1−αn JJrnxn
, n=0, 1, 2,..., (1.3) whereJis the duality mapping onEandJr=(J+rA)−1Jfor allr >0. Kohsaka and Taka- hashi [7] proved that ifA−10= ∅, limn→∞αn=0,∞n=0αn= ∞, and limn→∞rn= ∞, then the sequence generated by (1.3) converges strongly to an element ofA−10.
Copyright©2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:1 (2004) 37–47 2000 Mathematics Subject Classification: 47H09, 47H05, 47J25 URL:http://dx.doi.org/10.1155/S1687182004310089
On the other hand, Reich [13] studied an iteration sequence of nonexpansive map- pings in a Banach space which was first introduced by Mann [8]:x0∈Cand
xn+1=αnxn+1−αn
Sxn, n=0, 1, 2,..., (1.4) whereSis a nonexpansive mapping from a closed convex subsetCof Einto itself and {αn} ⊂[0, 1]. He proved that ifF(T) is nonempty and∞n=0αn(1−αn)= ∞, then the sequence generated by (1.4) converges weakly to some fixed point ofS.
Motivated by Kohsaka and Takahashi [7], and Reich [13], our purpose in this paper is to prove weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces which were first introduced by Butnariu et al. [3] and further studied by the authors [10]. For this purpose, we consider the following iterative sequence:x0∈C and
xn+1=ΠCJ−1αnJxn+1−αnJTxn
, n=0, 1, 2,..., (1.5) whereTis a relatively nonexpansive mapping fromCinto itself andΠCis the generalized projection ontoC. Notice that ifEis a Hilbert space andS=T, then the sequences (1.4) and (1.5) are equivalent. We prove that if F(T) is nonempty and the duality mapping J is weakly sequentially continuous, then the sequence{xn}converges weakly to a fixed point ofT and if the interior ofF(T) is nonempty, then{xn}converges strongly to a fixed point ofT. Using these results, we also consider the convex feasibility problem and a proximal-type algorithm for monotone operators in Banach spaces.
2. Preliminaries
LetEbe a Banach space with norm · and letE∗be the dual ofE. Then we denote by x,x∗the pairing betweenx∈Eandx∗∈E∗. When{xn}is a sequence inE, we denote the strong convergence and the weak convergence of{xn}tox∈Ebyxn→xandxnx, respectively.
A Banach spaceEis said to be strictly convex if(x+y)/2<1 for allx,y∈Ewith x = y =1 andx=y. It is also said to be uniformly convex if limn→∞xn−yn =0 for any two sequences{xn},{yn}inEsuch thatxn=yn=1 and limn→∞(xn+yn)/2= 1. The following result was proved by Xu [19].
Proposition2.1 [19]. Letr >0and letEbe a Banach space. IfEis uniformly convex, then there exists a continuous, strictly increasing, and convex functiong: [0,∞)→[0,∞)with g(0)=0such that
λx+ (1−λ)y2≤λx2+ (1−λ)y2−λ(1−λ)gx−y
(2.1) for allx,y∈Br= {z∈E:z ≤r}andλwith0≤λ≤1.
LetU= {x∈E:x =1}be the unit sphere ofE. The norm ofEis said to beGˆateaux differentiableif for eachx,y∈U, the limit
limt
→0
x+ty − x
t (2.2)
exists. In this case,Eis calledsmooth. The norm ofEis said to beFr´echet differentiableif for eachx∈U, the limit is attained uniformly fory∈U. It is also said to beuniformly smoothif the limit is attained uniformly forx,y∈U. The (normalized) duality mapping JfromEtoE∗is defined by
Jx=
x∗∈E∗:x,x∗= x2=x∗2 (2.3) forx∈E. We say thatJisweakly sequentially continuousif for a sequence{xn} ⊂E,xn x, thenJxn∗ Jx, where∗ denotes the weak∗convergence. We list several well-known properties of the duality mapping:
(1) ifEis smooth, thenJis single valued and norm-to-weak∗continuous;
(2) ifEis Fr´echet differentiable, thenJis norm-to-norm continuous;
(3) ifEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofE.
For more details, see [17]. Assume thatEis smooth. Then the functionV:E×E→Ris defined by
V(x,y)= x2−2 x,J y+y2 (2.4) forx,y∈E. From the definition ofV, we have that
x − y2
≤V(x,y)≤
x+y2
(2.5) forx,y∈E. The functionValso has the following property:
V(y,x)=V(z,x) +V(y,z) + 2 z−y,Jx−Jz (2.6) forx,y,z∈E. The following result was proved by Kamimura and Takahashi [5].
Proposition2.2 (Kamimura and Takahashi [5]). Letr >0and letEbe a uniformly convex and smooth Banach space. Then
gy−z
≤V(y,z) (2.7)
for all y,z∈Br= {w∈E:w ≤r}, where g: [0,∞)→[0,∞)is a continuous, strictly increasing, and convex function withg(0)=0.
LetCbe a nonempty closed convex subset ofE. Suppose thatEis reflexive, strictly convex, and smooth. Then, for anyx∈E, there exists a unique pointx0∈Csuch that
Vx0,x=min
y∈CV(y,x). (2.8)
Following Alber [1], we denote such anx0byΠCx. The mappingΠC is called thegen- eralized projectionfromEontoC. It is easy to see that in a Hilbert space, the mapping ΠCis coincident with the metric projection. Concerning the generalized projection, the following are well known.
Proposition2.3 (Alber [1]; see also Kamimura and Takahashi [5]). LetCbe a nonempty closed convex subset of a smooth Banach spaceEandx∈E. Then
x0=ΠCx⇐⇒
x0−y,Jx−Jx0
≥0 for eachy∈C. (2.9) Proposition2.4 (Alber [1]; see also Kamimura and Takahashi [5]). LetEbe a reflexive, strictly convex, and smooth Banach space, letCbe a nonempty closed convex subset ofE, and letx∈E. Then
Vy,ΠCx+VΠCx,x≤V(y,x) for eachy∈C. (2.10) LetTbe a mapping fromCinto itself. We denote byF(T) the set of fixed points ofT.
A point pinCis said to be anasymptotic fixed pointofT[12] ifCcontains a sequence {xn}which converges weakly topsuch that the strong limn→∞(xn−Txn)=0. The set of asymptotic fixed points ofT is denoted by ˆF(T). We say that the mapping T is called relatively nonexpansive[3,10] ifF(T)=F(T) andˆ
V(p,Tx)≤V(p,x) for eachx∈C, p∈F(T). (2.11) 3. Main results
In this section, we discuss the weak and strong convergence of (1.5). To prove our results, we need the following proposition.
Proposition3.1. LetEbe a uniformly convex and smooth Banach space, letCbe a non- empty closed convex subset of E, and let T be a relatively nonexpansive mapping fromC into itself such thatF(T)is nonempty. Let{αn}be a sequence of real numbers such that 0≤αn≤1. Suppose{xn}is the sequence generated byx0∈Candxn+1=ΠCJ−1(αnJxn+ (1−αn)JTxn),n=0, 1, 2,....Then{ΠF(T)xn}converges strongly to some fixed point ofT, whereΠF(T)is the generalized projection fromContoF(T).
Proof. We know thatF(T) is closed and convex (see [10]). So, we can define the general- ized projectionΠF(T)ontoF(T). Letp∈F(T). FromProposition 2.4and the convexity of · 2, we have
Vp,xn+1
=Vp,ΠCJ−1αnJxn+1−αn JTxn
≤Vp,J−1αnJxn+1−αn JTxn
= p2−2p,αnJxn+1−αn JTxn
+αnJxn+1−αn JTxn2
≤ p2−2αn p,Jxn
−21−αn
p,JTxn
+αnxn2+1−αnTxn2
=αn p2−2p,Jxn
+xn2
+1−αn p2−2p,JTxn
+Txn2
=αnVp,xn
+1−αn
Vp,Txn
≤αnVp,xn
+1−αn
Vp,xn
=Vp,xn .
(3.1)
Hence, limn→∞V(p,xn) exists and, in particular,{V(p,xn)}is bounded. Then, by (2.5), {xn} is also bounded. This implies that{Txn} is bounded. Letun=ΠF(T)xn for each n∈N∪ {0}. Then, we have
Vun,xn+1
≤Vun,xn
. (3.2)
It follows from (2.10) that Vun+1,xn+1
=VΠF(T)xn+1,xn+1
≤Vun,xn+1
−Vun,ΠF(T)xn+1
. (3.3)
Combining this with (3.2), we obtain Vun+1,xn+1
≤Vun,xn
. (3.4)
It follows that{V(un,xn)}converges. Then, from (3.3), Vun,un+1
≤Vun,xn
−Vun+1,xn+1
. (3.5)
By induction, we have
Vun,un+m
≤Vun,xn
−Vun+m,xn+m
(3.6) for eachm∈N. UsingProposition 2.2, we have, form,nwithn > m,
gum−un≤Vum,un
≤Vum,xm
−Vun,xn
, (3.7)
whereg: [0,∞)→[0,∞) is a continuous, strictly increasing, and convex function with g(0)=0. Then the properties ofg yield that{un}is a Cauchy sequence. SinceEis com- plete andF(T) is closed,{un}converges strongly to some pointuinF(T).
Now, we can prove a weak convergence theorem.
Theorem3.2. LetEbe a uniformly convex and uniformly smooth Banach space, letCbe a nonempty closed convex subset ofE, letT be a relatively nonexpansive mapping fromC into itself such thatF(T)is nonempty, and let{αn}be a sequence of real numbers such that 0≤αn≤1andlim infn→∞αn(1−αn)>0. Suppose{xn}is the sequence generated by (1.5).
IfJis weakly sequentially continuous, then{xn}converges weakly to some fixed point ofT.
Proof. As in the proof ofProposition 3.1, we know that{xn}and{Txn}are bounded. Put r=supn∈N∪{0}{xn,Txn}. SinceEis a uniformly smooth Banach space,E∗is a uni- formly convex Banach space (see [17,18] for more details). Therefore, byProposition 2.1, there exists a continuous, strictly increasing, and convex functiong: [0,∞)→[0,∞) with g(0)=0 such that
λx∗+ (1−λ)y∗2≤λx∗2+ (1−λ)y∗2−λ(1−λ)gx∗−y∗ (3.8)
for eachx∗,y∗∈Br= {z∗∈E∗:z∗ ≤r}andλwith 0≤λ≤1. Letp∈F(T). We have Vp,xn+1
=Vp,ΠCJ−1αnJxn+1−αn JTxn
≤Vp,J−1αnJxn+1−αn JTxn
= p2−2p,αnJxn+1−αn JTxn
+αnJxn+1−αn JTxn2
≤ p2−2αn p,Jxn
−21−αn
p,JTxn +αnxn2+1−αnTxn2−αn
1−αn
gJxn−JTxn
=αnVp,xn
+1−αn
Vp,Txn
−αn 1−αn
gJxn−JTxn
≤Vp,xn
−αn
1−αngJxn−JTxn,
(3.9)
and hence
αn 1−αn
gJxn−JTxn≤Vp,xn
−Vp,xn+1
. (3.10)
Since{V(p,xn)}converges and lim infn→∞αn(1−αn)>0, it follows that
nlim→∞gJxn−JTxn=0. (3.11) Then the properties ofgyield that
nlim→∞Jxn−JTxn=0. (3.12)
SinceJ−1is uniformly norm-to-norm continuous on bounded sets, we obtain
nlim→∞xn−Txn=nlim
→∞J−1Jxn
−J−1JTxn=0. (3.13) This implies that if there exists a subsequence{xni}of{xn}such thatxnivfor some v∈E, then, by the definition ofT,vis a fixed point ofT.
Letun=ΠF(T)xnfor eachn∈N∪ {0}. It follows from (2.9) that un−z,Jxn−Jun
≥0 (3.14)
for eachz∈F(T). Let{xni}be a subsequence of{xn}such that{xni}converges weakly tov. Then we havev∈F(T). By (3.14), we have
uni−v,Jxni−Juni
≥0. (3.15)
FromProposition 3.1, we know that{un}converges strongly to someu∈F(T) andJ is weakly sequentially continuous. Lettingi→ ∞, we have
u−v,Jv−Ju ≥0. (3.16)
On the other hand, from the monotonicity ofJ, we have
u−v,Ju−Jv ≥0. (3.17)
Combining this with (3.16), we have
u−v,Ju−Jv =0. (3.18)
Using the strict convexity ofE, we obtain u=v. Therefore,{xn}converges weakly to
u=limn→∞ΠF(T)xn. This completes the proof.
Next, we also consider the strong convergence of (1.5). We can prove the following theorem without the assumption of “weakly sequentially continuous” in the duality map- pingJ.
Theorem3.3. LetEbe a uniformly convex and uniformly smooth Banach space, letCbe a nonempty closed convex subset ofE, letTbe a relatively nonexpansive mapping fromCinto itself, and let{αn}be a sequence of real numbers such that0≤αn≤1andlim infn→∞αn(1− αn)>0. Suppose{xn}is the sequence generated by (1.5). If the interior ofF(T)is nonempty, then{xn}converges strongly to some fixed point ofT.
Proof. Since the interior ofF(T) is nonempty, there existp∈F(T) andr >0 such that
p+rh∈F(T) (3.19)
wheneverh ≤1. By (2.6), we have, for anyu∈F(T), Vu,xn
=Vxn+1,xn
+Vu,xn+1
+ 2xn+1−u,Jxn−Jxn+1
. (3.20)
This implies
xn+1−u,Jxn−Jxn+1 +1
2Vxn+1,xn
=1 2
Vu,xn
−Vu,xn+1. (3.21) We also have
xn+1−p,Jxn−Jxn+1
=xn+1−(p+rh) +rh,Jxn−Jxn+1
=xn+1−(p+rh),Jxn−Jxn+1
+rh,Jxn−Jxn+1. (3.22) On the other hand, sincep+rh∈F(T), as in the proof ofProposition 3.1, we have that
Vp+rh,xn+1
≤Vp+rh,xn
. (3.23)
From (3.21), this inequality is equivalent to 0≤
xn+1−(p+rh),Jxn−Jxn+1 +1
2Vxn+1,xn
. (3.24)
Then, by (3.21), we have rh,Jxn−Jxn+1
≤xn+1−p,Jxn−Jxn+1 +1
2Vxn+1,xn
=1 2
Vp,xn
−Vp,xn+1 ,
(3.25)
and hence
h,Jxn−Jxn+1
≤ 1 2r
Vp,xn
−Vp,xn+1
. (3.26)
Sincehwithh ≤1 is arbitrary, we have Jxn−Jxn+1≤ 1
2r
Vp,xn
−Vp,xn+1
. (3.27)
So, ifn > m, then
Jxm−Jxn=Jxm−Jxm+1+Jxm+1− ··· −Jxn−1+Jxn−1−Jxn
≤
n−1 i=m
Jxi−Jxi+1≤ 1 2r
n−1 i=m
Vp,xi
−Vp,xi+1
= 1 2r
Vp,xm
−Vp,xn .
(3.28)
We know that{V(p,xn)}converges. So,{Jxn}is a Cauchy sequence. SinceE∗is complete, {Jxn}converges strongly to some point inE∗. SinceE∗has a Fr´echet differentiable norm, thenJ−1is continuous onE∗. Hence,{xn}converges strongly to some pointuinC. As in the proof ofTheorem 3.2, we also have thatxn−Txn →0. So, we haveu∈F(T), where
u=limn→∞ΠF(T)xn.
4. Applications
In this section, using Theorems3.2and3.3, we give some applications. We first consider the problem of weak convergence concerning nonexpansive mappings in a Hilbert space.
Theorem4.1 (Browder and Petryshyn [2]). LetCbe a nonempty closed convex subset of a Hilbert spaceH, letT be a nonexpansive mapping fromC into itself such thatF(T)is nonempty, and letλbe a real number such that0< λ <1. Suppose that{xn}is given by x0∈Cand
xn+1=λxn+ (1−λ)Txn, n=0, 1, 2,.... (4.1) Then{xn}converges weakly touinF(T), whereu=limn→∞PF(T)xnandPF(T)is the metric projection fromContoF(T).
Proof. Letαn=λfor eachn∈N∪ {0}. It is clear that lim infn→∞αn(1−αn)=λ(1−λ)>0.
We show that ifT is nonexpansive, thenT is relatively nonexpansive. It is obvious that F(T)⊂F(T). Ifˆ u∈F(Tˆ ), then there exists{xn} ⊂Csuch thatxnuandxn−Txn→0.
SinceTis nonexpansive,Tis demiclosed. So, we haveu=Tu. This impliesF(T)=F(Tˆ ).
Further, in a Hilbert spaceH, we know that
V(x,y)= x−y2 (4.2)
for everyx,y∈H. So,Tx−T y ≤ x−yis equivalent toV(Tx,T y)≤V(x,y). There- fore,Tis relatively nonexpansive. UsingTheorem 3.2, we obtain the desired result.
We also consider the strong convergence concerning nonexpansive mappings in a Hilbert space. For related results, see Moreau [11], and Kirk and Sims [6].
Theorem4.2. LetCbe a nonempty closed convex subset of a Hilbert spaceH, letT be a nonexpansive mapping fromCinto itself, and letλ be a real number such that0< λ <1.
Suppose that{xn}is given byx0∈Cand
xn+1=λxn+ (1−λ)Txn, n=0, 1, 2,.... (4.3) If the interior ofF(T)is nonempty, then{xn}converges strongly touinF(T), whereu= limn→∞PF(T)xnandPF(T)is the metric projection fromContoF(T).
Next, we apply Theorems3.2and3.3to the convex feasibility problem. Before giving them, we introduce the following lemma which was proved by Reich [12].
Lemma 4.3 (Reich [12]). Let E be a uniformly convex Banach space with a uniformly Gˆateaux-differentiable norm, let{Ci}mi=1be a finite family of closed convex subsets ofE, and letΠibe the generalized projection fromEontoCifor eachi=1, 2,...,m. Then
Vp,ΠmΠm−1···Π2Π1x≤V(p,x) (4.4) for eachp∈F(Πˆ mΠm−1···Π2Π1),x∈E, andF(Πˆ mΠm−1···Π2Π1)= ∩mi=1Ci.
As direct consequences ofLemma 4.3and Theorems3.2 and3.3, we can prove the following two results.
Theorem4.4. LetEbe a uniformly convex and uniformly smooth Banach space, let{Ci}mi=1
be a finite family of closed convex subsets ofEsuch that∩mi=1Ciis nonempty, letΠibe the generalized projection fromEontoCifor eachi=1, 2,...,m, and let{αn}be a sequence of real numbers such that0≤αn≤1andlim infn→∞αn(1−αn)>0. Suppose that{xn}is given byx0∈Eand
xn+1=J−1αnJxn+1−αn
JΠmΠm−1···Π2Π1xn
, n=0, 1, 2,.... (4.5) IfJ is weakly sequentially continuous, then {xn} converges weakly tou in∩mi=1Ci, where u=limn→∞Π∩mi=1CixnandΠ∩mi=1Ciis the generalized projection fromEonto∩mi=1Ci.
Proof. PutT=ΠmΠm−1···Π2Π1. It is clear thatF(T)⊂F(T) andˆ ∩mi=1Ci⊂F(T). By Lemma 4.3, we have that T is a relatively nonexpansive mapping and F(T)= ∩mi=1Ci. ApplyingTheorem 3.2,{xn}converges weakly tou=limn→∞Π∩mi=1Cixn. Theorem4.5. LetEbe a uniformly convex and uniformly smooth Banach space, let{Ci}mi=1
be a finite family of closed convex subsets ofE, letΠibe the generalized projection fromEonto Cifor eachi=1, 2,...,m, and let{αn}be a sequence of real numbers such that0≤αn≤1 andlim infn→∞αn(1−αn)>0. Suppose that{xn}is given byx0∈Eand
xn+1=J−1αnJxn+1−αn
JΠmΠm−1···Π2Π1xn
, n=0, 1, 2,.... (4.6) If the interior of ∩mi=1Ci is nonempty, then{xn} converges strongly touin∩mi=1Ci, where u=limn→∞Π∩mi=1CixnandΠ∩mi=1Ciis the generalized projection fromEonto∩mi=1Ci.
LetAbe a multivalued operator with the domainD(A)= {x∈E:Ax= ∅}and the graphG(A)= {(x,x∗)∈E×E∗:x∗∈Ax}. The operatorAis said to bemonotoneif
x−y,x∗−y∗≥0 for eachx,x∗,y,y∗∈G(A). (4.7) The operatorAis maximal monotone ifAis monotone, and for any monotone operatorB fromEtoE∗withG(A)⊂G(B), we haveA=B. We know that ifAis maximal monotone, thenA−10 is closed and convex. The following result is also well known.
Theorem4.6 (Rockafellar [15]). LetEbe a reflexive, strictly convex, and smooth Banach space and letAbe a monotone operator from Eto E∗. ThenAis maximal if and only if R(J+rA)=E∗for allr >0.
LetEbe a reflexive, strictly convex, and smooth Banach space and letAbe a maximal monotone operator fromEtoE∗. UsingTheorem 4.6and the strict convexity ofE, we obtain that for everyr >0 andx∈E, there exists a uniquexr∈D(A) such that
Jx∈Jxr+rAxr. (4.8)
IfJrx=xr, then we can define a single-valued mappingJr:E→D(A) byJr=(J+rA)−1J.
Such a Jr is called theresolvent of A. We know that Jr is relatively nonexpansive (see [10, 12, 14]), andA−10=F(Jr) for all r >0 (see [17,18]). As direct consequences of Theorems3.2and3.3, we also have the following two results.
Theorem4.7. LetEbe a uniformly convex and uniformly smooth Banach space, letAbe a maximal monotone operator fromEtoE∗such thatA−10is nonempty, letJrbe the resolvent ofA, where r >0, and let {αn}be a sequence of real numbers such that0≤αn≤1and lim infn→∞αn(1−αn)>0. Suppose the sequence{xn}is given byx0∈Eand
xn+1=J−1αnJxn+1−αn JJrxn
, n=0, 1, 2,.... (4.9) IfJ is weakly sequentially continuous, then{xn}converges weakly touinA−10, whereu= limn→∞ΠA−10xnandΠA−10is the generalized projection fromEontoA−10.
Theorem4.8. LetEbe a uniformly convex and uniformly smooth Banach space, letAbe a maximal monotone operator fromEtoE∗, letJr be the resolvent ofA, wherer >0, and let{αn}be a sequence of real numbers such that0≤αn≤1andlim infn→∞αn(1−αn)>0.
Suppose the sequence{xn}is given byx0∈Eand xn+1=J−1αnJxn+1−αn
JJrxn
, n=0, 1, 2,.... (4.10) If the interior ofA−10is nonempty, then{xn}converges strongly touinA−10, whereu= limn→∞ΠA−10xnandΠA−10is the generalized projection fromEontoA−10.
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Shin-ya Matsushita: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan
E-mail address:[email protected]
Wataru Takahashi: Department of Mathematical and Computing Sciences, Tokyo Institute of Tech- nology, Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan
E-mail address:[email protected]
