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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Uniform asymptotic stability in functional differential equations


Author: T. A. Burton
Journal: Proc. Amer. Math. Soc. 68 (1978), 195-199
MSC: Primary 34K20
DOI: https://doi.org/10.1090/S0002-9939-1978-0481371-5
MathSciNet review: 0481371
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Abstract: The classical uniform asymptotic stability result for a system of functional differential equations

$\displaystyle x' = F(t,{x_t})$ ($ 1$)

calls for a Liapunov functional $ V(t,\phi )$ satisfying $ W(\vert\phi (0)\vert) \leqslant V(t,\phi ) \leqslant {W_1}(\vert\phi (0)\vert)... ...vert\vert\phi \vert\vert\vert),{V'_{(1)}} \leqslant - {W_3}(\vert\phi (0)\vert)$, and $ \vert f(t,{x_t})\vert$ bounded for $ \vert\vert\vert{x_t}\vert\vert\vert$ bounded. We show that it is not necessary to require $ \vert f(t,{x_t})\vert$ bounded. Here, $ \vert\vert\vert \cdot \vert\vert\vert$ is the $ {L^2}$-norm.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0481371-5
Keywords: Functional differential equations, Liapunov functionals, uniform asymptotic stability
Article copyright: © Copyright 1978 American Mathematical Society

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