Asymptotic Behavior of Linear Integrodifferential Systems

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Abstract
We consider the system <!-- MATH $({\text{L)}}y'(t) = Ay(t) + \int_{ - \infty }^t {B(t - s)y(s)ds,y(t) = f(t),t \leqslant 0}$ --> where is an -vector and and are <!-- MATH $n \times n$ --> matrices. System <!-- MATH $({\text{L)}}$ --> generates a semigroup given by <!-- MATH ${T_t}f(s) = y(t + s;f)$ --> for bounded, continuous and having a finite limit at . Under hypotheses concerning the roots of <!-- MATH $\det (\lambda I - A - \hat B(\lambda ))$ --> , where <!-- MATH $\hat B(\lambda )$ --> is the Laplace transform, various results about the asymptotic behavior of are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If $ B(t) \in {L^1}[0,\infty )$ and $ {(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $ \operatorname{Re} \lambda > 0$, then for every $ \epsilon > 0$, there is an $ {M_{\epsilon}}$ such that <!-- MATH $||{T_t}f|| \leqslant {M_{\epsilon}}{e^{\epsilon t}}||f||$ --> . Theorem 2. If $ {(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $ \operatorname{Re} \lambda > - \alpha (\alpha > 0)$ and if $ B(t){e^{\alpha t}} \in {L^1}[0,\infty )$, then the solution to $ ({\text{L)}}$ is exponentially asymptotically stable.