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Proceedings of the American Mathematical Society

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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} $.

References [Enhancements On Off] (What's this?)

  • [1] T. A. Burton, Stability theory for Volterra equations, J. Differential Equations 32 (1979), 101-118. MR 532766 (80f:45016)
  • [2] R. Grimmer and G. Seifert, Stability properties of Volterra integrodifferential equations, J. Differential Equations 19 (1975), 142-166. MR 0388002 (52:8839)
  • [3] S. I. Grossman and R. K. Miller, Nonlinear Volterra integrodifferential systems with $ {L^1}$-kernels, J. Differential Equations 13 (1973), 551-566. MR 0348417 (50:915)
  • [4] J. J. Levin, The asymptotic behavior of the solution of a Volterra equation, Proc. Amer. Math. Soc. 14 (1963), 534-541. MR 0152852 (27:2824)

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Keywords: Integrodifferential equations, stability, Liapunov functional
Article copyright: © Copyright 1980 American Mathematical Society

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