Hindawi Publishing Co po a ion
Fixed Poin Theo y and Applica ions
Volume 2010, A icle ID 169837, 16 pages
doi:10.1155/2010/169837
Resea ch A icle
Fixed Poin s o Single- and Se -Valued
Mappings in Uni o mly Con ex Me ic Spaces
wi h No Me ic Con exi y
Ra a Esp´
ınola,1Au o a Fe n ´
andez-Le ´
on,1and Bo˙
zena Pia¸ ek2
1Depa amen o de An´
alisis Ma em´
a ico, Uni e sidad de Se illa, P.O. Box 1160, 41080 Se illa, Spain
2Ins i u e o Ma hema ics, Silesian Uni e si y o Technology, 44-100 Gliwice, Poland
Co espondence should be add essed o Ra a Esp´
ınola, [email p o ec ed]
Recei ed 20 Ap il 2009; Accep ed 28 May 2009
Academic Edi o : Mohamed A. Khamsi
Copy igh q2010 Ra a Esp´
ınola e al. This is an open access a icle dis ibu ed unde he C ea i e
Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in
any medium, p o ided he o iginal wo k is p ope ly ci ed.
We s udy he exis ence o ixed poin s and con e gence o i e a es o asymp o ic poin wise
con ac ions in uni o mly con ex me ic spaces. We also s udy he exis ence o ixed poin s o se -
alued nonexpansi e mappings in he same class o spaces. Ou esul s do no assume con exi y o
he me ic which makes a big diffe ence when s udying he exis ence o ixed poin s o se - alued
mappings.
1. In oduc ion
This pape is mo i a ed by he ecen pape 1.In1 he au ho s s udy diffe en ques ions
ela ed o ixed poin s o asymp o ic poin wise con ac i e/nonexpansi e mappings in
CAT0spaces. CAT0spaces a e s udied in 1as a e y signi ican example wi hin he
class o uni o mly con ex me ic spaces  he eade can consul 2 o de ails on CAT0
spaces. In ou p esen pape we p opose o conside simila ques ions on uni o mly con ex
me ic spaces unde he mildes addi ional condi ions we may impose. Mo e p ecisely, we
will wo k wi h uni o mly con ex me ic spaces wi h ei he a mono one modulus o con exi y
in he sense i s gi en in 3o a lowe semicon inuous om he igh modulus o con exi y
see Sec ion 2 o p ope de ini ions. Fo a ecen su ey on he exis ence o ixed poin s
in geodesic spaces, he eade may check 4, o ecen achie emen s on ela ed opics he
eade may also check 5.
The no ion o asymp o ic poin wise con ac ions was in oduced in 6. Then i was
also s udied in 7whe e, by means o ul apowe echniques, diffe en esul s abou he
2 Fixed Poin Theo y and Applica ions
exis ence o ixed poin s and con e gence o i e a es we e p o ed. In 8new p oo s we e
p esen ed bu his ime a e applying only elemen a y echniques. Ve y ecen ly, in 1, hese
echniques we e applied in CAT0, whe e he au ho s a end o he B uha -Ti s inequali y o
CAT0spaces in o de o ob ain such esul s. In he p esen pape we show ha ac ually mos
o hose esul s s ill hold o gene al uni o mly con ex me ic spaces unde mild condi ions
on he modulus o con exi y. In Sec ion 3 we ocus on single- alued mappings and, in
pa icula , on mappings which a e asymp o ically poin wise con ac i e/nonexpansi e o
s udy he exis ence o ixed poin s, con e gence o Pica d’s i e a es, and he s uc u e o
hei se s o ixed poin s. As a echnical esul we need o show ha bounded sequences
in hese spaces ha e a unique asymp o ic cen e which, as a by-p oduc , leads o Ki k’s Fixed
Poin Theo em. In Sec ion 4 we s udy diffe en p oblems ega ding se - alued mappings
in hese spaces. The main echnical difficul y o achie e simila esul s o hose shown in
1is ha now we canno coun on he exis ence o ixed poin s o nonexpansi e se -
alued mappings o he kind o spaces we deal wi h. Finding ixed poin o se - alued
nonexpansi e mappings in uni o mly con ex me ic spaces was i s s udied by Shimizu and
Takahashi 9, whe e he exis ence o ixed poin s was gua an eed unde s onge condi ions
on he modulus o con exi y and he addi ional condi ion o me ic con exi y o he space.
The ac ha we do no ha e ha he me ic a e con ex will make he p oblem mo e
complica ed and his will ake us o impose new condi ions on he modulus o con exi y
which we will ela e wi h he geome y o he space.
2. Basic De ini ions and Resul s
We in oduce nex some basic de ini ions.
De ini ion 2.1. Le X, dbe a me ic space. A mapping T:X→Xis called a poin wise
con ac ion i he e exis s a mapping α:X→0,1such ha
dTx,Ty≤αxdx,y2.1
o any y∈X.
I is p o ed in 8see also 6 ha i Kis a weakly compac con ex subse o a
Banach space and T:K→Kis a poin wise con ac ion, hen Thas a unique ixed poin and
he sequence o he i e a es o Tcon e ges o he ixed poin o any x∈K. As i is poin ed ou
in 1, he uniqueness o ixed poin s and con e gence o i e a es o hese mappings di ec ly
ollow i exis ence is gua an eed.
De ini ion 2.2. Le X, dbe a me ic space. Le T:X→Xbe a mapping, and le αn:X→
0,∞ o each n∈Nbe such ha
dTnx,Tny≤αnxdx, y o any y∈X.2.2
Then
iTis called an asymp o ic poin wise con ac ion i {αn}con e ges poin wise o α:
X→0,1;
Fixed Poin Theo y and Applica ions 3
iiTis called an asymp o ic poin wise nonexpansi e mapping i lim sup αnx≤1 o
any x∈X;
iiiTis called a s ongly asymp o ic poin wise con ac ion i lim sup αnx≤k,wi h
0<k<1, o any x∈X.
In his pape we will mainly wo k wi h uni o mly con ex geodesic me ic space. Since
he de ini ion o uni o m con exi y equi es he exis ence o midpoin s, he wo d geodesic is
edundan and so, o simplici y, we will usually omi i .
De ini ion 2.3. A geodesic me ic space X, dis said o be uni o mly con ex i o any >0and
any ε∈0,2 he e exis s δ∈0,1such ha o all a, x, y ∈Xwi h dx,a≤ ,dy,a≤
and dx,y≥ε i is he case ha
dm, a≤1−δ , 2.3
whe e ms ands o any midpoin o any geodesic segmen x, y. A mapping δ:0,∞×
0,2→0,1p o iding such a δδ , ε o a gi en >0andε∈0,2is called a modulus
o uni o m con exi y.
No ice ha his de ini ion o uni o m con ex me ic spaces is weake han he one
used in 9in wo ways. Fi s , we do no impose ha he me ic is con ex and, second, ou
modulus o con exi y does depend on he wo a iables and εwhile i is assumed o depend
only on εin 9.
De ini ion 2.4. Le X, dbe a me ic space, hen he me ic is said o be con ex i o any x, y
and zin X,andma midpoin in be ween xand y,
dz, m≤1/2dz, xdz, y.2.4
I is easy o see ha uni o mly con ex me ic spaces a e uniquely geodesic, ha is, o
each wo poin s he e is jus one geodesic joining hem. The e o e midpoin s and geodesic
segmen s x,yjoining wo poin s a e unique. In his case he e is a na u al way o de ine
con exi y. A subse Co a uniquelygeodesic space is said o be con ex i x, y⊆C o any
x,y ∈C. Fo mo e abou geodesic spaces he eade may check 2.
To ob ain ou esul s we will need o impose addi ional condi ions on he modulus
o con exi y. Following 3,10we conside he no ion o mono one modulus o con exi y as
ollows.
De ini ion 2.5. I a uni o mly con ex me ic space Xadmi s a modulus o con exi y δsuch
ha i dec eases wi h  o each ixed ε hen we say ha δis a mono one modulus o
con exi y o X.
In he same way we de ine a lowe semicon inuous om he igh modulus o
con exi y as ollows.
De ini ion 2.6. I a uni o mly con ex me ic space Xadmi s a modulus o con exi y δsuch
ha i is lowe semicon inuous om he igh wi h espec o  o each ixed ε hen we say
δis a lowe semicon inuous om he igh modulus o con exi y o X.
4 Fixed Poin Theo y and Applica ions
Le Xbe a me ic space and Fa amily o subse s o X. Then, ollowing 1, we say ha
Fde ines a con exi y s uc u e on Xi i con ains he closed balls and is s able by in e sec ion.
Le Xbe a me ic space and Fa con exi y s uc u e on X.Gi enΦ:X→0,∞,we
say ha Φis F-con ex i {x:Φx≤ }∈F o any ≥0.
I we conside a bounded sequence {xn}in X,wea eable ode inea unc ion ·,x
n,
called ype, such ha o each x
x,xnlim sup
n→∞
dxn,x.2.5
The asymp o ic cen e o a bounded sequence wi h espec o a subse Co Xis hen de ined as
AC{xn}x∈X: x,xn≤ y,xn o any y∈C.2.6
I he asymp o ic cen e is aken wi h espec o X hen i is simply deno ed by A{xn}.
De ini ion 2.7. We say ha a con exi y s uc u e is T-s able i ypes a e F-con ex.
In 1 he ollowing de ini ion o compac ness o con exi y s uc u e was conside ed.
De ini ion 2.8. Gi en Fa con exi y s uc u e, we will say ha Fis compac i any amily
Aαα∈Γo elemen s o Fhas nonemp y in e sec ion p o ided ∩α∈F/
∅ o any ini esubse
F⊂Γ.
In ou pape we will a he use he idea o compac ness gi en in 11. No ice ha his
second no ion o compac ness is weake han he p e ious one.
De ini ion 2.9. Gi en Fa con exi y s uc u e, we will say ha Fis nes ed compac i any
dec easing chain Aαα∈Γo nonemp y bounded elemen s o Fhas nonemp y in e sec ion.
A e y impo an p ope y gi en in 3abou comple e uni o mly con ex me ic
spaces wi h mono one modulus o con exi y is ha dec easing sequences o nonemp y
bounded closed and con ex subse s o hese spaces ha e nonemp y in e sec ion. As a
consequence, we ha e ha i Fs ands o he collec ion o nonemp y closed and con ex
subse s o a comple e uni o mly con ex me ic space wi h mono one modulus o con exi y,
hen Fis a nes ed compac con exi y s uc u e.
Rema k 2.10. I is no ha d o see ha he same emains ue i he mono one condi ion on he
modulus is eplaced by lowe semicon inui y om he igh .
3. Asymp o ic Poin wise Con ac ions in Uni o mly Con ex
Me ic Spaces
In his sec ion we gi e diffe en esul s o he abo e de ined mappings in uni o mly con ex
me ic spaces. Al hough, o exposi o y easons, ou esul s will be usually p o ed only o
uni o mly con ex me ic spaces wi h a mono one modulus o con exi y, hey also hold when
he e is a lowe semicon inuous modulus o con exi y. Some indica ions abou diffe ences in
bo h cases will be gi en. We begin wi h a echnical esul .
Fixed Poin Theo y and Applica ions 5
P oposi ion 3.1. Le X, dbe a comple e uni o mly con ex me ic space wi h a mono one (o lowe
semicon inuous om he igh ) modulus o con exi y δ , ε. Conside he amily Fo all nonemp y
closed and con ex subse s o X.ThenFde ines a nes ed compac and T-s able con exi y s uc u e on
X.
P oo . I only emains o be p o ed ha Fis T-s able. Le {xn}be a bounded sequence in X
and conside he ype de ined by {xn}. We need o show ha C {x: x,xn≤ }∈F o
any posi i e . I is immedia e o see ha C is closed and nonemp y. To see ha C ∈Fis also
con ex, conside xand y o be wo diffe en poin s in C . The e is no es ic ion i we assume
ha lim sup dy,xn≤lim sup dx, xn 1≤ .Le mbe he midpoin o he segmen x,y
and ake ε1dx, y/ 1, hen, by uni o m con exi y, we ha e ha
dm, xn≤1−δmaxdx,xn,dy,xn,ε
1maxdx, xn,dy,xn
<maxdx, xn,dy,xn,
3.1
and so,
lim sup dm, xn≤lim sup maxdx, xn,dy,xn 1≤ . 3.2
Hence, m∈C .
The ollowing heo ems we e p o ed in 1unde he hypo hesis o compac ness on
he con exi y s uc u e. We s a e i , howe e , unde he hypo hesis o nes ed compac ness
since his is all i is ac ually equi ed in he p oo s gi en in 1.
Theo em 3.2. Le Xbe a bounded me ic space. Assume ha he con exi y s uc u e AMis nes ed
compac . Le T:X→Xbe a poin wise con ac ion. Then Thas a unique ixed poin x0. Mo eo e
he o bi {Tnx}con e ges o x0, o each x∈X.
Theo em 3.3. Le Xbe a bounded me ic space. Assume ha he con exi y s uc u e AMis nes ed
compac . Le T:X→Xbe a s ongly asymp o ic poin wise con ac ion. Then Thas a unique ixed
poin x0. Mo eo e he o bi {Tnx}con e ges o x0, o each x∈X.
Now he nex co olla y ollows.
Co olla y 3.4. The abo e heo ems hold o comple e bounded uni o mly con ex me ic spaces wi h
ei he mono one o lowe semicon inuous om he igh modulus o con exi y.
The ollowing lemma is immedia e.
Lemma 3.5. Le Xbe a me ic space and Fa nes ed compac con exi y s uc u e on Xwhich is
T-s able. Then o any ype ·,x
n, he eexis sx0∈Xsuch ha
x0,x
nin { x, xn:x∈X}.3.3
As a di ec consequence o P oposi ion 3.1 and he p e ious lemma we ge he
ollowing esul o asymp o ic poin wise con ac ions. We omi he de ails o i s p oo as
i ollows simila pa e ns as in 1, Theo em 4.2.

6 Fixed Poin Theo y and Applica ions
Theo em 3.6. Le X, dbe a comple e uni o mly con ex me ic space wi h a mono one (o lowe
semicon inuous om he igh ) modulus o con exi y δ , ε. Suppose Xis bounded. Then e e y T:
X→Xasymp o ic poin wise con ac ion has a unique ixed poin x0. Mo eo e , he o bi {Tnx}
con e ges o x0 o each x∈X.
Nex we show some consequences o P oposi ion 3.1 and Lemma 3.5. The cases
o mono one and lowe semicon inuous om he igh modulus o con exi y a e shown
sepa a ely as hey equi e diffe en p oo s.
Co olla y 3.7. Le Xbe a comple e uni o mly con ex me ic space wi h a mono one modulus o
con exi y and {xn}a bounded sequence in X. Then he se o asymp o ic cen e s o {xn}is a single on.
P oo . Le uand be wo diffe en poin s in A{xn},and le mbe he midpoin o u, .Le
 u, xn u, xn,c 1,and ε1du, /c. By he uni o m con exi y, he e exis s
N∈Nsuch ha o e e y n≥N,
dm, xn≤1−δmax{du, xn,d , xn},ε
1 max{du, xn,d , xn}
≤1−δc,ε1 max{du, xn,d ,xn}.
3.4
I we le ngo o in ini e, we ob ain ha m, xn≤1−δc,ε1 < ,which is clea ly a
con adic ion.
Rema k 3.8. This co olla y has been i s p o ed in 12, P oposi ion 3.3 o a ce ain class o
uni o mly con ex hype bolic spaces wi h mono one modulus o con exi y.
Now we show he lowe semicon inuous case.
Co olla y 3.9. Le Xbe a comple e uni o mly con ex me ic space wi h a lowe semicon inuous om
he igh modulus o con exi y and {xn}a bounded sequence in X. Then he se o asymp o ic cen e s
o {xn}is a single on.
P oo . Le uand be wo diffe en poin s in A{xn}and le mbe he midpoin o u, .Le
 u, xn  ,xn,εdu, / 1,and le us ix p∈N. Then max{du, xn,d ,xn}≤
p−1 o each nla ge enough. By he uni o m con exi y,
dm, xn≤1−δ p−1,ε p−13.5
o he same nas abo e and inally
m, xn≤1−δ p−1,ε p−1.3.6
Now i suffices o obse e ha
δ p−1,ε≥1
2δ , ε,
1−δ p−1,ε≤1−1
2δ , ε.
3.7
Fixed Poin Theo y and Applica ions 7
o pla ge enough. Combining i wi h 3.6and aking limp→∞ we ob ain m, xn< as in
he o me co olla y, and hus he con adic ion.
Ano he consequence is Ki k Fixed Poin Theo em in uni o mly con ex me ic spaces.
Co olla y 3.10. Le Xbe a comple e uni o mly con ex geodesic me ic space wi h a mono one
(o lowe semicon inuous om he igh ) modulus o con exi y. Suppose Xis bounded, hen any
nonexpansi e mapping T:X→Xhas a ixed poin .
P oo . Conside x∈Xand {Tnx} he sequence o i s i e a es. Le ωbe he only asymp o ic
cen e o {Tnx}in X. Then, by he nonexpansi eness o T, i ollows ha Tω,Tnx ≤
ω,Tnx and so, Tωω.
Now we p esen a coun e pa o 1, Theo em 5.1.
Theo em 3.11. Le X, dbe a comple e uni o mly con ex me ic space wi h a mono one (o lowe
semicon inuous om he igh ) modulus o con exi y δ , ε.Le Cbe a bounded closed con ex
nonemp y subse o X. Then any T:C→Casymp o ic poin wise nonexpansi e mapping has a
ixed poin , and he se o ixed poin s o T,FixT, is closed and con ex.
P oo . Le x∈Cand conside xnTnx.F omCo olla y 3.7, we know ha AC{xn}is a
single on. Le ωbe he only poin in ha se , ha is, ωis such ha ω, xnin { u, xn:u∈
C}. We wan o show ha {Tmω}is a Cauchy sequence. Suppose his is no he case. Then
he e exis s a sepa a ed subsequence {Tmiω}o {Tmω}, ha is, he e exis s ε>0 such ha
dTmkω,Tmhω ≥ε o e e y k/
hin N.
Le mkh be he midpoin o he segmen Tmkω,Tmhω,cdiamCand ε1ε/c.
The uni o m con exi y o he space, oge he wi h i s mono one cha ac e , implies ha o
e e y kand hin N
dmkh,x
n≤1−δmax{dTmhω,x
n,dTmkω,x
n},ε
1 max{dTmhω,x
n,dTmkω,x
n
}
≤1−δc,ε1 max{dTmhω,x
n,dTmkω,x
n}.
3.8
No ice ha , by de ini ion o T,
Tmω,x
n≤αmω ω,xn.3.9
Then, i we le ngo o in ini y,
ω,xn≤ mkh,x
n
≤1−δc,ε1 max{ Tmkω,x
n, Tmhω,x
n}
≤1−δc,ε1 max{αmkω ω,xn,α
mhω ω,xn}.
3.10
Since Tis poin wise asymp o ic nonexpansi e, hen
ω,xn≤1−δc,ε1 ω, xn,3.11
8 Fixed Poin Theo y and Applica ions
and so ω,xn0, which is a con adic ion since, in i ue o 3.9, hisimplies ha Tmω
con e ges o ω. The e o e, {Tmω}is a Cauchy sequence and i s limi , again by 3.9,isω.
Then, om he con inui y o T,Tωω.
In consequence, FixTis nonemp y. Now, since Tis con inuous, FixTis closed. We
show nex ha FixTis also con ex. Le u, be wo diffe en poin s in FixTand w he
midpoin o he segmen u, . We need o show ha w∈FixT. Now, since Tis poin wise
asymp o ic nonexpansi e,
du, Tnw dTnu,Tnw ≤αnwdu, wαnwdu, 
23.12
and, equally,
d ,Tnw dTn ,Tnw ≤αnwd ,wαnwdu, 
2.3.13
The e o e, o ε>0, he e exis s n0such ha i n≥n0 hen
Tnw∈Bu, du, 
2ε∩B , du, 
2εDε,3.14
bu , om he p oo o P oposi ion 2.2 in 3, he diame e s o he se s Dε end o 0 as ε ends
o 0 and so lim Tnww, which p o es wis a ixed poin o T.
Rema k 3.12. The p oo o he lowe semicon inuous case ollows in a simila way bu
ollowing he easoning o Co olla y 3.9.
In 1a demiclosed p inciple is also gi en o asymp o ic poin wise nonexpansi e
mappings in CAT0spaces. Nex we show ha an equi alen esul is also possible o
uni o mly con ex me ic spaces. Following 1we de ine
{xn}Cωi and only i ω, xnin
x∈C x,xn,3.15
whe e Cis a closed and con ex subse o a uni o mly con ex me ic space con aining he
bounded sequence {xn}. No ice ha his de ini ion does no depend on he se Cwhen he
space Xis a comple e CAT0space. This is due o he ac ha he asymp o ic cen e o
a bounded sequence o a comple e CAT0space belongs o he closed con ex hull o he
sequence, which easily ollows om he e y well-known ac ha he me ic p ojec ion on o
closed con ex subse s o a comple e CAT0space is nonexpansi e see 2 o de ails. Recall
ha he exis ence and uniqueness o such a ω∈Cin a comple e uni o mly con ex me ic
spaces wi h mono one modulus o con exi y is gua an eed by Co olla y 3.7.
P oposi ion 3.13. Le X, dbe a comple e uni o mly con ex me ic space wi h a mono one modulus
o con exi y δ , ε.Le Cbe a bounded closed con ex nonemp y subse o X.Le T:C→Can
asymp o ic poin wise nonexpansi e mapping. Le {xn}∈Cbe an app oxima e ixed poin sequence,
ha is, limn→∞dxn,Txn  0, and such ha xnω o a ce ain ω∈C.ThenTωω.
Fixed Poin Theo y and Applica ions 9
P oo . Since {xn}is an app oxima e ixed poin sequence, hen we ha e ha
x,xnlim sup
n→∞
dx,Tmxn  x,Tmxn 3.16
o any m≥1see No e Added in P oo a he end o he pape . In consequence, since
Tmx,Tmxn ≤αmx x,xn o x∈C,3.9holds o any x.
The e o e, pa icula izing o ω, we ha e ha lim supm→∞ Tmω,x
n ω, xn.
Nowweclaim ha Tmω→ωas m→∞. Suppose on he con a y ha he e exis an ε>0
and a subsequence {Tmkω}o {Tmω}such ha dTmkω,ω≥ε o e e y k∈N.Le ωmk
be he midpoin o he geodesic segmen Tmkω,ω,cdiamCand ε1ε/c. By uni o m
con exi y, o e e y k, we ha e ha
dωmk,x
n≤1−δmax{dω, xn,dTmkω,x
n},ε
1 max{dω, xn,dTmkω,x
n}
≤1−δc,ε1 max{dω,xn,dTmkω,x
n}.
3.17
I we conside he uppe limi o he abo e inequali y when n→∞,wege
ω,xn≤ ωmk,x
n≤1−δc,ε1 max{ ω,xn, Tmkω,x
n}.3.18
I we do he same when k→∞, we inally ob ain ha ω, xn≤1−δc,ε1 ω,xn.
The e o e ω,xn0,and he exis ence o ixed poin ollows he same as in Theo em 3.11.
Rema k 3.14. The p oo o he lowe semicon inuous case ollows in a simila way bu
ollowing he easoning o Co olla y 3.9.
4. Fixed Poin s o Se -Valued Mappings
In his sec ion we p esen ixed poin s heo ems o se - alued mappings de ined on
uni o mly con ex me ic spaces. Resul s s a ed o uni o mly con ex me ic space wi h a
mono one modulus o con exi y also hold i he e is a lowe semicon inuous om he igh
modulus o con exi y. P oo s o his second case will be omi ed as hey a e based on echnical
esul s al eady p o ed o bo h kinds o modulus in Sec ion 3. The Hausdo ffme ic on he
closed and bounded pa s o a me ic space Xis de ined as ollows. I Uand Va e bounded
and closed subse s o a me ic space X, hen
HU, V in {ε>0:U⊆NεV,V⊆NεU},4.1
whe e NεV{y∈X:dis y,Vin {dy,x:x∈V}<ε}.Le Cbe a subse o a me ic
space X. A mapping T:C→2Xwi h nonemp y bounded closed alues is nonexpansi e i
HTx,Ty≤dx, y4.2
o all x, y ∈C. Ou main goal in his sec ion is o s udy i gi en Xis a bounded uni o mly
con ex me ic space wi h mono one modulus o con exi y, hen e e y nonexpansi e
16 Fixed Poin Theo y and Applica ions
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ol. 2006, A icle ID 43591, 10 pages, 2006.
20T. Zam i escu, “On he cu locus in Alexand o spaces and applica ions o con ex su aces,” Paci ic
Jou nal o Ma hema ics, ol. 217, no. 2, pp. 375–386, 2004.