Asymptotic behavior of the solutions of an integrodifferential system期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Asymptotic behavior of the solutions of an integrodifferential system

Authors:	Deh-phone Kung Hsing

Institution:	(1) Kingston, Rhode Island, U.S.A.

Abstract:	Summary We consider the system(L): $y'(t) = \sum\limits_{i = 1}^\infty {A_i y(t - \tau _i )} + \int\limits_{ - \infty }^t {B(t - s)y(s)ds}$ , t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each A_i, B(t) are n × n matrices. System(L) generates a semigroup by means of T_tf(s)=y (t+s, f), f(s) ∈ BC_l(− ∞, 0]. Under some hypotheses concerning the roots ofdet $\lambda 1 - \mathop \Sigma \limits_{i = 1}^\infty A_i exp - \lambda \tau _i ] - \widehat{B(\lambda )}] = 0$ where $\widehat{B(\lambda )}$ is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose ∥B(t)∥ ɛ L₁0, ∞), $\left\\| {B(t)} \right\\|\varepsilon L_1 0,\infty ),\mathop \Sigma \limits_{i = 1}^\infty \left\\| {A_i } \right\\|< \infty$ thendet $\left {\lambda I - \mathop \Sigma \limits_{i = 1}^\infty A_i \exp - \lambda \tau _i ] - \widehat{B(\lambda )}} \right] \ne 0$ forRe λ>0 iff for every ɛ>0 there is an M_ɛ>0 such that ∥T_tf∥_l ⩽ ⩽ M_ɛ exp ɛt]∥f∥_l for t ⩾ 0. Corollary 3.1.1: suppose $\mathop \Sigma \limits_{i = 1}^\infty \left\\| {A_i } \right\\|\exp a\tau _i ]< \infty$ exp at]B(t) ∈ ∈ L₁0, ∞) for some a>0 anddet $\left {\lambda I - \mathop \Sigma \limits_{i = 1}^\infty A_i \exp - \lambda \tau _i ] - \widehat{B(\lambda )}} \right] \ne 0$ forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable. Entrata in Redazione il 21 marzo 1975. The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper.

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