# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Asymptotic behavior of linear integrodifferential systemsHTML articles powered by AMS MathViewer

by Viorel Barbu and Stanley I. Grossman
Trans. Amer. Math. Soc. 173 (1972), 277-288 Request permission

## Abstract:

We consider the system $({\text {L)}}y’(t) = Ay(t) + \int _{ - \infty }^t {B(t - s)y(s)ds,y(t) = f(t),t \leqslant 0}$ where $y(t)$ is an $n$-vector and $A$ and $B(t)$ are $n \times n$ matrices. System $({\text {L)}}$ generates a semigroup given by ${T_t}f(s) = y(t + s;f)$ for $f$ bounded, continuous and having a finite limit at $- \infty$. Under hypotheses concerning the roots of $\det (\lambda I - A - \hat B(\lambda ))$, where $\hat B(\lambda )$ is the Laplace transform, various results about the asymptotic behavior of $y(t)$ 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 $||{T_t}f|| \leqslant {M_{\epsilon }}{e^{\epsilon t}}||f||$. 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.
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