## An integro-differential equation

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- by T. A. Burton
- Proc. Amer. Math. Soc.
**79**(1980), 393-399 - DOI: https://doi.org/10.1090/S0002-9939-1980-0567979-6
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## Abstract:

The vector equation \[ xโ(t) = A(t)x(t) + \int _0^t {C(t,s)D(x(s))x(s)ds + F(t)} \] is considered in which*A*is not necessarily a stable matrix, but $A(t) + G(t,t)D(0)$ is stable where

*G*is an antiderivative of

*C*with respect to

*t*. Stability and boundedness results are then obtained. We also point out that boundedness results of Levin for the scalar equation $uโ(t) = - \int _0^t {a(t - s)g(u(s))ds}$ can be extended to a vector system $xโ(t) = - \int _0^t {H(t,s)x(s)ds}$.

## References

- T. A. Burton,
*Stability theory for Volterra equations*, J. Differential Equations**32**(1979), no.ย 1, 101โ118. MR**532766**, DOI 10.1016/0022-0396(79)90054-8 - Ronald Grimmer and George Seifert,
*Stability properties of Volterra integrodifferential equations*, J. Differential Equations**19**(1975), no.ย 1, 142โ166. MR**388002**, DOI 10.1016/0022-0396(75)90025-X - S. I. Grossman and R. K. Miller,
*Nonlinear Volterra integrodifferential systems with $L^{1}$-kernels*, J. Differential Equations**13**(1973), 551โ566. MR**348417**, DOI 10.1016/0022-0396(73)90011-9 - J. J. Levin,
*The asymptotic behavior of the solution of a Volterra equation*, Proc. Amer. Math. Soc.**14**(1963), 534โ541. MR**152852**, DOI 10.1090/S0002-9939-1963-0152852-8

## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**79**(1980), 393-399 - MSC: Primary 45J05
- DOI: https://doi.org/10.1090/S0002-9939-1980-0567979-6
- MathSciNet review: 567979