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Transactions of the American Mathematical Society
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
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0542880-9
PII: S 0002-9947(1979)0542880-9
Keywords: Stability, Liapunov functions, ordinary differential equations, delay equations
Article copyright: © Copyright 1979 American Mathematical Society