Weak pullback a ac o s o se alued
p ocesses
T. Ca aballo a, P.E. Kloeden b,∗, P. Ma ´ın–Rubio a
aDp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa,
Apdo. de Co eos 1160, 41080 Se illa, Spain
bFB Ma hema ik, Johann Wol gang Goe he Uni e si ¨a , D-60054 F ank u am
Main, Ge many
Abs ac
Weak pullback a ac o s a e de ined o nonau onomous se alued p ocesses and
hei exis ence and uppe semi con inuous con e gence unde pe u ba ion is es ab-
lished. Unlike s ong pullback a ac o s, in a iance and pullback a ac ion he e a e
equi ed only o a leas one ajec o y a he han all ajec o ies a each s a ing
poin . The concep is use ul in, o example, con inuous ime con ol sys ems and
is ela ed o ha o iabili y.
Key wo ds: se alued p ocesses, pullback a ac ion, weak in a iance
1991 MSC: 34D45, 37B25, 37B75, 58C06
1 In oduc ion
Typical and impo an examples o se alued p ocesses a e dynamical sys-
ems wi hou uniqueness gene a ed by o dina y di e en ial equa ions wi hou
uniqueness and o dina y di e en ial con ol sys ems (i.e., ˙x= ( , x, u) whe e
u∈U) o , mo e gene ally, inclusion equa ions (i.e., ˙x∈F(x)). Ob iously,
con ol sys ems ha e mo e signi ican applica ions and hus p o ide a powe -
ul mo i a ion o s udying dynamical sys ems wi hou uniqueness, al hough
his o ically he o iginal mo i a ion came om o dina y di e en ial equa ions
wi hou uniqueness. Many in e es ing sys ems a e in ac nonau onomous,
∗Co esponding au ho
Email add esses: [email p o ec ed] (T. Ca aballo),
[email p o ec ed] (P.E. Kloeden), [email p o ec ed]
(P. Ma ´ın–Rubio).
P ep in submi ed o Jou nal o Ma hema ical Analysis and Applica ions
al hough mos concep s ha e been de eloped only in he mo e con enien se -
ing o au onomous sys ems. I is o p ac ical impo ance as well as in ellec ual
in e es o see how such concep s gene alize o nonau onomous sys ems.
In many applica ions, physical, economical and indus ial, such p oblems a e
ypically s a ed on ini e ime in e als, e.g. in Viabili y Theo y (see Aubin
[3,4] and Aubin and F ankowska [7] and he e e ences he e), cap u e basins
ep esen he poin s om which a leas one ajec o y eaches he a ge in
ini e ime, hus in a weak sense. This a ises in minimal ime con ol p oblems
(c . Sain -Pie e [19]), o in acking con ol, c . Chen e al. [10], whe e he a -
ge is he g aph o a single o mul i- alued map (ob ained by condi ion (2.8),
see hei p oo o Theo em 2.2); and in con ollabili y heo y (e.g. Johnson and
Ne u ka [11]), in which he d ama ical in luence o pa ame e s on he con-
ollabili y o he sys em is pa icula ly wo h no ing. A he same ime, he
asymp o ic beha io o such weakly in a ian sys ems has also been in ensi ely
in es iga ed wi h many meaning ul in e p e a ions in biology such as pe sis-
ence and ex inc ion, and applica ions in popula ion gene ics (c . Vuille mo
[22]), in minimiza ion p oblems (c . A ouch and Comine i [1]) and s abi-
liza ion in Mechanics (c . A ouch and Cza necki [2]). A ac o s p o ide an
impo an means o cha ac e izing he long ime beha iou o dynamical sys-
ems. They ha e been ex ensi ely in es iga ed, in pa icula , global a ac o s
in he au onomous case and i s pullback e sion o gene al nonau onomous
si ua ions [9], which a e known as s ong a ac o s o se alued sys ems. In
he au onomous con ex , Szeg¨o and T eccani [21] in oduced he concep o a
weak a ac o o he con inuous ime se alued semig oup gene a ed by di -
e en ial inclusions. The key di e ence he e is ha only a leas one ajec o y
o each s a ing poin mus be a ac ed o o emain in he weak a ac o
a he han all ajec o ies as in he case o he usual (s ong) a ac o . The
concep o a weak a ac o has been ound o be e y use ul o au onomous
con ol sys ems as well as o some op imiza ion sys ems, so he co esponding
concep should hus also be o p ac ical use ulness in he nonau onomous case.
In his pape we will in oduce he concep o weak pullback a ac o o
he se alued p ocesses gene a ed by di e en ypes o nonau onomous dy-
namical sys ems such as o dina y di e en ial equa ions wi hou uniqueness,
nonau onomous con ingen o inclusion equa ions, nonau onomous o dina y
di e en ial con ol sys ems, e c. He e he a ac o consis s o a amily o se s
in a ian , i.e. ca ied in o each o he unde he dynamics. Thus o wa d con-
e gence is o a mo ing a ge , whe eas pullback con e gence is o a ixed
a ge , a pa icula membe o he amily. Al hough simila concep s we e
in oduced ecen ly in [15] o nonau onomous di e ence inclusions, he ech-
niques needed he e o p o e he exis ence o he weak pullback a ac o in he
con inuous ime case a e somewha di e en and mo e complica ed, in pa -
icula , equi ing Ba bashin’s esul s on he compac ness o se o ajec o ies
and i s gene aliza ion o a single se alued p ocess o a se alued con e gen
2
sequence o p ocesses. Mo eo e , we es ablish ou esul s o a mo e gene al
Banach s a e space, hus emo ing a long s anding es ic ion o locally com-
pac s a e spaces in ea lie publica ions.
In sec ions 2 and 3 we in oduce he usual no a ion o he se alued ame-
wo k, and he analogous ool o semi lows and semig oups h ough wha a e
known as he se alued p ocess o gene al(ized) dynamical sys ems. In sec-
ion 4 we es ablish ou main esul s and highligh some o hei ea u es wi h
se e al examples in Sec ion 5. P oo s a e gi en a he end o he pape .
2 Te minology
Le be gi en a gene al Banach space (X, k·k). Recall ha
dis (x, A) = min
a∈Akx−ak
is he dis ance o a poin x∈X om a nonemp y compac se Aand ha
he Hausdo sepa a ion H∗(A, B) o nonemp y compac subse s A,Bo X
is de ined as
H∗(A, B) := max
a∈Adis (a, B) = max
a∈Amin
b∈Bka−bk,
while H(A, B) = max {H∗(A, B), H∗(B, A)}is he Hausdo me ic on he
space (X) o nonemp y compac subse s o X.
De ine an open –neighbou hood o A∈(X) by N(A) = {x∈X: dis (x, A)<
}and closed –neighbou hood o Aby N[A] = {x∈X: dis (x, A)≤}.
A mapping F:X7→ (X) is uppe semi con inuous a x0i o all ε > 0 he e
exis s a δ=δ(ε, x0)>0 such ha F(x)⊂Nε(F(x0)) o all x∈Nδ({x0}) o
al e na i ely i
lim
xn→x0H∗(F(xn), F(x0)) = 0
o all sequences xn→x0.
Fo any A∈(X) de ine F(A) := ∪a∈AF(a) and de ine he se composi ion o
wo mappings F,G:X7→ (X) as F◦G(x) := F(G(x)) o all x∈X. No e
ha F◦Gis uppe semi con inuous and compac alued i Fand Ga e (see
[6]).
3
3 Se alued p ocesses
Ba bashin [8] in es iga ed se alued gene alized o gene al dynamical sys-
ems gene a ed by o dina y di e en ial equa ions wi hou uniqueness. Roxin
[18] showed ha nonau onomous con ingen o inclusion equa ions gene a ed
nonau onomous gene al dynamical sys ems, as did nonau onomous o dina y
di e en ial con ol sys ems in which case he called he gene a ed sys em a
gene al con ol sys em [17]. See also [12,13]. We will use he name se al-
ued p ocess o all such nonau onomous se alued sys ems wi hou assumed
backwa ds ex endabili y in ime.
De ini ion 1 A se alued p ocess on a s a e space Xis de ined in e ms o
an a ainabili y se mapping ( , 0, x0)7→ Φ( , 0, x0) o all ≥ 0in Rand x0
∈Xwhich sa is ies he ollowing p ope ies:
1. Compac ness Φ( , 0, x0)is a nonemp y compac subse o X o all ≥ 0
in Rand all x0∈X;
2. Ini ial condi ion
Φ( 0, 0, x0) = {x0}
o all 0∈Rand x0∈X;
3. Time e olu ion
Φ( 2, 0, x0) = Φ ( 2, 1,Φ( 1, 0, x0))
o all 0≤ 1≤ 2in Rand all x0∈X;
4. Con inui y in ime
lim
s→ H(Φ(s, 0, x0),Φ( , 0, x0)) = 0
o all s, ≥ 0and all 0∈Rand x0∈X;
5. Uppe semi con inui y in ini ial condi ions
lim
(n)
0→ 0,x(n)
0→x0
H∗Φ( , (n)
0, x(n)
0),Φ( , 0, x0)= 0
uni o mly in ∈[T0, T1] o any T0< T1<∞wi h T0≥ (n)
0, 0and o all
0∈Rand x0∈X.
Simple examples o di e en ial equa ions wi hou uniqueness (e.g., Example 1
on page 122 o [13]) show ha Condi ion 5 canno in gene al be s eng hened
4
o con inui y in he ini ial a iables, i.e., wi h he Hausdo me ic Hins ead
o he semi–me ic H∗.
De ini ion 2 A ajec o y o a se alued p ocess Φis a single alued mapping
φ: [T0, T1]→Xwhich sa is ies
φ( )∈Φ( , s, φ(s)) o all T0≤s≤ ≤T1
o some T0< T1in R. A ajec o y φis called an en i e ajec o y i i is
de ined on all o R.
T ajec o ies a e in ac con inuous unc ions. See Lemma 6.1 in [17], o The-
o em 4.2 in [12] o he sys ems wi hou assumed backwa ds ex endabili y
unde conside a ion he e.
Ba bashin [8] p o ed 1exis ence o a leas one ajec o y φ: [ 0, 1]→Xwi h
φ( 0) = x0and φ( 1) = x1 o any x0and x1wi h x1∈Φ( 1, 0, x0). Ba bashin
[8] also p o ed a esul on he compac ness o ajec o ies o a se alued p ocess
Φ. The ollowing gene aliza ion is due o Roxin [17]; see also [12]).
Theo em 3 (Ba bashin) Le Bbe a nonemp y compac subse o Xand le
φn: [ 0, 1]→Xbe a sequence o ajec o ies o a se alued p ocess Φwi h
φn( 0)∈B o gi en 0< 1∈R. Then he e exis s a subsequence φnjand a
ajec o y ¯
φ: [ 0, 1]→Xo Φwi h ¯
φ( 0)∈Bsuch ha φnj( )→¯
φ( )as nj
→ ∞ uni o mly in ∈[ 0, 1].
We will also s a e, p o e and use a u he gene aliza ion o his heo em o
sequences o ajec o ies belonging o a sequence o uppe semi con inuously
con e gen se alued p ocesses; see Theo em 16 in Sec ion 6.
4 Weak a ac o s o se alued p ocesses
Fo se alued sys ems a ising om con ol sys ems, one is o en in e es ed in
si ua ions whe e jus one o a ew a he han all ajec o ies emana ing om
each s a ing poin sa is y a gi en p ope y. Szeg¨o and T eccani [21] in oduced
concep s o weak in a iance and weak a ac o s o such si ua ions in he
au onomous case.
Ou aim in his pape is o in oduce and in es iga e pullback e sions o
hese weak concep s o se alued p ocesses. As wi h he s ong concep s o
1Al hough he wo ked in ini e dimensional spaces, he ex ension o a gene al Ba-
nach space Xis s aigh o wa d due o he ac ha he cons uc ed objec s a e
con ained in he compac in eg al unnel.
5

in a iance and a ac ion, i is also less es ic i e he e o conside amilies o
se s a he han indi idual se s.
De ini ion 4 A amily A={A , ∈R}o nonemp y compac subse s o X
is said o be weakly posi i ely in a ian o a se alued p ocess Φon Xi o
e e y 0∈Rand e e y x0∈A 0 he e exis s a ajec o y φ: [ 0,∞)→Xo
Φwi h φ( 0) = x0such ha φ( )∈A o all ≥ 0.
I is called weakly in a ian i , o e e y 0∈Rand e e y x0∈A 0, he e is
an en i e ajec o y φwi h φ( 0) = x0and φ( )∈A o all ∈R.
De ini ion 5 A weakly in a ian amily A={A , ∈R}o nonemp y com-
pac subse s o Xis called a weak pullback a ac o o a se alued p ocess Φ
on Xi i is weakly pullback a ac ing, i.e., o any 0∈R, any nonemp y
bounded subse Do Xand any sequence dn∈D he e exis sequences o pos-
i i e numbe s τn→ ∞ as n→ ∞ and ajec o ies φn: [ 0−τn, 0]→Xo Φ
wi h φn( 0−τn) = dnsuch ha
lim
n→∞ dis (φn( 0), A 0)=0.(1)
No e ha a s ong pullback a ac o , when i exis s, is also a weak pullback
a ac o . Now, one o ou main esul s will be o show ha he exis ence o
a weak pullback a ac o ollows om ha o a mo e easily de e mined weak
pullback abso bing amily o se s.
De ini ion 6 A weakly posi i ely in a ian amily B={B , ∈R}o nonemp y
compac subse s o Xis called a weak pullback abso bing amily o a se alued
p ocess Φon Xi o 0∈Rand any bounded subse Do X he e exis s a
T 0,D ∈R+such ha o each τn≥T 0,D and dn∈D he e exis s a ajec o y
φn: [ 0−τn, 0]→Xo Φwi h
φn( 0−τn) = dnand φn( 0)∈B 0.
No e ha by he weak posi i e in a iance o B he ajec o ies φncan be ex-
ended, using he conca ena ion p ope y gi en by he ime e olu ion p ope y
3, o emain in B o ≥ 0, i.e. φn( )∈B o each ≥ 0.
Theo em 7 Le Φbe a se alued p ocess wi h a weak pullback abso bing am-
ily B. Then Φhas a maximal weak pullback a ac o A={A , ∈R} ela i e
o B, which is uniquely de e mined by
A 0={a0∈X; he e exis τn→ ∞ as n→ ∞,
and ajec o ies φn: [ 0−τn, 0]→X(2)
wi h φn( )∈B o ∈[ 0−τn, 0]
6
and lim
n→∞ φn( 0) = a0
o each 0∈R.
Rema k 8 A weak pullback a ac o consis s o ajec o ies ha exis and
emain in B o he en i e ime se R, bu i does no necessa ily con ain all
such ajec o ies. See Lemma 13 in he Sec ion 6.
Rema k 9 As well as being weakly in a ian , a weak pullback a ac o is
also nega i ely s ongly in a ian , i.e., sa is ies A ⊂Φ ( , 0, A 0) o all ≥
0and 0∈R.
Rema k 10 The uniqueness and maximali y o a weak a ac o canno be
unde s ood in he usual sense, bu a he wi h espec o an abso bing am-
ily Bo se s unde discussion. This is an in insic p ope y o weak pullback
a ac o s and is no con adic ed by he exis ence o o he weak pullback a -
ac o s, wi h o wi hou in e sec ing componen se , wi h espec o di e en
amilies B. This is anspa en in he examples o weak pullback a ac o s o
nonau onomous di e ence equa ions in [15]. A simila example o se alued
p ocesses will be gi en in Sec ion 5.
The p oo o he ollowing basic con inui y p ope y o a weak pullback a -
ac o is no as immedia e a consequence o de ini ions as in he s ong case.
I is gi en in Sec ion 8.
P oposi ion 11 Le A={A , ∈R}be a weak pullback a ac o . Then he
se alued mapping 7→ A is con inuous.
Ou second objec i e is o p o e some esul s on he s uc u e o weak pullback
a ac o s o se alued p ocesses. In ac , we a e in e es ed in some kind o
uppe semi con inuous beha iou p oduced by some pe u ba ions appea ed
in he model.
Theo em 12 Suppose ha he se alued p ocess Φhas a weak pullback ab-
so bing amily B={B , ∈R}and suppose ha each pe u bed se alued
p ocess Φhas a weak pullback abso bing amily B={B
, ∈R} o  > 0
such ha
max
0≤δ≤1H∗(Φ( +δ, , x),Φ( +δ, , x)) ≤ o all ∈R, x ∈X(3)
and
H∗B
0, B 0≤ o all 0∈R.(4)
Then he maximal weak pullback a ac o s A={A
, ∈R}w. . . Bo he
7
pe u bed p ocesses Φcon e ge uppe semi con inuously o he maximal weak
pullback a ac o A={A , ∈R}w. . . Bo Φin he sense ha
lim
→0H∗A
0, A 0= 0.(5)
o each 0∈R.
The ollowing s uc u al condi ion on he unpe u bed se alued p ocess Φ
p o ides a simple (i a he s ong) condi ion o X=Rdensu ing he exis-
ence o a nea by uni o m weak pullback abso bing amily: assume ha B=
{B , ∈R}is a amily o nonemp y compac se s o Rdand ha he e exis s
aγ:R+→[0,1] such ha
min
y∈Φ( , 0,x)dis (y, B )≤γ( − 0) dis (x, B 0)
o all x∈Rdand ≥ 0in Rand ha o all bounded Dand all ixed ime :
lim
0→−∞ γ( − 0) sup
x∈D
dis (x, B 0) = 0.
We can ake N[B ] := {x∈Rd: dis (x, B )≤} o  > 0 small enough.
Then he amily N[B ] is weakly posi i ely in a ian and weakly pullback
abso bing.
5 Examples
Ou i s example in ol es nonau onomous se alued p ocess gene a ed by he
nonau onomous di e en ial inclusion
x0∈F( , x) := 








{−x}i < 0
{−x, 0}i ≥0
, x ∈R.
8
The se alued p ocess he e is gi en by
Φ( , 0, x0) :=





























nx0e−( − 0)oi 0≤ ≤0, x0∈R
hx0e−( − 0), x0ii 0 ≤ 0≤ , x0≥0
hx0, x0e−( − 0)ii 0 ≤ 0≤ , x0≤0
and he composi ion o hese cases. The amily A={A , ∈R}wi h A ≡
{0} o all ∈Ris s ongly in a ian and hence weakly in a ian . I is a weak
pullback (and o wa d) a ac o wi h espec o any abso bing amily se B
={B , ∈R}wi h componen se s B ≡[−R, R] o all ∈Rand any R≥
0.
As a second example we conside he nonau onomous se alued p ocess gen-
e a ed by he nonau onomous di e en ial inclusion
x0∈F( , x) := 








{−x}i < 0,
{−x, 1}i ≥0.
The se alued p ocess he e is gi en by
Φ( , 0, x0) =































nx0e−( − 0)oi 0≤ ≤0,
{x0}i 0 ≤ 0= ,
hx0e−( − 0), x0+ − 0ii 0 ≤ 0< , x0≥ − 0
e 0− −1,
hx0+ − 0, x0e−( − 0)ii 0 ≤ 0< , x0≤ − 0
e 0− −1.
and he composi ion o hese cases. The amily A={A , ∈R}wi h A ≡
{0} o all ∈Ris weakly in a ian , bu no s ongly in a ian . I is a weak
pullback (and o wa d) a ac o wi h espec o any abso bing amily se B
={B , ∈R}wi h componen se s B ≡[−R, R] o all ∈Rand any 0 ≤
R≤1.
Ou hi d example illus a es he ambigui y conce ning he exis ence and
uniqueness o weak pullback a ac o s alluded o in Rema k 10. I is based
9
0≤H∗A , Asnj
≤dis φnj( ), φnj(snj)
≤dis φnj( ),¯
φ( )+ dis ¯
φ( ),¯
φ(snj)+ dis ¯
φ(snj), φnj(snj)
≤dis φnj( ),¯
φ( )+ dis ¯
φ( ),¯
φ(snj)+ sup
−1≤s≤ +1
dis ¯
φ(s), φnj(s)
→0
as j→ ∞ by he uni o m con e gence o he subsequence in he i s and hi d
e ms and he con inui y o he ajec o y ¯
φin he second e m. Bu his is a
con adic ion, so we mus ha e H∗(A , As)→0 as s→ .
Secondly, conside he limi H∗(As, A )→0 as s→ . I his does no hold
he e would exis an 0>0 and a sequence sn→ such ha
0≤H∗(Asn, A ), n ∈N.
We will show ha his leads o a con adic ion.
Since Asnis compac , he e exis s an an∈Asnsuch ha
H∗(Asn, A ) = dis (an, A )≤dis (an, a)
o all a∈A . By he weak in a iance o A he e exis s a ajec o y φnwi h
φn(sn) = anand φn(s)∈As o all s∈Rand n∈N. Thus
H∗(Asn, A )≤dis (φn(sn), φn( )) , n ∈N.
Now an∈Asn⊂Φ ([ −1, + 1], −1, A −1), which is compac . He e we ha e
used he nega i e s ong in a iance o he weak pullback a ac o , see Re-
ma k 9. Thus we can apply Ba bashin’s Theo em o ob ain he exis ence o
a subsequence o ajec o ies φnjwhich con e ges o a ajec o y ¯
φuni o mly
on he in e al [ −1, + 1]. Thus
0≤H∗Asnj, A 
≤dis φnj(snj), φnj( )
≤dis φnj(snj),¯
φ(snj)+ dis ¯
φ(snj),¯
φ( )
≤sup
−1≤s≤ +1
dis φnj(s),¯
φ(s)+ dis ¯
φ(snj),¯
φ( )
→0
as j→ ∞ by he uni o m con e gence o he subsequence in he i s e m
16

and he con inui y o he ajec o y ¯
φin he second e m. Bu his is a con-
adic ion, so we mus ha e H∗(As, A )→0 as s→ .
Combining he wo cases gi es he desi ed esul , i.e., H(As, A )→0 as s→
.
9 P oo o Theo em 12
Le A={A , ∈R}be he weak pullback a ac o in Bgi en by (2) o
he unpe u bed se alued p ocess Φ and le A={A
, ∈R}be he weak
pullback a ac o in B o he pe u bed se alued p ocess Φ. Suppose o
some 0∈R ha
lim
→0H∗A
0, A 06= 0.
Then he e exis s an η0>0 and a subsequence j→0 as j→ ∞ such ha
H∗Aj
0, A 0≥η0(8)
o all j∈Z+. We will show ha his leads o a con adic ion.
Le aj∈Aj
0be such ha dis (aj, A 0) = H∗Aj
0, A 0, so dis (aj, A 0)≥
η0 o j∈Z+. This is possible since Aj
0is compac . By Lemma 13 he e is an
en i e ajec o y φjo he pe u bed se alued p ocess Φjsuch ha φj( )∈
Aj
⊂Bj
o each ∈Rwi h φj( 0) = aj.
Since he Bj
0and B 0a e compac wi h H∗Bj
0, B 0→0 as j→0, by
Lemma 14 he e exis s a con e gen subsequence a0
j=φ0
j( 0)→¯a0∈B 0as
0
j→0.
¿F om (8) we ha e
dis (¯a0, A 0)≥η0/2.(9)
By Theo em 16 (Gene alized Ba bashin Theo em) applied o he in e al
[ 0, 0+ 1], he e exis s a ajec o y ¯
φo Φ on [ 0, 0+ 1] wi h ¯
φ( 0) = ¯a0
and a subsubsequence φ00
jwi h φ00
j( )→¯
φ( ) as 00
j→0 uni o mly in ∈
[ 0, 0+ 1]. Mo eo e , by Lemma 14 we ha e ¯
φ( )∈B o each ∈[ 0, 0+ 1].
We epea his cons uc ion on successi e subin e als [ 0+n, 0+n+ 1] o n
= 1, 2, . . . o ob ain a ajec o y ¯
φo Φ on [ 0,∞) and a diagonal subsequence
17
(deno ed he same as be o e) φ00
jwi h φ00
j( )→¯
φ( )∈B as 00
j→0 o all
∈[ 0,∞). We can also wo k backwa ds in ime on successi e subin e als
[ 0−n−1, 0−n] o n= 1, 2, . . . o ob ain a ajec o y ¯
φo Φ on (−∞, 0] wi h
¯
φ( 0) = a0and a u he diagonal subsequence (deno ed he same as be o e)
φ00
jwi h φ00
j( )→¯
φ( )∈B as 00
j→0 o all ∈(−∞, 0].
Thus ¯
φis an en i e ajec o y o he unpe u bed se alued p ocess Φ wi h
¯
φ( )∈B o each ∈R. By Lemma 13 i ollows ha ¯
φ( )∈A o each
∈R. In pa icula , ¯
φ( 0)∈A 0. Howe e , his con adic s (9) and hence (8).
This con adic ion means ha he A
con e ge uppe semi con inuously o A
o each ∈R.
10 P oo o Theo em 16
Fo con enience, we conside wi hou loss o gene ali y he in e al [0,1] in-
s ead o [ 0, 1]. By assump ion, he e is a sequence o ajec o ies φjo Φj
on [ 0, 1] wi h φj(0) = x0,j →x0as j→0. W i e φ(0) = x0.
By he uppe semi con inuous con e gence (3) and he uppe semi con inui y
o Φ( , 0,·) uni o mly in ∈[0,1], we ha e
H∗(Φj( , 0, x0,j),Φ( , 0, x0)) ≤H∗(Φj( , 0, x0,j),Φ( , 0, x0,j))
+H∗(Φ( , 0, x0,j),Φ( , 0, x0))
≤j+H∗(Φ( , 0, x0,j),Φ( , 0, x0))
−→ 0 as j→0
uni o mly in ∈[0,1]. Hence o e e y  > 0 and aking jsu icien ly small,
Φj( , 0, x0,j)⊂N[Φ([0,1],0, x0)] ,
o all ∈[0,1]. The se Φ([0,1],0, x0) is compac by he con inui y o Φ(·,0, x)
because o he P ope ies 1 and 4 o a se alued p ocess, so om φj(1) ∈B∗,
he e exis s a con e gen subsequence φ0
j(1) = x1,j →x1∈Φ([0,1],0, x0) as
0
j→0. W i e φ(1) = x1. Mo eo e , φ(1) ∈Φ(1,0, x0). This ollows om he
ac ha
dis (φ(1),Φ(1,0, φ(0))) ≤

φ(1) −φ0
j(1)

+ dis φ0
j(1),Φ0
j(1,0, φ0
j(0))
+H∗Φ0
j(1,0, φ0
j(0)),Φ(1,0, φ(0))
=

φ(1) −φ0
j(1)

+H∗Φ0
j(1,0, φ0
j(0)),Φ(1,0, φ(0)),
18
since φ0
j(1) ∈Φ0
j(1,0, φ0
j(0)) o he ajec o ies φ0
jo Φ0
j. Thus
φ0
j(1) →φ(1), φ0
j(0) →φ(0) as 0
j→0.
Since he se alued mappings Φ0
j(1,0,·) and Φ(1,0,·) a e uppe semi con in-
uous and he Φ0
j(1,0,·) con e ge uppe semi con inuously o Φ(1,0,·) due o
(3), i ollows by Lemma 15 ha
H∗Φ0
j(1,0, φ0
j(0)),Φ(1,0, φ(0))−→ 0 as 0
j→0.
Thus dis (φ(1),Φ(1,0, φ(0))) = 0, i.e., φ(1) ∈Φ (1,0, φ(0)).
Conside he ime ins an =1
2. We epea he abo e a gumen on he in e al
[0,1
2], o cons uc φ1
2∈Φ1
2,0, φ(0)using a subsequence o he abo e one
ha con e ges a =1
2as well as a = 0 and 1. Using his same sequence
on he in e al [1
2,1] we also ob ain φ(1) ∈Φ1,1
2, φ 1
2.
The cons uc ion o φ( ) o dyadic ∈Sq=0,1,2,... np
2q:p= 0,1,2, . . . , qo
ollows ecu si ely, aking subsequences o he p e ious ones ha also con e ge
a he new poin s unde conside a ion. Suppose ha o a gi en qwe ha e
cons uc ed all o he φp
2qsuch ha
φp+ 1
2q∈Φp+ 1
2q,p
2q, φ p
2q, p = 0,1,2, . . . , q −1.(10)
Conside he ime ins an 2p+1
2q+1 , which is he midpoin o he in e al [ p
2q,p+1
2q].
The cons uc ion o φ2p+1
2q+1 wi h
φ2p+ 1
2q+1 ∈Φ2p+ 1
2q+1 ,p
2q, φ p
2q
and
φp+ 1
2q∈Φp+ 1
2q,2p+ 1
2q+1 , φ 2p+ 1
2q+1 
ollows exac ly he same as in he case o p= 0 and q= 1, i.e., o φ1
2 om
φ(0) and φ(1).
I ollows om he 2–pa ame e semig oup p ope y o Φ, i.e., he ime e o-
lu ion p ope y 3, and he inclusions (10), we ha e φ( )∈Φ ( , s, φ(s)) o all
19
dyadic s, ∈[0,1] wi h s≤ . As in he p oo o he o iginal Ba bashin The-
o em, he φ( ) o nondyadic a e de ined by a limi ing a gumen and he
ac ha φ( )∈Φ ( , s, φ(s)) o all s, ∈[0,1] wi h s≤ ollows om he
con inui y and uppe semi con inui y p ope ies o Φ. (See [12] o addi ional
de ails). Thus he unc ion φis a ajec o y o Φ wi h he s a ed p ope ies.
In pa icula , he unc ion 7→ φ( ) is con inuous since φis a ajec o y.
Acknowledgmen s This wo k was pa ially suppo ed by he DAAD (Ge -
many), he Minis e io de Ciencia y Tecnolog´ıa (Spain) and he Royal Socie y
o London. The i s and hi d au ho s would like o hank James Robinson
o his use ul commen s and hospi ali y a he Ma hema ics Ins i u e o he
Uni e si y o Wa wick.
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