Weak pullback attractors of setvalued processes

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. 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