## Asymptotic behavior of linear integrodifferential systems

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- by Viorel Barbu and Stanley I. Grossman PDF
- 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*.

## References

- R. K. Miller,
*Asymptotic stability properties of linear Volterra integrodifferential equations*, J. Differential Equations**10**(1971), 485–506. MR**290058**, DOI 10.1016/0022-0396(71)90008-8
S. I. Grossman and R. K. Miller, - Einar Hille and Ralph S. Phillips,
*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR**0089373** - Rodney D. Driver,
*Existence and stability of solutions of a delay-differential system*, Arch. Rational Mech. Anal.**10**(1962), 401–426. MR**141863**, DOI 10.1007/BF00281203 - J. J. Levin and J. A. Nohel,
*On a system of integro-differential equations occurring in reactor dynamics*, J. Math. Mech.**9**(1960), 347–368. MR**0117522**, DOI 10.1512/iumj.1960.9.59020 - Kôsaku Yosida,
*Functional analysis*, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR**0180824**

*Linear Volterra integro-differential systems with ${L^1}$ kernels*(submitted).

## Additional Information

- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**173**(1972), 277-288 - MSC: Primary 45M05
- DOI: https://doi.org/10.1090/S0002-9947-1972-0308712-2
- MathSciNet review: 0308712