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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Stability theory for functional-differential equations
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by T. A. Burton PDF
Trans. Amer. Math. Soc. 255 (1979), 263-275 Request permission

Abstract:

We consider a system of functional differential equations $x’ (t) = \mathcal {F} (t, x( \cdot ))$, together with a Liapunov functional $\mathcal {V} (t, x( \cdot ))$ with $\mathcal {V}’ \leqslant 0$. Most classical results require that $\mathcal {F}$ be bounded for $x( \cdot )$ bounded and that $\mathcal {F}$ depend on $x(s)$ only for $t - \alpha (t) \leqslant s \leqslant t$ where $\alpha$ is a bounded function in order to obtain stability properties. We show that if there is a function $H(t, x)$ whose derivative along $x’ (t) = \mathcal {F} (t, x( \cdot ))$ is bounded above, then those requirements can be eliminated. The derivative of H may take both positive and negative values. This extends the classical theorem on uniform asymptotic stability, gives new results on asymptotic stability for unbounded delays and unbounded $\mathcal {F}$, and it improves the standard results on the location of limit sets for ordinary differential equations.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 255 (1979), 263-275
  • MSC: Primary 34K20
  • DOI: https://doi.org/10.1090/S0002-9947-1979-0542880-9
  • MathSciNet review: 542880