Asymptotic instability of nonlinear differential equations

Transcription

1 Elecronic Journal of Differenial Equaions, Vol. 1997(1997), No. 16, pp ISSN: URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp (login: fp) or Asympoic insabiliy of nonlinear differenial equaions Rafael Avis & Raúl Naulin Absrac This aricle shows ha he zero soluion o he sysem x = A()x + f(, x), f(, ) = is unsable. To show insabiliy, we impose condiions on he nonlinear par f(, x) and on he fundamenal marix of he linear sysem y = A()y. Our resuls generalize he insabiliy resuls obained by J. M. Bownds, Havani-Pinér, and K. L. Chiou. 1 Inroducion Bownds [1] sudied sabiliy properies of he second order differenial equaions y + a()y =, (1) x + a()x = f(, x, x ),, (2) where a() is a coninuous real-valued funcion. I is proved in [1] ha, if (1) has a sable zero soluion and has anoher soluion wih he propery lim sup( y() + y () ) >, (3) hen, under suiable condiions on f, here exiss a soluion x o (2) which saisfies (3). Bownds [1] conjecured ha his resul is rue wihou he sabiliy assumpion for (1), conjecure ha was laer proven in [6]. This resul and some oher ideas from [6] have opened ineresing possibiliies in he sudy of asympoic insabiliy, as shown in [7]. This aricle concerns he generalizaion of he resuls given in [2] for he sysems y () = A()y(), (4) x () = A()x()+f(, x()), f(, ) =, (5) x () = A()x()+b(),, (6) 1991 Mahemaics Subjec Classificaions: 39A11, 39A1. Key words and phrases: Liapunov insabiliy, h-sabiliy. c 1997 Souhwes Texas Sae Universiy and Universiy of Norh Texas. Submied July 9, Published Ocober 15,

2 2 Asympoic insabiliy of nonlinear differenial equaions EJDE 1997/16 where A(), f(, x) andb() are coninuous funcions, and f(, x) saisfies f(, x) γ() x m, (7) where m is a posiive consan and γ is an inegrable funcion. The following wo heorems are proven in [2]. Theorem A [2] Assume ha he fundamenal marix, Φ, ofsysem(4) saisfies Φ()Φ 1 (s) K h() h(s), s, (8) for some consan K. If (7) is fulfilled wih γ L 1 [, ), and here exiss a soluion y() of (4) such ha lim sup y(), (9) hen here exiss a nonrivial soluion x() o (5) saisfying (9). Theorem B [2] If he linear sysem (4) has a soluion y such ha, < lim sup y(), (1) hen here exiss a soluion x() of (6) saisfying (1). Our goal is o exend Theorems A and B for funcions f(, x) for which (7) holds more general funcions γ. This generalizaion is obained by using he noion of h-asympoic insabiliy. 2 Preliminaries Le V n denoe one of he spaces R n or C n. In his space x denoes a fixed norm of a vecor x, and A denoes he corresponding marix-norm of marix A. Throughouhis aricle, he funcion h is assumed o be posiive and coninuous, he inerval [, + ) is denoed by J, and we use he following noaion: x h =sup x() h() C h ={x:j V n : xis coninuous and x h < }, B h [, 1] := {x C h : x h 1 }, L 1 h = {x : J V n x() : ds < }. h() The following definiions are aken from [8]. Definiion 1 We say ha he null soluion o (5) is:

3 EJDE 1997/16 Rafael Avis & Raúl Naulin 3 h-unsable on J iff here exis an ε>and J, such ha for each δ>, here exis an iniial condiion ξ δ and a δ >, such ha h( ) 1 ξ δ <δ, and h( δ ) 1 x( δ,,ξ δ ) ε. Asympoically h-unsable on J iff x =is h-unsable or here exiss a J, such ha for any δ>, here exiss ξ V n such ha h( ) 1 ξ δ <δ, and lim sup h() 1 x(,,ξ δ ) >. 3 h-asympoic insabiliy Theorem 1 Assume ha he fundamenal marix of sysem (4) saisfies (8), and he funcion f(, x) in (5) saisfies (7) wih γ L 1 h 1 m. If here exiss a soluion of (4), such ha < lim sup y() <, (11) h() hen here exiss a nonrivial soluion x of (5) wih Propery (11). Proof. From (11), we may assume ha for a fixed ε, wih <ε<1, h() 1 y() 1 ε,. (12) Since γ L 1 h, here exiss a posiive 1 m, such ha K h(s) m 1 γ(s) ds < ε,, (13) where K is he same consan ha appears in (8). We find a soluion o (5) by finding a soluion o he inegral equaion x() =y() Φ() on he se B h [, 1]. For x B h [, 1], define U(x)() =y() Using (7), (8), and (12), we obain h() 1 U(x)() 1 ε+ Φ 1 (s)f(s, x(s)) ds,, Φ()Φ 1 (s)f(s, x(s)) ds. (14) h() 1 Φ()Φ 1 (s) γ(s) x(s) m ds.

4 4 Asympoic insabiliy of nonlinear differenial equaions EJDE 1997/16 For we obain h() 1 U(x)() 1 ε+ 1 ε + K 1 ε + ε =1. h() 1 Φ()Φ 1 (s) γ(s) h(s)h 1 (s)x(s) m ds h(s) m 1 γ(s) ds Hence U : B h [, 1] B h [, 1]. Now, we prove ha U is coninuous in he following sense: Suppose ha a sequence {x n } in C h converges uniformly o x on each compac subinerval of J, henu(x n ) converges uniformly o U(x) on each compac subinerval of J. ForafixedT >, we will show he uniform convergence of {U(x n )} on [,T]. Choose 1 >T, such ha > 1 implies K h(s) 1 γ(s) ds ε 4. (15) By he uniform convergence of {x n } on he inerval [, 1 ], here exiss a posiive ineger N = N(ε, 1 ), such ha n N implies [ 1 f(s, x n (s)) f(s, x(s)) ε 2K 1 sup h() ] 1, s [, 1 ]. (16) [, 1] For [,T]wewrie where h() 1 [U(x n )() U(x)()] I 1 +I 2 +I 3, (17) I 1 = I 2 = I 3 = 1 1 h() 1 Φ()Φ 1 (s) f(s, x n (s)) f(s, x(s)) ds h() 1 Φ()Φ 1 (s) f(s, x n (s)) ds 1 h() 1 Φ()Φ 1 (s) f(s, x(s)) ds. From (7) and (13) we obain I 2 ε 4 and I 3 ε 4. From (15) we have I 1 ε 2. These esimaes and (17) yield h() 1 [U(x n )() U(x)()] ε, [,T], which proves he uniform convergence of U(x n )ou(x)on[,t]. Now, we prove ha he se of funcions U(B h [, 1]) is equiconinuous a each poin [, ).

5 EJDE 1997/16 Rafael Avis & Raúl Naulin 5 For each x B h [, 1], he funcion z() =U(x)() is a soluion of he nonhomogeneous linear sysem z () =A()z()+f(, x()). Since h() 1 z() = h() 1 U(x)() 1 and f(, x()) is uniformly bounded on any finie -inerval, he se of all funcions z() =U(x)(), wih x B h [, 1], is equiconinuous a each poin of [, ). In his manner all he hypoheses of he Schauder-Tychonoff heorem [5] are saisfied. Consequenly, here exiss x B h [, 1] such ha x() =U(x)(), i.e. x saisfies he inegral equaion From (11) and we obain x() =y() Φ() lim Φ 1 (s)f(s, x(s)) ds. Φ()Φ 1 (s)f(s, x(s)) ds =, lim h() 1 [x() y()] =. (18) From (11) and (18) we conclude ha (11) is saisfied wih y replaced by x, and his proof is complee. Remarks Noe ha Theorem A follows from Theorem 1, by puing h() =1. Also noe ha under he condiions of Theorem 1, if we assume ha lim sup h() =, hen he rivial soluion of (5) is unsable in he sense of Liapounov. Le us consider Equaion (5) wih A() = ( 1 1 In his case he fundamenal marix Φ for sysem (4) saisfies Φ()Φ 1 (s) /s, s. Assume ha f(, x) saisfies (7) wih m 1 γ L 1. Then, according o Theorem 1, Equaion (5) yields a soluion x saisfying lim sup 1 x() >. This propery implies insabiliy in he sense of Liapounov. Noe ha his resul canno be obained from Perron s heorem [3], from Coppel s insabiliy heorem [4], or from Theorem A. Our nex goal is o generalize Theorem B. ).

6 6 Asympoic insabiliy of nonlinear differenial equaions EJDE 1997/16 Theorem 2 If here exiss a soluion y of (4) saisfying < lim sup h() 1 y(), (19) hen here exiss a soluion x of (6) wih he same propery. Proof. Noe ha every soluion x() of (6) has he form x() =Φ()c+Φ() Le y() = Φ()cbe a soluion ha saisfies (19). If lim sup h() 1 Φ() we muliply (2) by h() 1 o obain lim sup h() 1 x() > lim sup h() 1 y() lim sup h() 1 Φ() Φ 1 ()b(s)ds. (2) h(s) 1 Φ 1 (s)b(s)ds =, (21) h(s) 1 Φ 1 (s)b(s) ds. From (19) and (21) i follows ha lim sup h() 1 x() belongs o (, ]. Therefore, (19) is saisfied wih y replaced by x. On he oher hand, if < lim sup h() 1 Φ() h(s) 1 Φ 1 (s)b(s)ds, (22) he asserion of his heorem follows independenly of (19). Acknowledgmens. The auhors express heir graiude o Consejo de Invesigación of Universidad de Oriene for he financial suppor of Proyeco CI /95. References [1] Bownds J. M., Sabiliy implicaions on he asympoic beween of second order differenial equaions, Proc. Amer. Mah. Soc., 39, (1973). [2] Chiou K. L., Sabiliy implicaions on asympoic behavior of non linear sysems, Inerna. J. Mah and Mah Sci., Vo. 5, No 1, (1982). [3] Coddingon E. A., and Levinson N., Theory of Ordinary Differenial Equaion, New York, McGraw-Hill (1955).

7 EJDE 1997/16 Rafael Avis & Raúl Naulin 7 [4] Coppel W.A., On he sabiliy of ordinary differenial equaions, J. London Mah. Soc., 39, (1964). [5] CoppelW.A.,Sabiliy and Asympoic Behavior of Differenial Equaions, D. C. Heah and Company, Boson (1965). [6] Havani L., Pinér L., On perurbaion of unsable second order linear differenial equaions, Proc. Amer. Soc., 61, (1976). [7] Naulin R., Insabiliy of nonauonomous differenial sysems, o appear in Differenial Equaions and Dynamical Sysems (1997). [8] Pino M., Asympoic inegraion of a sysem resuling from he perurbaion of an h-sysem, J. Mah. Anal. and App., 131, (1988). Rafael Avis & Raúl Naulin Deparameno de Maemáicas, Universidad de Oriene Cumaná 611 A-285. Venezuela address: rnaulin@cumana.sucre.udo.edu.ve