NEW APPROACH TO DIFFERENTIAL EQUATIONS WITH COUNTABLE IMPULSES

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1 1 9 NEW APPROACH TO DIFFERENTIAL EQUATIONS WITH COUNTABLE IMPULSES Hong-Kun ZHANG Jin-Guo LIAN Jiong SUN Received: 1 January 2007 c 2006 Springer Science + Business Media, Inc. Absrac This paper provides a new approach o sudy he soluions of a class of generalized Jacobi equaions associaed wih he linearizaion of cerain singular flows on Riemannian manifolds wih dimension n + 1. A new class of generalized differenial operaors is defined. We invesigae he kernel of he corresponding maximal operaors by applying operaor heory. I is shown ha all nonrivial soluions o he generalized Jacobi equaion are hyperbolic, in which here are n dimension soluions wih exponenial-decaying ampliude. Key words 1 Inroducion counable impulses, differenial operaor, hyperbolic This paper is moivaed by he sudy of hyperboliciy of mulidimensional semi-dispersing billiards on Riemannian manifold wih boundaries, see [1 4], [5]. Hyperboliciy is defined in he language of he linearized sysem. I requires esablishing exponenial growh or decay of soluions of he linearized sysem. Choose a parallel frame fields along a ypical billiard rajecory γ, hen any variaion vecor field J will saisfy he following equaions: J () = A()J(), ( i, i+1 ) J( i + 0) = J( i 0) (1) J ( i + 0) = J ( i 0) + B i J( i + 0) where A() is he curvaure operaor of he billiard able a γ(), and B i is he collision operaor of he boundary a γ( i ). For a deailed descripion of hese formulas, see [6]. The differenial operaor L we sudy is associaed wih he above generalized Jacobi equaions. hence we ask wheher soluions o Lf = 0 have he corresponding propery under cerain assumpions. In [6], here is a seup for sudying differenial operaors wih jump condiions. However, because of very srong assumpions on he coefficien marices, is applicaion o Hongkun ZHANG Deparmen of Mahemaics and Saisics, Universiy of Massachuses, Amhers, MA 01003, USA. hongkunz@gmail.com. Jin-Guo LIAN Deparmen of Mahemaics and Saisics, Universiy of Massachuses, Amhers, MA 01003, USA. lianjinguo@gmail.com. Jiong SUN Deparmen of Mahemaics, Inner Mongolia Universiy, Hohho, Inner Mongolia, , China. This research is suppored by he Naional Naural Science Foundaion of USA (NSF-DMS )

2 2 Hong-Kun ZHANG Jin-Guo LIAN Jiong SUN dynamical sysems wih singulariies (especially billiard flows) is very resriced. In fac, we realized laer ha for he dynamical sysems we are ineresed in, he se of rajecories saisfying he assumpions of [6] has measure zero. In his paper we sudy he differenial operaors wih relaxed condiions on A() and B i. We inroduce a new operaor and invesigae he exponenial behavior of funcions in he kernel of he operaors and hope ha his mehod can be applied furher o he analysis of billiard flows, which carry dynamical informaion. In summary, his is a paper in differenial equaions, moivaed by ergodic heory. While he resuls of his paper are preliminary, and hey give some hope ha he srucure of differenial operaor heory may be used laer o analyze some ineresing problems in dynamical sysems. Acknowledgmen This paper is wrien in he memory of professor Rober Kauffman. The firs auhor would like o express her deep graiude o hank Prof. Kauffman for simulaing her ineres in he sudy of differenial operaors associaed wih Jacobi fields and numerous fruiful discussions wih him. 2 Saemen of Resuls Le I = [0, ), define a sricly increasing sequence J = { i } + i=0, such ha 0 = 0, lim n n = + and se I = I \ J. Consider a differenial equaion wih counable impulses: f () + A()f() = 0 I (2) f + ( i ) = f ( i ) i J f +( i ) = f ( i ) + B i f + ( i ) i J where {B i } i=1 is a sequence of symmeric marices and A is a piecewise coninuous funcion from I ino he space of n n symmeric marices wih disconinuiies conained only in J and f : I C n are complex vecor-valued funcions. Now we make assumpions for he paper: (h1) There exiss b > 0 such ha sup ( i i 1 ) b; i N (h2) For any i = 0, 1,..., any ( i, i+1 ), boh B i and A() are posiive semi-definie; (h3) There exis c > 0 and a subsequence {i k } N wih such ha 1 lim inf n n ( ik ik 1) δ (3) k n B ik c. Theorem 1 Le f be a real-valued, non-consan soluion of (2). 1. If f is bounded, hen here exis λ 1 > 0, T 1 > 0, c 1 > 0 such ha for all [T 1, ): f() c 1 e λ1.

3 NEW APPROACH TO DIFFERENTIAL EQUATIONS WITH COUNTABLE IMPULSES 3 2. If f is unbounded, hen here exiss λ 2 > 0, T 2 > 0, and c 2 > 0 such ha for all [T 2, ): f() c 2 e λ2. Here boh λ 1 and λ 2 are consans independen of f. Proof Le I, unless oherwise specified. Since for any real soluion f o Lf = 0, By assumpion so (f f) () = 2f () f(); (f f) () = 2 f 2 () + 2(A()f() f()) 0. (f f )( i ) (f f )( i 0) = (B i f f)( i 0) 0, (f f) ( i ) (f f) ( i 0). By he preceding, we see ha (f f) is nondecreasing on [0, ). The nondecreasing naure of (f f ) shows direcly ha f is bounded if and only if is magniude is non-increasing, since f is coninuous on R +. Noe ha i+1 f 2 ( i+1 0) = f 2 ( i ) + 2 f f () d i f 2 ( i ) + 2 ( i+1 i ) f f ( i ). Moreover, again for f bounded, i now follows ha for k large, 0 (f f) ( ik 0) + c f 2 ( ik 0) (f f) ( ik 1) + 2bc(f f) ( ik 1) + c f 2 ( ik 1) = (1 + 2bc)(f f) ( ik 1) + c f 2 ( ik 1). This implies ha Since we also have (f f) ( ik 1) c 1 + 2bc f 2 ( ik 1). Le (f f) ( ik 0) c f 2 ( ik 0). κ 1 = hen we have proved ha for any k N c 1 + 2bc (f f) ( ik 1) κ 1 f 2 ( ik 1), and (f f) ( ik 0) κ 1 f 2 ( ik 0).

4 4 Hong-Kun ZHANG Jin-Guo LIAN Jiong SUN For [ ik 1, ik ), If hen Bu on he oher hand, if (f f) () (f f) ( ik 0) κ 1 f 2 ( ik 0) = κ 1 [ f 2 () + ik (f f) (s)ds] = κ 1 [ 1 2 f 2 () f 2 () f 2 () + ik ik (f f) (s)ds 0, (f f) () κ 1 2 f 2 (). (f f) (s)ds]. since ik b, so we have 1 2 f 2 () + ik (f f) (s)ds 0, (f f) () 1 2b f 2 (). Le κ = min{κ 1 /2, 1/2b}, hen we ge for all [ ik 1, ik ), k > 0: (f f) () κ f 2 (). By Gronwall s inequaliy, we know ha on each [ ik 1, ik ), k N: f() 2 e λ( i k 1) f( ik 1) 2 hold for any λ (0, κ). Since by assumpion, here exiss N, such ha for any n > N, n ( ik ik 1) nδ 2 δ 2b. k=1 Since for any > in, here exiss k N, such ha ik < ik+1. Thus we have Therefore f 2 () f 2 ( ik ) e λ(i k i k 1) f( ik 1) 2 e λ δ b f( i1 ) 2 e λ δ b f( 1 ) 2. f() e λ1 f( 1 ) hold for any λ 1 = κδ/b and any [ 1, + ). The conclusion abou ( unbounded soluions follows from a compleely parallel argumen. In his case, of course, f 2) is evenually posiive. Bu T2 may no be he same as 1, since he unbounded soluions may no be nondecreasing from he beginning. I is very imporan o deermine he dimension of exponenially decreasing (or increasing) funcions in he kernel of L, since i is closely relaed o he so called Lyapunov exponens o

5 NEW APPROACH TO DIFFERENTIAL EQUATIONS WITH COUNTABLE IMPULSES 5 he dynamics relaed o he equaions in (1). In his paper we use operaor heory o prove he following main heorem. Theorem 2 The differenial equaion given by (2) wih counable impulses has exacly n linearly independen exponenially increasing soluions and n linearly independen exponenially decreasing soluions. 3 Proof of Theorem 2 To prove he main heorem, we inroduce a class of generalized differenial operaors on an L 2 space induced by he sum of aom measures. Consider a special Radon measure µ on I, i.e., he sum of δ measures defined on J : µ = δ i. i J Denoe f ± () = f( ± 0). Definiion 1 Fix an n N. Denoe by F he se of all measurable funcions f from I o C n such ha f and f are absoluely coninuous on each open inerval of I. Le L be he linear operaor defined by (Lf)( i ) = f +( i ) + f ( i ) + B i f( i ), i J (4) wih domain D = {f F f () + A()f() = 0, I}, where A is a piecewise coninuous funcion from I ino he space of n n posiive semi-definie marices wih disconinuiies only conained in J and for each i J, B i is a posiive definie, n n real marix. Le he maximal operaor L M of L be he resricion of L I on D M, wih D M = {f D L 2 (I, dµ) Lf L 2 (I, dµ)}. Definiion 2 defined by For L defined as above, le p() = Le M be he linear operaor (Mf)( i ) = (plpf)( i ) = p 2 ( f +( i ) + f ( i ) + B i f( i )), i J (5) wih domain Definiion 3 D = {f F (pf) () + A()pf() = 0, I}. Le he maximal operaor M M of M be he resricion of M on D M, wih D M = {f D L 2 (I, dµ) Mf L 2 (I, dµ)}. Le ˆD be he se of all f D M such ha f and f vanish a he lef end poin of I, supp(f) is compac. Denoe by ˆM he resricion of M on ˆD, define he minimal operaor M 0 of M o be he smalles closed operaor in L 2 (I, dµ) which exends ˆM, and denoe by D 0 he domain of M 0. Lemma 1 Le M 0 and M M be he minimal and maximal operaor of M, respecively. Then M 0 M M and for any f D 0, f and f vanish a 0. Proof Le {f n } be a Cauchy sequence of funcions from ˆD converging o f wih M 0 f n converging o M 0 f in L 2 (I, dµ). Since Mf n 0 on I, i follows ha, on any compac inerval [α, β] I, f n and f n boh converge uniformly.

6 6 Hong-Kun ZHANG Jin-Guo LIAN Jiong SUN For simpliciy, from now on, we denoe f = pf and g = pg. Lemma 2 M 0 is a symmeric operaor. Proof Clearly, M 0 is densely defined in L 2 (I, dµ). For any f, g in he domain of M 0, by Lemma 1, we know ha f( 0 ) = f ( 0 +) = 0, and g( 0 ) = g ( 0 +) = 0. Wihou loss of generaliy, we may assume ha f and g are real vecor valued funcions. Then ( f g)() = ( f + g)( 0 ) + 0 i=1 (B i f g)(i ) [Mf, g] = p(b i f g)(i ) + ( f g)( 0 +) + ( f g)( i+1 ) ( f g)( i +) i=0 = ( f + g)( 0 ) + i=0 (B i f g)(i ) + [A f, g] = [f, Mg] i=1 Thus by definiion, M 0 is symmeric. Corollary 1 M M M 0 Lemma 3 [Hardy s Inequaliy] Le h(x) be nonnegaive inegrable funcion, define and ake q > 1. Then 0 ( F () F () = 0 h(s)ds ) q q d < ( q 1 )q 0 h() q d Lemma 4 The minimal operaor M 0 has closed range, and, for all f D (M 0 ), [Mf, f] 1 4 f 2.. Proof Inegraion by pars shows ha, for f C 0 (R + ), [M 0 f, f] = [ f, f ] [ ] + A f, f 1 [f, f] 4 where he righ hand inequaliy follows from Hardy s inequaliy. Corollary 2 The minimal operaor is one-o-one. In he following secions, from Lemma 5 o Lemma 6, we will le M o be defined as in Definiion 4, bu choose I o be a compac inerval J = [α, β], wih j1 1 < α < j1 < j2 < β < j2+1. Again we denoe M 0 and M M o be he corresponding minimal and maximal operaors, respecively. Lemma 5 For h L 2 (J, dµ), he equaion Mf = h has a soluion f in he domain of M 0 if and only if h is orhogonal o all soluions of M M g = 0. i.e. Ker(M M ) = R(M 0 ). Proof Assume ha for h L 2 (J, dµ), here exiss f D 0, such ha Mf = h. For any g Ker(M M ), use he same calculaion as in Lemma 2, he following hold: [h, g] J = [M 0 f, g] J = [f, Mg] J = 0.

7 NEW APPROACH TO DIFFERENTIAL EQUATIONS WITH COUNTABLE IMPULSES 7 Conversely, if h is orhogonal o all soluions of M M g = 0, choose f such ha Mf = h and f( j1 ) = f ( j1 ) = 0. We need o show ha f( j2 ) = f ( j2 +) = 0. By basic calculaions, for any g Ker(M M ), [Mf, g] J [f, Mg] J = ( f g )( j2 +) ( f g)( j2 ). On he oher hand, since M M g = 0 and [h, g] J = 0, we have [Mf, g] J [f, Mg] J = 0. Choose g Ker(M M ), s.. g( j ) 0, bu g ( j +) = 0, hen we know ha f ( j +) = 0. Similarly, we can show f( j ) = 0. Corollary 3 M 0 is he resricion of M o he domain: {f D M : f( j1 ) = f ( j1 ) = f( j2 ) = f ( j2 +) = 0} Lemma 6 If we resric M on a compac inerval J, hen M 0 = M M Proof By Corollary 1, we only need o show M 0 M M. Le f be in he domain of M 0. Since M M is surjecive for compac J, so here is a funcion g in D M such ha M M g = M 0 f. Thus M 0 (f g) = 0. For any u in he range of M 0, here exiss v in he domain of M 0, such ha M 0 v = u. We also have [f g, u] J = [f g, M 0 v] J = [M 0 (f g), v] J = 0. Therefore f g is orhogonal o he range of M 0. Bu by Lemma 5, f g is in he null space of M M. Hence f D M and M M f = M0 f. Thus we have proved M0 M M. Lemma 7 If I = [0, ), we also have M0 = M M. Proof By Corollary 1, we only need o show M0 M M. If g is in he domain of M0, hen on any compac subinerval J = [α, β] of I, we have [Mf, g] = [f, M 0 g], f D. Thus he resricion of g o J is in he domain of ˆM. Bu by Lemma 6, he resricion of M0 g o J mus agree wih M M g. Since J is arbirary, he heorem is proved. Lemma 8 [5][Theorem IV.1.2] Le T be a closed and densely defined operaor from a Hilber space H 1 ino a Hilber space H 2. Denoe he range of T by R(T ). Then he following saemens are equivalen: (1) R(T ) is closed; (2) R(T ) is closed; (3) R(T ) is he orhogonal complemen of he null space of T ; (4)R(T ) is he orhogonal complemen of he null space of T. Lemma 9 Boh M 0 and M M are closed and have closed range. Furhermore, M M is surjecive. Proof By Corollary 2, we know ha he null space of he minimal operaor is {0}. From he definiion of M 0, i is easy o show ha i is a closed operaor wih closed range. I follows from Lemma 6 and Lemma 8 ha he range of M M is closed and is in fac he whole space D L 2 (I, dµ). Lemma 10 [7] If T is a closed symmeric operaor ha is semi-bounded, hen T has equal deficiency indices. Furhermore, if T is a posiive symmeric operaor, hen is quadraic form Q is a closable quadraic form and is closure is he quadraic form of a unique selfadjoin operaor ˆT (called he Friedrichs exension of T ). ˆT is a posiive exension of T wih he same lower bound. H is he only self-adjoin exension of T whose domain is conained in he domain of ˆQ (he closure of Q).

8 8 Hong-Kun ZHANG Jin-Guo LIAN Jiong SUN Lemma 11 M 0 has a self-adjoin Friedrichs exension H saisfies: [Hf, f] 1 [f, f], f D(H). 4 Furhermore, for any f D(H), f vanishes a 0. Proof By Lemma 10, M 0 has equal deficiency indices. I follows from Von Neumann s Theorem ha M 0 always has self-adjoin exensions. Furhermore, by Lemma 4, M 0 is posiive and symmeric. I follows from he above lemma ha M 0 has a unique selfadjoin exension he Friedrichs exension H associaed o Q defined by Q(f, g) = [f, M 0 g] for f, g D 0. Moreover, H is bounded from below by he same bound 1/4 as M 0. To prove he las saemen, we use he closedness propery of ˆQ. From he consrucion of ˆQ (see [7] Theorem X.23), ˆQ is he closed form on he Hilber space obained by he compleion of D 0 under he inner produc [f, g] 1 := Q(f, M 0 g) + [f, g]. For any f D(H), here exis {f k } D 0, such ha [f k f, f k f] 1 0 as k. From he fac cq(f k f, f k f) Q(f k f, M 0 (f k f)), i follows ha f k ( i ) f ( i ) for any i J, which gives he resul. Lemma 12 The Friedrichs exension H equals Ĥ, where Ĥ is he resricion of he maximal operaor M M o {f D M f(0) = 0}. Proof By Lemma 11, H is posiive definie and surjecive. I is clearly conained in Ĥ. Bu he elemenary heory of Fredholm operaors shows ha an n dimensional exension of H has an n dimensional null space. By he above heorem, M M has exacly an n dimensional null space. Hence, every elemen of D M differs from an elemen of he domain of H by a piecewise smooh compacly suppored funcion which does no vanish a 0. For any f, g D(Ĥ), [Ĥf, g] = (B i f g)(i ) + ( ( f) g + A f g) = [f, Ĥg]. I i J Thus, Ĥ is symmeric. Since H Ĥ, we have Ĥ (Ĥ) H. This implies ha he wo operaors are he same. Now we use he resuls obained from he above general operaor heorem o prove he main Theorem 2. Corollary 4 f Ker(L M ) is non-increasing if and only if g = f is a non-increasing soluion o Mg = 0. Noice ha soluions of he differenial equaions (2) are in fac he kernel of L. By Theorem 1, hose decreasing soluions decay exponenially. So hey belong o he kernel of L M. I is enough o show dim(ker(l M )) = n. I is sufficien o consider real soluions. By he assumpion ha A() 0 for each I, we have 1 2 (f f) () = f 2 () + (A()f() f()) 0. On he oher hand, if i J, hen (f f )( i +) (f f )( i ) = (B i f f)( i ). This implies ha f f is an increasing funcion on I. Choose f such ha i vanishes a 0, bu f does no. Then by Theorem 1, he soluion f has exponenially increasing ampliude. Clearly dim(ker(l)) 2n. The se {f Ker(L) f(0) = 0, f (0+) 0} is n dimensional, which means ha here are a mos n dimensional bounded soluions for L M f = 0. Thus dim(ker(l M )) n. This means ha here are a mos n linearly independen non-increasing soluions for Mf = 0. If H is he Friedrichs exension of M 0, hen by Lemma 11, H is self-adjoin and posiive definie. Furhermore, for any f in D(H), f(0) = 0.

9 NEW APPROACH TO DIFFERENTIAL EQUATIONS WITH COUNTABLE IMPULSES 9 We choose n compacly suppored piecewise smooh funcions f 1, f 2,..., f n on I such ha he space spanned by {f i (0) i = 1, 2,..., n} has dimension n. Then hey are linearly independen mod D(H). I is clear ha f i D M for i = 1, 2,..., n. This forces he Fredholm index o increase by n. Since H is already surjecive, by Lemma 11, his produces an n dimensional L 2 (I, dµ) soluion space o M M f = 0. So dim(ker(m M )) n, which implies ha dim(ker(m M )) = n. This complees he proof of he main Theorem 2. 4 Conclusions and Fuure Works In applicaion o billiard dynamics, if we consider billiard ables o be Riemannian manifolds wih boundaries, assuming hey have all non-posiive secional curaures [9]. Thus A() 0 in equaion (1), and B i s are symmeric and corresponding o he curvaure operaor a he boundary. A Jacobi field along a billiard rajecory γ is coninuous iself, bu is direcive has jumps a disconinuiies of he rajecory. I follows from he above heorem ha as long as hose B i s are bounded from below by a posiive number c for a sequence of {i k } k=1, hen here are n dimensional sable Jacobi fields and n dimensional unsable Jacobi fields along γ. Furhermore, his propery is also rue for rajecories running ino corners or cusps by he seup of our main heorem. References [1] L. A. Bunimovich, Y. G. Sinai and N. I. Chernov, Markov pariions for wo-dimensional billiards, Russ. Mah. Surv. 1990, 45: [2] P. Balin, N. I. Chernov, D. Szasz and I. P. Toh, Geomery of mulidimensional dispersing billiards, Aserisque, 2003, 286: [3] N. I. Chernov and H. K. Zhang, Billiards wih polynomial mixing raes, Nonlineariy, 2005, 18: [4] N. I. Chernov and H. K. Zhang, A family of chaoic billiards wih variable mixing raes, Sochasics and Dynamics, 2005, 5: [5] S. Goldberg S., Unbounded Linear Operaors, New York: Dover Publicaions, [6] R. M. Kauffman and H. K. Zhang, A Class of Ordinary Differenial Operaors wih Jump Boundary Condiions, in Lecure Noes In Pure and Applied Mahemaics, Marcel Dekker, Inc. New York, 2003, 234: [7] M. Reed and B. Simon, Mehods of Modern Mahemaical Physics II, Academic Press, New York, [8] D. Szasz, Hard Ball Sysems and he Lorenz Gas, Encyclopaedia of Mahemaical Sciences, Springer, 2000, 101. [9] H. K. Zhang, Exponenial behavior of soluions o a class of differenial equaions, Aca Scieniarum Nauralium Universiais Neimongol, 2005, 36(6):