An integro-differential equation

Burton, T. A.

doi:10.1090/S0002-9939-1980-0567979-6

An integro-differential equation
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by T. A. Burton PDF

Proc. Amer. Math. Soc. 79 (1980), 393-399 Request permission

Abstract:

The vector equation \[ x’(t) = A(t)x(t) + \int _0^t {C(t,s)D(x(s))x(s)ds + F(t)} \] 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 $u’(t) = - \int _0^t {a(t - s)g(u(s))ds}$ can be extended to a vector system $x’(t) = - \int _0^t {H(t,s)x(s)ds}$.

References

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