Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Uniform asymptotic stability in functional differential equations
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by T. A. Burton
Proc. Amer. Math. Soc. 68 (1978), 195-199
DOI: https://doi.org/10.1090/S0002-9939-1978-0481371-5

Abstract:

The classical uniform asymptotic stability result for a system of functional differential equations \begin{equation}\tag {$1$} x’ = F(t,{x_t})\end{equation} calls for a Liapunov functional $V(t,\phi )$ satisfying $W(|\phi (0)|) \leqslant V(t,\phi ) \leqslant {W_1}(|\phi (0)|) + {W_2}(|||\phi |||),{V’_{(1)}} \leqslant - {W_3}(|\phi (0)|)$, and $|f(t,{x_t})|$ bounded for $|||{x_t}|||$ bounded. We show that it is not necessary to require $|f(t,{x_t})|$ bounded. Here, $||| \cdot |||$ is the ${L^2}$-norm.
References
  • Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, vol. 9, Mathematical Society of Japan, Tokyo, 1966. MR 0208086
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Bibliographic Information
  • © Copyright 1978 American Mathematical Society
  • 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