Olaru Ion Marian. In 1968, Vasilios A. Staikos [6] studied the equation:

Transcription

1 ACTA UNIVERSITATIS APULENSIS No 11/2006 Proceedings of he Inernaional Conference on Theory and Applicaion of Mahemaics and Informaics ICTAMI Alba Iulia, Romania THE ASYMPTOTIC EQUIVALENCE OF THE DIFFERENTIAL EQUATIONS WITH MODIFIED ARGUMENT Olaru Ion Marian Absrac. This paper reas he asympoic equivalence of he equaions x () = A()x() and x () = A()x() + f(, x(g())) using he noion of ϕ- conracion Mahemaics Subjec Classificaion: 34K05, 47H Inroducion In 1964,W.A. Coppel [1] proposed an ineresing applicaion of Massera and Schäfer Theorem ([4],p. 530) obaining he necessary and sufficien condiions for he exisence of a leas one soluions for he equaions x () = A()x() + b() (1) for every b() funcion. More precisely, hey consider b C, C being he class of he coninuous and bounded funcions defined on R + = [0, ) wih he norm b = sup b(), R + where is he euclidian norm of R n. W.A. Coppel([2],Ch.V) reaed he case when b L 1, L 1 represens he Banach space of he Lebesgue inegrable funcions on R + wih he norm b L 1 = R + b() d. Using W.A.Coppel mehod in 1966 R.Coni [3] sudied he same problem for he paricular case when b L p, 1 p, L p being he space of he funcions wih b() p inegrable on R + wih he norm b L p = { R+ b() p d In 1968, Vasilios A. Saikos [6] sudied he equaion: } 1 p x = A()x + f(, x), (2) where he funcion f belongs o a class of funcions defined on R + and saisfies some resricive condiions. 211.

2 All along he menioned paper he auhors consider he subspace X 1 of he poins in R n which are he values of he bounded soluions for he equaions x = A()x (3) a momen = 0 and X 2 R n is a supplemenary subspace R n = X 1 X2. The fundamenal condiions which was inerpolaed in W.A. Coppel paper, for equaion (1) o have a leas one bounded soluion is he exisence of projecors P 1 and P 2 and a consan K > 0 such ha when b C, 0 X()P 1 X 1 (s) ds + X()P 2 X 1 (s) ds K, (4) { X()P1 X 1 (s) K, 0 s X()P 2 X 1 (s) K, 0 s, (5) when b L 1. In heir paper R. Coni and V.A. Saikos replaced condiions (4), and (5) wih ( 0 for p 1 and X()P 1 X 1 (s) p ds + X()P 2 X 1 (s) p p K, (6) sup X()P 1 X 1 (s) + sup X()P 2 X 1 (s) K, (7) 0s s for p =. In [6] Pavel Talpalaru consider he equaion and he perurbed equaion x = A()x (8) y = A()y + f(, y), (9) where x, y, f are vecors in R n, A() M n n,coninuous in relaion o and y for 0, y <. 212

3 He demonsraed ha under some condiions (see Theorem 2.1 from [7]) for all he bounded x() soluions of he equaion (8) here exiss a leas one y() bounded soluion (9), such ha he nex relaion ake place : lim x() y() = 0. (10) Nex we inroducing he noion of ϕ-conracion and comparison funcion by: Definiion 1.1.[8]ϕ : R + R + is a sric comparison funcion if ϕ saisfies he following: i) ϕ is coninuous. ii)ϕ is monoone increasing. iii) lim ϕ n () 0, for all > 0. n iv) -ϕ(),for. Le (X, d) be a meric space and f : X X an operaor. Definiion 1.2.[8] The operaor f is called a sric ϕ-conracion if: (i) ϕ is a sric comparison funcion. (ii)d(f(x), f(y)) ϕ(d(x, y)), for all x, y X. In [8] I.A Rus give he following resul: Theorem 1.1.Le (X, d) be an complee merical space, ϕ : R + R + a comparison funcion and f : X X a ϕ-conracion.then f, is Picard operaor. Nex we using he following lema: Lemma 1.1.[6] We suppose ha X() is a coninuous and inverible marix for 0 and le P an projecor;if here exiss a consan K > 0 such ha { 0 } 1 X()P X 1 (s) q q hen here exiss N > 0 such ha K for 0, (11) X()P Nexp( qk 1 1 q 1 1 q ) for 0 (12) 213

4 2. Main resuls Le 0 0. We consider he equaion: and perurbed equaion x () = A()x(), 0 (13) y () = A()y() + f(, y(g())), 0, (14) under condiions: (a) A M n n, coninuous on [ 0, ); (b) g : [ 0, ) [ 0, ), coninuous; (c) f C([ 0, ) S), where S = {y R n y < }. We noe wih C α, he space of funcions coninuous and bounded defined on [α, ). Theorem 2.1. Le X() be a fundamenal marix of equaion (13). We suppose ha: (i) There exiss he projecors P 1, P 2 and a consan K > 0 such ha ( 0 X()P 1 X 1 (s) q ds + X()P 2 X 1 (s) q q ds K, for 0, q > 1; (ii) There exiss ϕ : R + R +, comparison funcion, and λ L p ([ 0, ) such ha f(, y) f(, y) λ()ϕ( y y ), for all 0, y, y S; (iii) f(, 0) L p ([ 0, )). Then, for every soluion bounded x() of equaion (13), here exiss a unique soluion bounded y() of equaion (14) such ha lim x() y() = 0 (15) Proof. For x C 0 we consider he operaor T y() = x()+ 0 X()P 1 X 1 (s) f(s, y(g(s)))ds 214 X()P 2 X 1 (s) f(s, y(g(s)))ds

5 We show ha he space C 0 is invarian for he operaor T.If y C 0, hen f(, y(g()) f(, y(g())) f(, 0) + f(, 0) λ()ϕ( y ) + f(, 0). From: X()P 2 X 1 (s)f(s, y(g(s))) ds 0 ( ϕ( y ) 0 q ( X()P 2 X 1 (s) q ds ( + X()P 2 X 1 (s) q 0 [( Kϕ( y ) 0 ( q 0 ( ds ds + ] we have ha he definiion of T is corec. Le x a bonded soluion for he equaion (13) and y C 0. Then: r+ + 0 T y() x() + 0 X()P 1 X 1 (s)f(s, y(g(s))) ds + + X()P 2 X 1 (s)f(s, y(g(s))) ds X()P 1 X 1 (s) f(s, y(g(s))) f(s, 0 ds+ X()P 2 X 1 (s) f(s, y(g(s))) f(s, 0) ds+ ( ( r + 2K ϕ( y ) 0 ds + ( We show ha he operaor T is ϕ-conracion. T y() T y() X()P 1 X 1 (s) f(s, 0) ds+ X()P 2 X 1 (s) f(s, 0) ds ) < X()P 1 X 1 (s) f(s, y(g(s))) f(s, y(g(s))) ds+ 215

6 + X()P 2 X 1 (s) f(s, y(g(s))) f(s, y(g(s))) ds ( 2K ds ϕ( y y ) 0 We choose 0 such ha λ(s) p ds 1. 2K 0 From Theorem 1.1 we obain ha here exiss a unique soluions of equaion (14). Le y() be soluion of (14) coresponden o x().then 0 x() y() X()P 1 X 1 (s)f(s, y(g(s))) ds+ X()P 2 X 1 (s)f(s, y(g(s))) ds = I 1 +I 2. 1 I 1 = 0 X()P 1 X 1 (s)f(s, y(g(s))) ds + 1 X()P 1 ε 0 0 X()P 1 X 1 (s)f(s, y(g(s))) ds ( X 1 (s) f(s, y(g(s))) ds+kϕ( y ) ( We choice 1 0 such ha λ(s) p 1 1 p 3K By using lema (1.1), we obain ha I1 < ε. I 2 For I 2 we have: X()P 1 X 1 (s)f(s, y(g(s))) ds 1 ( +K 1 ( ε, and 3Kϕ( y 1 X()P 2 X 1 (s) f(s, y(g(s))) f(s, 0) ds+ X()P 2 X 1 (s) f(s, 0) ds ( Kϕ( y ) ( + K

7 References [1] W.A. Coppel,On he sabiliy of ordinary differenial equaions,j. London Mah. Soc,39 (1964),pp [2] W.A. Coppel, Sabiliy and asympoic behavior of differenial equaions, Heah Mah. Monographs Boson,1965. [3] R.Coni, On he boundlessness of soluions of differenial equaions, Funkcialaj Ekvacioj,9 (1966), pp [4]J.L. Massera,J.J Scäffer, Linear differenial equaions and funcional analysis,annals of Mah., 67(1958),pp [5] I.A. Rus, Princiipii si aplicaii ale eoriei puncului fix,ediura Dacia,Cluj-Napoca,1979. [6] V.A. Saikos, A noe on he boundens of soluions of ordinary differenial equaions,boll.u.m.i, S IV,1 (1968), pp [7]Pavel Talpararu, Quelques problemes concernan l equivalence asympoique des sysemes differeniels,boll. U.M.I (4)1971 pp [8]I.A.Rus, Generalized conracions, Seminar on fixed poin heory,no 3,1983, Olaru Ion Marian Deparmen of Mahemaics Universiy of Sibiu Address Sr.Dr I.Raiu No 5-7 olaruim@yahoo.com 217