Hindawi Publishing Co po a ion
Fixed Poin Theo y and Applica ions
Volume 2010, A icle ID 418030, 8pages
doi:10.1155/2010/418030
Resea ch A icle
B owde ’s Con e gence o Uni o mly
Asymp o ically Regula Nonexpansi e Semig oups
in Hilbe Spaces
Gena o L ´
opez Acedo1and Tomona i Suzuki2
1Depa amen o de An´
alisis Ma em´
a ico, Facul ad de Ma em´
a icas, Uni e sidad de Se illa,
41080 Se illa, Spain
2Depa men o Ma hema ics, Kyushu Ins i u e o Technology, Toba a, Ki akyushu 804-8550, Japan
Co espondence should be add essed o Gena o L´
opez Acedo, [email p o ec ed]
Recei ed 6 Oc obe 2009; Accep ed 14 Oc obe 2009
Academic Edi o : Tomas Dominguez Bena ides
Copy igh q2010 G. L´
opez Acedo and T. Suzuki. 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 gi e a sufficien and necessa y condi ion conce ning a B owde ’s con e gence ype heo em
o uni o mly asymp o ically egula one-pa ame e nonexpansi e semig oups in Hilbe spaces.
1. In oduc ion
Le Cbe a closed con ex subse o a Hilbe space E. A mapping Ton Cis called a
nonexpansi e mapping i Tx −Ty≤x−y o all x, y ∈C. We deno e by FT he se
o ixed poin s o T. B owde , see 1, p o ed ha FTis nonemp y p o ided ha Cis, in
addi ion, bounded. Ki k in a e y celeb a ed pape , see 2, ex ended his esul o he se ing
o e lexi e Banach spaces wi h no mal s uc u e.
B owde 3ini ia ed he in es iga ion o an implici me hod o app oxima ing ixed
poin s o nonexpansi e sel -mappings de ined on a Hilbe space. Fix u∈C, he s udied he
implici i e a i e algo i hm
z  u 1− Tz .1.1
Namely, z , ∈0,1, is he unique ixed poin o he con ac ion x→ u 1− Tx,x∈C.
B owde p o ed ha lim →0z Pu, whe e Puis he elemen o FTnea es o u. Ex ensions
o he amewo k o Banach spaces o B owde ’s con e gence esul s ha e been done by
many au ho s, including Reich 4, Takahashi and Ueda 5, and O’Ha a e al. 6.
2 Fixed Poin Theo y and Applica ions
A amily o mappings {T : ≥0}is called a one-pa ame e s ongly con inuous
semig oup o nonexpansi e mappings nonexpansi e semig oup, o sho on Ci he ollowing
a e sa is ied.
NS1Fo each ≥0, T is a nonexpansi e mapping on C.
NS2Ts Ts◦T  o all s, ≥0.
NS3Fo each x∈C, he mapping → T x om 0,∞in o Cis s ongly con inuous.
The e a e many pape s conce ning he exis ence o common ixed poin s o {T : ≥0};
see, o ins ance, 7–13. As a ma e o ac , B owde 8p o ed ha i Cis bounded, hen
 ≥0FT  is nonemp y.
B owde ’s ype con e gence heo em o nonexpansi e semig oups is p o ed in 11,
14–18and o he s. Fo example, he ollowing heo em is p o ed in 17.
Theo em 1.1 see 17.Le Cbe a closed con ex subse o a Hilbe space E.Le {T : ≥0}
be a nonexpansi e semig oup on Csuch ha  ≥0FT  /
∅.Le {αn}and { n}be sequences in R
sa is ying
C10<α
n<1and 0≤ n;
C2limn nlimnαn/ n0,whe e1/0∞.
Fix u∈Cand de ine a sequence {xn}in Cby
xnαnu1−αnT nxn.1.2
Then {xn}con e ges s ongly o he elemen o  ≥0FT  nea es o u.
We no e ha C1is needed o de ine {xn}.
A nonexpansi e semig oup {T : ≥0}on Cis said o be uni o mly asymp o ically
egula u.a. .i o e e y ≥0 and o e e y bounded subse Ko C,
lim
s→∞sup
x∈K
Ts x−Tsx01.3
holds. The ollowing is p o ed by Dom´
ınguez Bena ides e al. 16;seealso15.
Theo em 1.2 see 16.Le E,C, and {T : ≥0}be as in Theo em 1.1. Assume ha {T : ≥
0}is u.a. . Le {αn}and { n}be sequences in Rsa is ying (C1) and
D2limnαn0and limn n∞.
Fix u∈Cand de ine a sequence {xn}in Cby 1.2.Then{xn}con e ges s ongly o he elemen o
 ≥0FT  nea es o u.
The e is an in e es ing diffe ence be ween Theo ems 1.1 and 1.2, ha is,{ n}in
Theo em 1.1 con e ges o 0 and { n}in Theo em 1.2 di e ges o ∞. By he way, e y ecen ly,
Akiyama and Suzuki 14gene alized Theo em 1.1. They eplaced C2o Theo em 1.1 by
Fixed Poin Theo y and Applica ions 3
he ollowing:
C2{ n}is bounded;
C3limnαn/ n−τ0 o all τ∈0,∞.
They also showed ha he conjunc ion o C2and C3is bes possible; see also 18.
In his pape , mo i a ed by he p e ious conside a ions, we gene alize Theo em 1.2
conce ning {αn}and { n}. Also, we will show ha ou new condi ion is bes possible.
2. Main Resul s
We deno e by N he se o all posi i e in ege s and by R he se o all eal numbe s. Fo ∈R,
we deno e by   he maximum in ege no exceeding .
The ollowing p oposi ion plays an impo an ole in his pape .
P oposi ion 2.1. Le Cbe a se o a sepa a ed opological ec o space E.Le {T : ≥0}be a amily
o mappings on Csuch ha Ts◦T Ts  o all s, ∈0,∞. Assume ha {T : ≥0}is
asymp o ic egula , ha is,
lim
s→∞
T sx−Tsx02.1
o all ∈0,∞and x∈C.Then
FT  
s≥0
FTs 2.2
holds o all ∈0,∞.
P oo . Fix ∈0,∞. I is ob ious ha FT  ⊃sFTs holds. Le z∈Cbe a ixed poin
o T . Fo e e y h∈0,∞, we ha e
Thz−zlim
n→∞
Th◦T nz−T nz
lim
n→∞
Thn z−Tn z
lim
s→∞
Thsz−Tsz
0,
2.3
and hence zis a common ixed poin o {T : ≥0}.
I is well known ha e e y Hilbe space has he Opial p ope y.
P oposi ion 2.2 Opial 19.Le Ebe a Hilbe space. Le {xn}be a sequence in Econ e ging
weakly o z0∈H. Then he inequali y lim in nxn−z≤lim in nxn−z0implies zz0.
We gene alize Theo em 1.2.
4 Fixed Poin Theo y and Applica ions
Theo em 2.3. Le Cbe a closed con ex subse o a Hilbe space E.Le {T : ≥0}be a
u.a. . nonexpansi e semig oup on Csuch ha  ≥0FT  /
∅.Le {αn}and { n}be sequences in
Rsa is ying (C1) and
D2limnαnlimnαn/ n0.
Fix u∈Cand de ine a sequence {xn}in Cby 1.2.Then{xn}con e ges s ongly o he elemen o
 ≥0FT  nea es o u.
P oo . Pu FT ≥0FT .Le be he elemen o FTnea es o u. Since
xn− 1−αnT nxnαnu− 
≤1−αnT nxn− αnu− 
≤1−αnxn− αnu− ,
2.4
we ha e xn− ≤u− . The e o e {xn}is bounded. Hence {T xn:n∈N, ≥0}is also
bounded.
We pu
M:sup{T xn−u:n∈N, ≥0}<∞.2.5
Le { n}be an a bi a y subsequence o {n}. Then he e exis s a subsequence {gn}o {n}
such ha {x ◦gn}con e ges weakly o x. We choose a subsequence {hn}o {n}such ha
τ:lim
n→∞ ◦g◦hnlim sup
n→∞
◦gn.2.6
Pu yjx ◦g◦hj,βjα ◦g◦hj,and sj ◦g◦hj. We will show x∈FT, di iding he
ollowing h ee cases:
iτ∞,
ii0<τ<∞,
iiiτ0.
In he i s case, we ix ≥0. Fo sufficien ly la ge j∈N, we ha e
T x−yj≤T x−T yjT yj−yj
≤x−yjβjT yj−u1−βjT yj−Tsjyj
≤x−yjβjM1−βjTsj− yj−yj
≤x−yjβjM1−βjβjTsj− yj−u1−βj2Tsj− yj−Tsjyj
≤x−yjβj2−βjM1−βj2Tsj−  yj−Tsj− yj,
2.7
Fixed Poin Theo y and Applica ions 5
and hence
lim in
j→∞ 
T x−yj
≤lim in
j→∞ 
x−yj
.2.8
By he Opial p ope y, we ob ain T xx.Thusx∈FT.
In he second case, we ha e
Tτx−yj≤Tτx−TsjxTsjx−TsjyjTsjyj−yj
≤Tτx−Tsjxx−yjβjTsjyj−u
≤Tτ−sjx−T0xx−yjβjM,
2.9
and hence
lim in
j→∞ 
Tτx−yj
≤lim in
j→∞ 
x−yj
.2.10
By he Opial p ope y, we ob ain Tτxx.ByP oposi ion 2.1,weob ainx∈FT.
In he hi d case, we ix ≥0. Fo sufficien ly la ge j∈N, we ha e
T x−yj≤T x−T /sjsjxT /sjsjx−T /sjsjyj

 /sj−1

k0
Tksjyj−Tk1sjyjT0yj−yj
≤T − /sjsjx−T0xx−yj
 /sjTsjyj−yjT0yj−TsjyjTsjyj−yj
≤T − /sjsjx−T0xx−yj
 /sjTsjyj−yjyj−TsjyjTsjyj−yj
T − /sjsjx−T0xx−yj /sj2Tsjyj−yj
T − /sjsjx−T0xx−yj /sj2βjTsjyj−u
≤maxTsx−T0x:0≤s≤sjx−yj βj/sj2βjM.
2.11
Hence 2.8holds. Thus we ob ain x∈FT.
We nex p o e ha {yj}con e ges s ongly o . Since
βj
yj− 

21−βjyj−Tsjyj− −Tsj ,y
j− 
βju− ,yj− ,
yj−Tsjyj− −Tsj ,y
j− 
≥yj− 2−Tsjyj−Tsj yj− ≥0,
2.12

6 Fixed Poin Theo y and Applica ions
we ob ain yj− 2≤u− , yj− . Since u− ,x − ≤0, we ha e

yj− 

2≤u− , yj− 
u− ,yj−xu− ,x − 
≤u− ,yj−x,
2.13
and hence {yj}con e ges s ongly o . Since {x n}is a bi a y, we ob ain ha {xn}
con e ges s ongly o .
Using 20, Theo em 7, we ob ain he ollowing Mouda i’s ype con e gence heo em;
see 21.
Co olla y 2.4. Le E,C,{T : ≥0},{αn},and { n}be as in Theo em 2.3.Le Φbe a con ac ion
on C; ha is, he e exis s ∈0,1such ha Φx−Φy≤ x−y o x, y ∈C. De ine a sequence
{xn}in Cby
xnαnΦxn1−αnT nxn.2.14
Then {xn}con e ges s ongly o he unique poin z∈Csa is ying P◦Φzz,whe ePis he me ic
p ojec ion om Con o  ≥0FT .
We will show ha D2is bes possible.
Example 2.5. Pu E2N, ha is,Eis a Hilbe space consis ing o all he unc ions x om
Nin o Rsa is ying k∈N|xk|2<∞wi h inne p oduc x, yk∈Nxkyk. De ine a
bounded closed con ex subse Co Eby
Cx∈E:0≤xk≤pk,2.15
whe e pk2−k/2. De ine a u.a. . nonexpansi e semig oup {T : ≥0}on Cby
T xkmaxxk− pk2,0.2.16
Le {ek}be he canonical basis o Eand pu u∞
k1pkek.Le {αn}and { n}be sequences in
Rsa is ying C1and de ine {xn}in Cby 1.2. Then {xn}con e ges o a common ixed poin
o {T : ≥0}only i limnαnlimnαn/ n0.
P oo . Fo α∈0,1and ≥0, we de ine xα, by
xα, αu 1−αT xα, .2.17
Fixed Poin Theo y and Applica ions 7
We no e
xα, k⎧
⎪
⎨
⎪
⎩
αpk,i α≤ pk,
1 pk− pk
αpk,i α≥ pk.
2.18
So, xα, k≥αpk. I is ob ious ha  ≥0FT   {0}. We assume limnxnlimnxαn,
n
Pu 0. Then
0lim
n→∞
xn1
p1
≥lim
n→∞αn.2.19
A guing by con adic ion, we assume lim supnαn/ n>0. Then he e exis κ∈Nand a
subsequence { n}o {n}such ha
α n
n
≥2pκ.2.20
Since limnx nκ0, we ha e
0lim
n→∞
x nκ
pκ
lim
n→∞
1 npκ− npκ
α n
≥lim sup
n→∞ 1− npκ
α n≥1
2>0,
2.21
which is a con adic ion. The e o e we ob ain limnαn/ n0.
By Theo em 2.3 and Example 2.5, we ob ain he ollowing.
Theo em 2.6. Le Ebe an in ini e-dimensional Hilbe space. Le {αn}and { n}be sequences in R
sa is ying (C1). Then he ollowing a e equi alen :
ilimnαnlimnαn/ n0,
iii Cis a bounded closed con ex subse Co E,{T : ≥0}is a u.a. . nonexpansi e
semig oup on C,u∈C, and {xn}is a sequence in Cde ined by 1.2, hen{xn}con e ges
s ongly o he elemen o  ≥0FT  nea es o u.
Compa e D2wi h he conjunc ion o C2and C3. We can ell ha he diffe ence
be ween bo h condi ions is u.a. .
Acknowledgmen s
The i s au ho was pa ially suppo ed by DGES, G an MTM2006-13997-C02-01 and Jun a
de Andaluc´
ıa, G an FQM-127. The second au ho is suppo ed in pa by G an s-in-Aid o
Scien i ic Resea ch om he Japanese Minis y o Educa ion, Cul u e, Spo s, Science and
Technology.
8 Fixed Poin Theo y and Applica ions
Re e ences
1F. E. B owde , “Fixed-poin heo ems o noncompac mappings in Hilbe space,” P oceedings o he
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