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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An integro-differential equation


Author: T. A. Burton
Journal: Proc. Amer. Math. Soc. 79 (1980), 393-399
MSC: Primary 45J05
MathSciNet review: 567979
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Abstract: The vector equation

$\displaystyle 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} $.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0567979-6
PII: S 0002-9939(1980)0567979-6
Keywords: Integrodifferential equations, stability, Liapunov functional
Article copyright: © Copyright 1980 American Mathematical Society