tailieunhanh - Báo cáo " Asymptotic equilibrium of the delay differential equation "

In this paper, we show that if the operator $A(\cdot)$ is strongly continuous on Hilbert space $\mathbb H,$ $A(t)=A^*(t),$ $\sup\limits_{\|h\|\le 1}\int\limits_{T}^{+\infty}\|A(t)h\|dt\le q | VNU Journal of Science Mathematics - Physics 23 2007 92-98 Asymptotic equilibrium of the delay differential equation Nguyen Minh Man Nguyen Truong Thanh Department of Mathematics Hanoi University of Mining and Geolory Dong Ngac Tu Liem Hanoi Vietnam Received 16 April 2007 received in revised form 15 August 2007 Abstract. In this paper we show that if the operator A - is strongly continuous on Hilbert space H A t A t sup A t h dt q 1 then the equation l h 1 T dt x t A V x t r Vt 0 r is a given positive constant is asymptotic equilibrium. 1. Introduction Consider the delay differential equation d . . dtx t A t x t r t 0 1 where r is a given positive constant A - E C R L H H is a Hilbert space. We will show a condition for the asymptotic equilibrium of Eq 1 by extending some results obtained from the equation d dtx t A t x t t 0 2 Finding conditions for the asymptotic equilibrium of a differential equation is considered by many mathematicians. Some of the mathmaticians are L. Cezari A. Winter A. Ju. Levin Nguyen The Hoan .etc. In a paper L. Cezari asserted that If A t H E L1 R Rn then Eq 2 is asymptotic equilibrium The result was devoloped by A. Winter 1954 see 2 and A. Ju. Levin 1967 see 3 . However those results were restricted to finite dimentional spaces. Nguyen The Hoan extended them into any Hilbert spaces. From this result we extend on Eq 1 and obtain a similar result theorem Corresponding author. Tel. 84-4-8387564. E-mail ngmman@ 92 . Man . Thanh VNU Journal of Science Mathematics - Physics 23 2007 92-98 93 2. Preliminaries The section will be devoted entirely to the notation and concept of asymptotic equilibrium of differential equations. Almost all results of this section are more or less known. However for the reader s convenience we will quote them here and even verify several results which seem to be obviuous but not available in the mathematical literature. Thoughout this paper we will use the following notations H is a given hilbert .