On the asymptotic behavior of delay differential equations a n d i t s r e l a t i o n s h i p w i t h Co- s e m i g r o u p
Main Article Content
Abstract
A b s t r a c t . In this paper, we study the asymptotic behavior o f linear diíĩerential equations
under nonlinear pcrturbation. Lct’s considcr the dclay diíĩerential equations:
^ = A x + f(t,Xị),
whcre t G / ĩ + , A e C(E)} f : R+ X E — ► E and {T(t))t> 0 is Co-semigroup be generated by
A. Wc will give some suffìcicnt conditions for uniĩormly stable and asymptotic equivalence of
above equations.
References
D.D. Chau, On the asymptotic cquivalence of lincar diíĩirential cquations in llilbert spaccs, VNU Journaỉ o f Science.
Mathematics-Physics 18, No2 (2002) 8 .
[2] J. Kato, The asymptotíc cquivalencc of ĩunctional diíĩcrenttial cquations, J. DiJJerenttiaỉ Equai. 1 (1996) 306.
[3] E.v. Voskorcsenski, Asymptotic equivalcnce of systems of diíĩcrential equations, Resuỉts o f maỉhematic Science 40
(1985) 245 (in Russian).
[4] M. Svec, Ịtegral and asymptotic equivelence o f two systems o f diỊỊrential equations, Equadiíĩ. Proceedings of ứìe fif\h
Czechoslovak confrecc on diíĩirenlial cquations and Their Application held in Bratislava 1981, Tcubner, Lcipzig, 1982,
p 329 338.
[5] N. Lcvinson, The asymptolic bchavior of systems of linear diíĩcrcntal equations, Amer. J. Math. 63 (1946) 1.
[6 ] H. Inaba, A semigroup - approach to the strong crgodic theorem of the multistatc slable population proccss, Mathematical
Popuỉation Studies 1 (1988) 49.
[7] H. Inaba, Asymptơtic propcrtics of thc inhomogeneuos Lotka - Von Focrster system, Mathcmatical Populaíion Studies
I (1988) 247.
[8 ] C.M. Macrcus, F.R. Waugh, R.M. VVestcvelt, Nonlincar d>namics and stability of analogc ncural networks, Physìca D
51 (1991) 234.
G.F. Wcbb, Thcory of nonlincar agc- dcpcndcnt population dynamics Pure and applied mathematics, a program o f
m o n o g r a p h s . t extbooks. L e c t u r e Notes. 1985.
[ 1 0 ] K J . E ngel, R. N a g c l , One-parameter semigroup f o r Line ar Evoỉution E q uaíions, S p r i n g e r - V c r l a g , N e w Y o r k , B e r l i n ,
L o n d o n , Paris, T o k y o . H o n g k o n g , B a r c c l o n a , H c i d c ỉ b e r g , M i l a n , S i n g a p o r e , 2 0 0 0 .
[ 1 1 ] A . Pazy, Semigroup oJ ỉinear operators and appỉications to p a r í i a ỉ dijjerential e q u a t i o n s , S p r i n g e r - V c r l a g N e w Y o r k
Inc. 1983.
[12] K.G. Valcev, O.A. Raoulukov, ỉnỹmite system o f (lijỵerential equations, Science publishing housc Anma-Ata, 1974 (in
Russian).
[13] N.T. Hoan, Asymptotic equivalcnce of systcms of diíTcrcntial equalions, IZV.Acad. Nauk ASSR 2 (1975) 35 (in Russian).
|14] C.K. Sung, H.G. Yoon, J.K. Nam. Asymptotic cquivalencc between to linear diíĩcrential systcms, Ann. Differential
Equation 13 ( 1 9 9 7 ) 4 4 .
Mathematics-Physics 18, No2 (2002) 8 .
[2] J. Kato, The asymptotíc cquivalencc of ĩunctional diíĩcrenttial cquations, J. DiJJerenttiaỉ Equai. 1 (1996) 306.
[3] E.v. Voskorcsenski, Asymptotic equivalcnce of systems of diíĩcrential equations, Resuỉts o f maỉhematic Science 40
(1985) 245 (in Russian).
[4] M. Svec, Ịtegral and asymptotic equivelence o f two systems o f diỊỊrential equations, Equadiíĩ. Proceedings of ứìe fif\h
Czechoslovak confrecc on diíĩirenlial cquations and Their Application held in Bratislava 1981, Tcubner, Lcipzig, 1982,
p 329 338.
[5] N. Lcvinson, The asymptolic bchavior of systems of linear diíĩcrcntal equations, Amer. J. Math. 63 (1946) 1.
[6 ] H. Inaba, A semigroup - approach to the strong crgodic theorem of the multistatc slable population proccss, Mathematical
Popuỉation Studies 1 (1988) 49.
[7] H. Inaba, Asymptơtic propcrtics of thc inhomogeneuos Lotka - Von Focrster system, Mathcmatical Populaíion Studies
I (1988) 247.
[8 ] C.M. Macrcus, F.R. Waugh, R.M. VVestcvelt, Nonlincar d>namics and stability of analogc ncural networks, Physìca D
51 (1991) 234.
G.F. Wcbb, Thcory of nonlincar agc- dcpcndcnt population dynamics Pure and applied mathematics, a program o f
m o n o g r a p h s . t extbooks. L e c t u r e Notes. 1985.
[ 1 0 ] K J . E ngel, R. N a g c l , One-parameter semigroup f o r Line ar Evoỉution E q uaíions, S p r i n g e r - V c r l a g , N e w Y o r k , B e r l i n ,
L o n d o n , Paris, T o k y o . H o n g k o n g , B a r c c l o n a , H c i d c ỉ b e r g , M i l a n , S i n g a p o r e , 2 0 0 0 .
[ 1 1 ] A . Pazy, Semigroup oJ ỉinear operators and appỉications to p a r í i a ỉ dijjerential e q u a t i o n s , S p r i n g e r - V c r l a g N e w Y o r k
Inc. 1983.
[12] K.G. Valcev, O.A. Raoulukov, ỉnỹmite system o f (lijỵerential equations, Science publishing housc Anma-Ata, 1974 (in
Russian).
[13] N.T. Hoan, Asymptotic equivalcnce of systcms of diíTcrcntial equalions, IZV.Acad. Nauk ASSR 2 (1975) 35 (in Russian).
|14] C.K. Sung, H.G. Yoon, J.K. Nam. Asymptotic cquivalencc between to linear diíĩcrential systcms, Ann. Differential
Equation 13 ( 1 9 9 7 ) 4 4 .