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Transactions of the American Mathematical Society

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

 
 

 

Stability theory for functional-differential equations


Author: T. A. Burton
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
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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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Keywords: Stability, Liapunov functions, ordinary differential equations, delay equations
Article copyright: © Copyright 1979 American Mathematical Society