ACTA UNIVERSITATIS APULENSIS No 11/2006 Proceedings of he Inernaional Conference on Theory and Applicaion of Mahemaics and Informaics ICTAMI 2005 - 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. 2000 Mahemaics Subjec Classificaion: 34K05, 47H10. 1. 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.
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
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
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
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 + 0 0 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() 0 0 0 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
+ 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. 1 1 216
References [1] W.A. Coppel,On he sabiliy of ordinary differenial equaions,j. London Mah. Soc,39 (1964),pp255-260. [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 23-26. [4]J.L. Massera,J.J Scäffer, Linear differenial equaions and funcional analysis,annals of Mah., 67(1958),pp 517-573. [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 256-261. [7]Pavel Talpararu, Quelques problemes concernan l equivalence asympoique des sysemes differeniels,boll. U.M.I (4)1971 pp 164-186. [8]I.A.Rus, Generalized conracions, Seminar on fixed poin heory,no 3,1983,1-130. Olaru Ion Marian Deparmen of Mahemaics Universiy of Sibiu Address Sr.Dr I.Raiu No 5-7 email:olaruim@yahoo.com 217