TAILIEUCHUNG - Báo cáo "On the asymptotic behavior of delay differential equations and its relationship with C0 - semigoup "

In this paper, we study the asymptotic behavior of linear differential equations under nonlinear perturbation. Let’s consider the delay differential equations: dx = Ax + f(t, xt ), dt where t ∈ R+ , A ∈ L(E), f : R+ × E −→ E and (T (t))t≥0 is C0 -semigroup be generated by A. We will give some sufficient conditions for uniformly stable and asymptotic equivalence of above equations. | VNU Journal of Science Mathematics - Physics 23 2007 63-69 On the asymptotic behavior of delay differential equations and its relationship with Co - semigoup Dang Dinh Chau Nguyen Bui Cuong Department of Mathematics Mechanics Informatics College of Science VNU 334 Nguyen Trai Hanoi Vietnam Received 15 November 2006 received in revised form 2 August 2007 Abstract. In this paper we study the asymptotic behavior of linear differential equations under nonlinear perturbation. Let s consider the delay differential equations dx I A Ax f t xt dt where t G R A G L E f R X E E and T t i 0 is C0-semigroup be generated by A. We will give some sufficient conditions for uniformly stable and asymptotic equivalence of above equations. 1. Introduction Consider the following delay differential equations Eq. xdt Ax t pf t x t 0 t 0 h 0 0 1 where x . G E A G L E E is a Banach space the operator f R X E E is continuous in t and satisfies all following conditions f t 0 0 2 Ilf t y t 0 f t z t Ớ H L sup y t 0 z t 0 . 3 -h ớ 0 In 1 proved that if Eq. 1 satisfies 2 and 3 with given initial condition x t p t h t 0 p . G C h 0 E then Eq. 1 has a unique solution on the half-line. In recent years much attentions have been devoted to the qualitative theory of solutions of differential equation with time delay see 1-5 . In this direction a particular attentions has been focused on extending the classical results on the asymptotic behavior of solutions of differential equations. In many applied models concerned to mechanics models of biology and population see 6-9 . In this paper we give some extending results for sufficient conditions of stable and asymptotic equivalence see 1-5 of linear delay differential equations under nonlinear perturbation in Banach space. The obtained results thank to use of the theories of general dynamic systems see 10 11 . Corresponding author. Tel. 84-4-8325854. E-mail chaudd@ 63 64 . Chau . Cuong VNU Journal of Science Mathematics - Physics

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