RESEARCH Open Access
Composi ions and con ex combina ions o
asymp o ically egula i mly nonexpansi e
mappings a e also asymp o ically egula
Heinz H Bauschke
1*
, Vic o ia Ma ín-Má quez
2
, Sa ah M Mo a
1
and Xian u Wang
1
* Co espondence: heinz.
[email p o ec ed]
1
Ma hema ics, Uni e si y o B i ish
Columbia, Kelowna, BC V1V 1V7,
Canada
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
Because o Min y’s classical co espondence be ween i mly nonexpansi e mappings
and maximally mono one ope a o s, he no ion o a i mly nonexpansi e mapping
has p o en o be o basic impo ance in ixed poin heo y, mono one ope a o
heo y, and con ex op imiza ion. In his no e, we show ha i ini ely many i mly
nonexpansi e mappings de ined on a eal Hilbe space a e gi en and each o hese
mappings is asymp o ically egula , which is equi alen o saying ha hey ha e o
“almos ha e” ixed poin s, hen he same is ue o hei composi ion. This
signi ican ly gene alizes he esul by Bauschke om 2003 o he case o p ojec o s
(nea es poin mappings). The p oo esides in a Hilbe p oduc space and i elies
upon he B ezis-Ha aux ange app oxima ion esul . By wo king in a sui ably scaled
Hilbe p oduc space, we also es ablish he asymp o ic egula i y o con ex
combina ions.
2010 Ma hema ics Subjec Classi ica ion: P ima y 47H05, 47H09; Seconda y 47H10,
90C25.
Keywo ds: asymp o ic egula i y, i mly nonexpansi e mapping, Hilbe space, maxi-
mally mono one ope a o , nonexpansi e mapping, esol en , s ongly nonexpansi e
mapping
1 In oduc ion and s anding assump ions
Th oughou his a icle,
Xis a eal Hilbe s
p
ace wi h inne
p
oduc ·,·

(1)
and induced no m || ⋅||. We assume ha
m∈
{
2, 3, 4, ...
}
and I:=
{
1, 2, ...,m
}.
(2)
Recall ha an ope a o T: X ®Xis i mly nonexpansi e (see, e.g., [1-3] o u he
in o ma ion) i (∀xÎX)(∀yÎX)||Tx -Ty||
2
≤〈x-y, Tx -Ty〉and ha ase - alued
ope a o A: X ⇉Xis maximally mono one i i is mono one,i.e., o all(x, x*)and(y,
y*) in he g aph o A, we ha e 〈x - y, x* - y*〉≥0 and i he g aph o Acanno be p op-
e ly enla ged wi hou des oying mono onici y (We shall w i e dom A={xÎX|Ax≠
Ø} o he domain o A, anA=A(X)=∪
xÎX
Ax o he ange o A,andg A o he
g aph o A.) These no ions a e equi alen (see [4,5]) in he sense ha i Ais maximally
mono one, hen i s esol en J
A
: = (Id + A)
-1
is i mly nonexpansi e, and i Tis i mly
Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53
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© 2012 Bauschke e al; licensee Sp inge . This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (h p://c ea i ecommons.o g/licenses/by/2.0), 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.
nonexpansi e, hen T
-1
- Id is maximally mono one. (He e and elsewhe e, Id deno es
he iden i y ope a o on X.) The Min y pa ame iza ion (see [4] and also [[1], Rema k
23.22(ii)]) s a es ha i Ais maximally mono one, hen
g
A={
(
JAx,x−JAx
)
|x∈X}.(3)
In op imiza ion, one main p oblem is o ind ze os o maximally mono one ope a-
o s– hese ze os may co espond o c i ical poin s o solu ions o op imiza ion p o-
blems. In e ms o esol en s, he co esponding p oblem is ha o inding ixed
poin s. Fo backg ound ma e ial in ixed poin heo y and mono one ope a o heo y,
we e e he eade o [1-3,6-16].
The aim o his no e is o p o ide app oxima e ixed poin esul s o composi ions
and con ex combina ions o ini ely many i mly nonexpansi e ope a o s.
The i s main esul (Theo em 4.6) subs an ially ex ends a esul by Bauschke [17]
on he composi ions o p ojec o s o he composi ion o i mly nonexpansi e map-
pings. The second main esul (Theo em 5.5) ex ends a esul by Bauschke, Mo a and
Wang [18] on he con ex combina ion o i mly nonexpansi e ope a o s om Eucli-
dean o Hilbe space.
The emainde o his sec ion p o ides he s anding assump ions used h oughou
he a icle.
E en hough he main esul s a e o mula ed in he gi en Hilbe space X, i will u n
ou ha he key space o wo k in is he p oduc space
Xm:= {x=
(
xi
)
i∈I|
(
∀i∈I
)
xi∈X}
.
(4)
This p oduc space con ains an embedding o he o iginal space X ia he diagonal
subspace
:= {x=
(
x
)
i∈I|x∈X}
.
(5)
We also assume ha we a e gi en m i mly nonexpansi e ope a o s T
1
,..., T
m
;equi a-
len ly, m esol en s o maximally mono one ope a o s A
1
,..., A
m
:
(∀i∈I)Ti=JA
i
=(Id+Ai)−1is i mly nonexpansi e
.
(6)
We now de ine a ious pe inen ope a o s ac ing on X
m
. We s a wi h he Ca e-
sian p oduc ope a o s
T:Xm→Xm:
(
xi
)
i∈I→
(
Tixi
)
i∈
I
(7)
and
A:Xm⇒Xm:
(
xi
)
i∈I→
(
Aixi
)
i∈I
.
(8)
Deno ing he iden i y on X
m
by Id, we obse e ha
J
A=
(
Id +A
)
−1=T1×···×Tm=T
.
(9)
O cen al impo ance will be he cyclic igh -shi ope a o
R
:Xm→Xm:
(
x1,x2...,xm
)
→
(
xm,x1,...,xm−1
)
(10)
and o con enience we se
M=Id −R
.
(11)
Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53
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We also ix s ic ly posi i e con ex coe icien s (o weigh s) (l
i
)
iÎI
, i.e.,
(∀i∈I)λi∈]0, 1[ and

i∈
I
λi=1
.
(12)
Le us make X
m
in o he Hilbe p oduc space
X:=Xm,wi hx,y=
i∈I
xi,yi.(13)
The o hogonal complemen o Δwi h espec o his s anda d inne p oduc is
known (see, e.g., [[1], P oposi ion 25.4(i)]) o be
⊥=

x=(xi)i∈I|
i∈I
xi=0
.
(14)
Finally, gi en a nonemp y closed con ex subse Co X, hep ojec o (nea es poin
mapping) on o Cis deno ed by P
C
. I is well known o be i mly nonexpansi e.
2 P ope ies o he ope a o M
In his sec ion, we collec se e al use ul p ope ies o he ope a o M, including i s
Moo e-Pen ose in e se (see [19] and e.g., [[1], Sec ion 3.2] o u he in o ma ion.).
To ha end, he ollowing esul –which is p obably pa o he olklo e–will u n ou
o be use ul.
P oposi ion 2.1 Le Y be a eal Hilbe space and le B be a con inuous linea ope a-
o om X o Y wi h adjoin B*and such ha an B is closed. Then he Moo e-Pen ose
in e se o B sa is ies
B†=P
a
nB∗◦B−1◦P
a
nB
.
(15)
P oo Take yÎY. De ine he co esponding se o leas squa es solu ions (see, e.g.,
[[1], P oposi ion 3.25]) by C:=B
-1
(P
an B
y). Since an Bis closed, so is an B* (see, e.
g., [[1], Co olla y 15.34]); hence,
a
U:=
(
Ke B
)
⊥= an B∗= anB
∗
. Thus, C=B
†
y+ ke
B=B
†
y+U
┴
. The e o e, since an B
†
= anB* (see, e.g., [[1], P oposi ion 3.28( )]), P
U
(C)=P
U
B
†
y=B
†
y, as claimed.
Be o e we p esen a ious use ul p ope ies o M, le us ecall he no ion o a ec angula
(which is also known as s a o 3* mono one, see [20]) ope a o . A mono one ope a o
B: X⇉Xis ec angula i (∀(x,y∗)∈dom B× an B)sup
(
z,z∗
)
∈g Bx−z,z∗−y<+
∞
.
Theo em 2.2 De ine
b
L
:⊥→X:y→
m−1

i
=1
m−i
mRi−1y
.
(16)
Then he ollowing hold.
(i) Mis con inuous, linea , and maximally mono one wi h dom M=X.
(ii) Mis ec angula .
(iii) ke M= ke M*=Δ.
(i ) an M= an M*=Δ
┴
is closed.
( ) an L=Δ
┴
.
( i) M◦L=Id|
⊥
.
Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53
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( ii) M−1:X⇒X:y→

Ly +,i y∈⊥;
∅,o he wise
.
( iii) M†=P⊥◦L◦P⊥=L◦P
⊥
.
(ix) M†=
m

k
=1
m−(2k−1)
2mRk−
1
.
P oo .(i):Clea ly,domM=Xand (∀xÎX)||Rx|| = ||x||. Thus, Ris nonexpansi e
and he e o e M=Id -Ris maximally mono one (see, e.g., [[1], Example 20.27]).
(ii): See [[1], Example 24.14] and [[17], S ep 3 in he p oo o Theo em 3.1] o wo
di e en p oo s o he ec angula i y o M.
(iii): The de ini ions o Mand Rand he ac ha R* is he cyclic le shi ope a o
eadily imply ha ke M= ke M*=Δ.
(i ), ( i), and ( ii): Le y=(y
1
, ..., y
m
)ÎX. Assume i s ha yÎ an M. Then he e
exis s x=(x
1
,...,x
m
) such ha y
1
=x
1
-x
m
,y
2
=x
2
-x
1
,...,andy
m
=x
m
-x
m-1
.I
ollows ha ∑
iÎI
y
i
= 0, i.e, yÎΔ
┴
by [[1], P oposi ion 25.4(i)]. Thus,
an M
⊆
⊥
.
(17)
Con e sely, assume now ha yÎΔ
┴
. Now se
x:= Ly =
m−
1

i
=1
m−i
mRi−1y
.
(18)
I will be no a ionally con enien o w ap indices a ound, i.e., y
m+1
=y
1
,y
0
=y
m
and
likewise. We hen ge
(∀i∈I)xi=m−1
m
yi+m−2
m
yi−1+···+1
m
yi+2
.
(19)
The e o e,

i
∈
I
xi=m−1
m
i
∈
I
yi+m−2
m
i
∈
I
yi+···+1
m
i
∈
I
yi=m−1
2
i
∈
I
yi=0
.
(20)
Thus xÎΔ
┴
and
an L
⊆
⊥.(21)
Fu he mo e,
(∀i∈I)xi−xi−1=m−1
m
yi−1
m
yi−1−1
m
yi−2−···− 1
m
yi+
1
(22a)
=yi−1
m

j
∈I
yj=yi
.
(22b)
Hence Mx =x-Rx =yand hus yÎ an M. Mo eo e , in iew o (iii),
M−1
y
=x+ke M=x+
.
(23)
We hus ha e shown
⊥⊆ an M
.
(24)
Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53
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Combining (17) and (24), we ob ain an M=Δ
┴
. We hus ha e e i ied ( i), and ( ii).
Since an Mis closed, so is an M* (by, e.g., [[1], Co olla y 15.34]). Thus (i ) holds.
( iii)&( ): We ha e seen in P oposi ion 2.1 ha
M†=P
a
nM∗◦M−1◦P
a
nM
.
(25)
Now le zÎX.Then,by(i ),y:= P
a
nMZ=P⊥Z∈
⊥
.By( ii),M
-1
y=Ly +Δ.So
M
†
z=P
an M∗
M
−1
P
an M
z=P
an M*
M
−1
y=P

⊥
(Ly +)=P

⊥
Ly =Ly =(L◦P

⊥
)
z
because
an L⊆Δ
┴
by (21). Hence ( iii) holds. Fu he mo e, by (i ) and e.g., [[1], P oposi ion
3.28( )], an L= anL◦P

⊥= anM†= anM∗=
⊥
and so ( ) holds.
(ix): No e ha P

⊥=Id −P

and ha P
Δ
=m
-1
∑
jÎI
R
j
. Hence
P⊥=Id −1
m
j
∈I
Rj
.
(26)
Thus, by ( iii) and (16),
M†=L◦P⊥=1
m
m−1

i=1
(m−i)Ri−1◦⎛
⎝
Id −1
m
j∈I
Rj
⎞
⎠
(27)
=1
m
m−1

i=1
(m−i)Ri−1−1
m2
m−1

i=1
(m−i)
j
∈I
Ri+j−1
.
(28)
Re-a anging his exp ession in e ms o powe s o Rand simpli ying leads o
M†=(Id −R)†=
m

k
=1
m−(2k−1)
2mRk−1
.
(29)
Rema k 2.3 Suppose ha
˜
L:
⊥→
X
sa is ies M◦˜
L=Id|
⊥
Then
M−1:X⇒X:y→


Ly+,i y∈⊥;
∅,o he wise
.
(30)
One may show ha M†=P

⊥◦˜
L◦P
⊥
and ha P

⊥◦
˜
L=
L
(see (16)). Conc e e
choices o ˜
L
and La e
⊥→X:
(
y1,y2,...,ym
)
→
(
y1,y1+y2,...,y1+y2+y3+...+ym
);
(31)
howe e , he ange o he la e ope a o is no equal Δ
┴
whene e X≠{0}.
Rema k 2.4 Deno ing he symme ic pa o Mby M+=1
2
M+1
2
M∗and de ining
he quad a ic o m associa ed wi h Mby qM:x→1
2
x, Mx

, we no e ha [[17], P opo-
si ion 2.3] implies ha
c
an M+= domq∗
M
=
⊥
.
Fac 2.5 (B ezis-Ha aux) (See [20] and also, e.g., [[1], Theo em 24.20].) Suppose A
and B a e mono one ope a o s on X such ha A + B is maximally mono one, dom A⊆
dom B, and B is ec angula . Then in an(A+B)=in ( an A+ an B)and
an(A+B)= an A+ anB.
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Applying he B ezis-Ha aux esul o ou gi en ope a o s Aand M, we ob ain he
ollowing.
Co olla y 2.6 The ope a o A+Mis maximally mono one and
an
(
A+M
)
=⊥+ an
A
P oo . Since each A
i
is maximally mono one and ecalling Theo em 2.2(i), we see ha
Aand Ma e maximally mono one. On he o he hand, dom M=X. Thus, by he well
known sum heo em o maximally mono one ope a o s (see, e.g., [[1], Co olla y 24.4
(i)]), A+Mis maximally mono one. Fu he mo e, by Theo em 2.2(ii) and (i ), Mis
ec angula and an M=Δ
┴
. The esul he e o e ollows om Fac 2.5.
3 Composi ion
We now use Co olla y 2.6 o s udy he composi ion. When m= 2, hen Theo em 3.1
( ) also ollows om [[21], p. 124].
Theo em 3.1 Suppose ha
(
∀i∈I
)
0∈ an
(
Id −Ti
)
.Then he ollowing hold.
(i) 0∈ an
(
A+M
)
.
(ii) (∀ε>0)(∃(b, x)ÎX×X)||b|| ≤εand x=T(b+Rx).
(iii) (∀ε>0)(∃(c, x)ÎX×X)||c|| ≤εand x=c+T(Rx).
(i ) (∀ε>0)(∃xÎX)(∀iÎI)||T
i-1
···T
1
x
m
-T
i
T
i-1
···T
1
x
m
-x
i-1
+x
i
|| ≤(2i-
1)ε, whe e x
0
=x
m
.
( ) (∀ε>0)(∃xÎX)||x-T
m
T
m-1
···T
1
x|| ≤m
2
ε.
P oo . (i): The assump ions and (3) imply ha
(
∀i∈I
)
0∈ an Ai. Hence, 0
∈
an
A
.
Ob iously, 0ÎΔ
┴
.I ollows ha 0
∈
⊥+ an
A
.Thus,byCo olla y2.6,
0∈ an
(
A+M
)
.
(ii): Fix ε>0. In iew o (i), he e exis s xÎXand bÎXsuch ha ||b|| ≤εand b
ÎAx +Mx. Hence b+Rx Î(Id +A)xand hus x=J
A
(b +Rx) =T(b +Rx).
(iii): Le ε>0. By (ii), he e exis s (b, x)ÎX×Xsuch ha ||b|| ≤εand x=T(b +
Rx). Se c=x-T(Rx) =T(b +Rx) -T(Rx). Then, since Tis nonexpansi e, ||c|| =
||T(b+Rx)-T(Rx)|| ≤||b|| ≤ε.
(i ): Take ε>0. Then, by (iii), he e exis s xÎXand cÎXsuch ha ||c|| ≤εand
x=c+T(Rx). Le iÎI. Then x
i
=c
i
+T
i
x
i-1
. Since ||c
i
|| ≤||c|| ≤εand T
i
is non-
expansi e, we ha e

TiTi−1···T1x0−xi

≤

TiTi−1···T1x0−Tixi−1

+

Tixi−1−xi

(32a)
≤

TiTi−1···T1x0−Tixi−1

+ε
.
(32b)
We hus ob ain induc i ely
TiTi−1···T1x0−xi≤iε.(33)
Hence,
Ti−1···T1x0−xi−1≤
(
i−1
)
ε
.
(34)
Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53
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The conclusion now ollows om adding (33) and (34), and ecalling he iangle
inequali y
( ): Le ε>0. In iew o (i ), he e exis s xÎXsuch ha
(
∀i∈I
)
Ti−1···T1xm−TiTi−1···T1xm−xi−1+xi≤
(
2i−1
)ε
(35)
whe e x
0
=x
m
.Nowse (∀iÎI)e
i
=T
i-1
···T
1
x
m
-T
i
T
i-1
···T
1
x
m
-x
i-1
+x
i
.Then
(∀iÎI)||e
i
|| ≤(2i-1)ε. Se x=x
m
. Then
m

i
=1
ei=
m

i
=1
Ti−1...T1xm−TiTi−1...T1xm−xi−1+x
i
(36)
=x−
TmTm
−1...
T
1x
.
(37)
This, (35), and he iangle inequali y imply ha
x−TmTm−1···T1x≤
m

i
=1
ei≤
m

i
=1
(2i−1)ε=m2ε
.
(38)
This comple es he p oo .
Co olla y 3.2 Suppose ha
(
∀i∈I
)
0∈ an
(
Id −Ti
)
.Then
0∈ an
(
Id −TmTm−1···T1
)
.
P oo . This ollows om Theo em 3.1( ).
Rema k 3.3 The con e se implica ion in Co olla y 3.2 ails in gene al: indeed, con-
side he case when X≠{0}, m=2,and ÎX {0}. Now se T
1
X®X: x ↦x+ and
se T
2
X®X: x ↦x- . Then 0/∈ an
(
Id −T1
)
={−
}
and 0/∈ an
(
Id −T2
)
={
}
how-
e e , T
2
T
1
= Id and an
(
Id −T2T1
)
={0
}
.
Rema k 3.4 Co olla y 3.2 is op imal in he sense ha e en i (∀iÎI)weha e0Î
an(Id - T
i
), we canno deduce ha 0 Î an(Id - T
m
T
m-1
···T
1
): indeed, suppose ha
X
=
R2
and m=2.Se C
1
: = epi exp and C2:= R×
{
0
}
. Suppose u he ha T1=PC
1
and T2=PC
2
.Then(∀iÎI)0Î an(Id - T
i
); howe e ,
0∈ an
(
Id −T2T1
)
an
(
Id −T2T1
)
.
4 Asymp o ic egula i y
The ollowing no ions ( aken om B uck and Reich’s seminal a icle [22]) will be e y
use ul o ob ain s onge esul s.
De ini ion 4.1 ((s ong) nonexpansi eness and asymp o ic egula i y) Le S: X ®
X. Then:
(i) Sisnonexpansi e i (∀xÎX)(∀yÎX)||Sx - Sy|| ≤||x - y||.
(ii) Siss ongly nonexpansi e i S is nonexpansi e and whene e (x
n
)
nÎN
and (y
n
)
nÎN
a e sequences in X such ha (x
n
-y
n
)
nÎN
is bounded and ||x
n
-y
n
||- ||Sx
n
-
Sy
n
|| ®0, i ollows ha (x
n
-y
n
)-(Sx
n
-Sy
n
)®0.
(iii) S is asymp o ically egula i (∀xÎX)S
n
x-S
n+1
x®0.
The ollowing esul illus a es ha s ongly nonexpansi e mappings gene alize he
no ion o a i mly nonexpansi e mapping.Inaddi ion, heclasso s ongly
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nonexpansi e mappings is closed unde composi ions. (In con as , he composi ion o
wo (necessa ily i mly nonexpansi e) p ojec o s may ail o be i mly nonexpansi e.)
Fac 4.2 (B uck and Reich) The ollowing hold.
(i) E e y i mly nonexpansi e mapping is s ongly nonexpansi e.
(ii) The composi ion o ini ely many s ongly nonexpansi e mappings is also s ongly
nonexpansi e.
P oo . (i): See [[22], P oposi ion 2.1]. (ii): See [[22], P oposi ion 1.1].
The sequences o i e a es and o di e ences o i e a es ha e s iking con e gence
p ope ies as we shall see now. In passing, we no e ha Fac 4.3(i) also appea s in
[[21], Theo em 3.7(b)] e en in ce ain Banach spaces.
Fac 4.3 (B uck and Reich) Le S: X ®X be s ongly nonexpansi e and le x ÎX.
Then he ollowing hold.
(i) The sequence (S
n
x-S
n+1
x)
nÎN
con e ges s ongly o he unique elemen o leas
no m in an
(
Id −S
)
.
(ii) I Fix S =Ø, hen ||S
n
x||®+∞.
(iii) I Fix S≠Ø, hen (S
n
x)
nÎN
con e ges weakly o a ixed poin o S.
P oo (i): See [[22], Co olla y 1.5]. (ii): See [[22], Co olla y 1.4]. (iii): See [[22], Co ol-
la y 1.3].
Suppose S: X ®Xis asymp o ically egula . Then, o e e y xÎX,0¬S
n
x-S
n+1
x
=(Id-S)S
n
xÎ an(Id - S) and hence 0∈ an
(
Id −S
)
. The opposi e implica ion ails
in gene al (conside S= - Id), bu i is ue o s ongly nonexpansi e mappings. Unde
he assump ion ha Sis i mly nonexpansi e, he ollowing esul also ollows om
[[23], Co olla y 2].
Co olla y 4.4 Le S: X ®X be s ongly nonexpansi e. Then S is asymp o ically egu-
la i and only i 0∈ an
(
Id −S
)
.
P oo .“⇒": Clea . “⇐": Fac 4.3(i).
Co olla y 4.5 Se S =T
m
T
m-1
···T
1
.Then S is asymp o ically egula i and only i
0∈ an
(
Id −S
)
.
P oo Since each T
i
is i mly nonexpansi e, i is also s ongly nonexpansi eby Fac 4.2
(i). By Fac 4.2(ii), Sis s ongly nonexpansi e. Now apply Co olla y 4.4. Al e na i ely,
0∈ an
(
Id −S
)
by Co olla y 3.2 and again Co olla y 4.4 applies.
We a e now eady o ou i s main esul . When m=2, hen heconclusionalso
ollows om [[21], p. 124].
Theo em 4.6 Suppose ha each T
i
is asymp o ically egula . Then T
m
T
m-1
···T
1
is
asymp o ically egula as well.
P oo . Theo em 3.1( ) implies ha 0∈ an
(
Id −TmTm−1···T1
)
. The conclusion hus
ollows om Co olla y 4.5.
As an applica ion o Theo em 4.6, we ob ain he main esul o [17].
Example 4.7 Le C
1
, ..., C
m
be nonemp y closed con ex subse s o X. Then he com-
posi ion o he co esponding p ojec o s, PC
m
PC
m
−1...PC1is asymp o ically egula .
Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53
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P oo . Fo e e y iÎI, he p ojec o PC
i
is i mly nonexpansi e, hence s ongly nonex-
pansi e, and Fix PC
i
=Ci=
∅
. Suppose ha (∀i∈I)Ti=PC
i
, which is hus asymp o i-
cally egula by Co olla y 4.4. Now apply Theo em 4.6.
5 Con ex combina ion
In his sec ion, we use ou ixed weigh s (l
i
)
iÎI
(see (12)) o u n X
m
in o a Hilbe p o-
duc space di e en om Xconside ed in he p e ious sec ions. Speci ically, we se
Y:=Xmwi h x,y=

i∈
I
λixi,yi

(39)
so ha ||x||
2
=∑
iÎI
l
i
||x
i
||
2
. We also se
Q:Xm→Xm:x→ (¯
x)i∈I,whe e
¯
x:=

i∈
I
λixi
.
(40)
Fac 5.1 (See [[1], P oposi ion 28.13].) In he Hilbe p oduc space Ywe ha e P
Δ
=
Q.
Co olla y 5.2 In he Hilbe p oduc space Y he ope a o Qis i mly nonexpansi e
and s ongly nonexpansi e. Fu he mo e,FixQ=Δ≠Ø, 0Î an(Id -Q), and Qis
asymp o ically egula .
P oo By Fac 5.1, he ope a o Qis equal o he p ojec o P
Δ
and hence i mly non-
expansi e. Now apply Fac 4.2(i) o deduce ha Qis s ongly nonexpansi e. I is clea
ha Fix Q=Δand ha 0Î an(Id -Q). Finally, ecall Co olla y 4.4 o see ha Qis
asymp o ically egula .
P oposi ion 5.3 In he Hilbe p oduc space Y he ope a o Tis i mly
nonexpansi e.
P oo . Since each T
i
is i mly nonexpansi e, we ha e (∀x=(x
i
)
iÎI
ÎY)(∀y=(y
i
)
iÎI
Î
Y)||T
i
x
i
-T
i
y
i
||
2
≤〈x
i
-y
i
,T
i
x
i
-T
i
y
i
〉⇒||Tx -Ty||
2
=∑
iÎI
l
i
||T
i
x
i
-T
i
y
i
||
2
≤∑
iÎI
l
i
〈x
i
-y
i
,T
i
x
i
-T
i
y
i
〉=〈x-y, Tx -Ty〉.
Theo em 5.4 Suppose ha
(
∀i∈I
)
0∈ an
(
Id −Ti
)
.Then he ollowing hold in he
Hilbe p oduc space Y.
(i) 0∈ an
(
Id −T
)
.
(ii) Tis asymp o ically egula .
(iii) Q○Tis asymp o ically egula .
P oo . (i): This ollows because (∀x=(x
i
)
iÎI
)||x-Tx||
2
=∑
iÎI
l
i
||x
i
-T
i
x
i
||
2
.
(ii): Combine Fac 4.2(i) wi h Co olla y 4.4.
(iii): On he one hand, Qis i mly nonexpansi e and asymp o ically egula by Co -
olla y 5.2. On he o he hand, Tis i mly nonexpansi e and asymp o ically egula
by P oposi ion 5.3 and Theo em 5.4(ii). Al oge he , he esul ollows om Theo-
em 4.6.
We a e now eady o ou second main esul , which conce ns con ex combina ions
o i mly nonexpansi e mappings. Fo u he esul s in his di ec ion-namely con ex
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