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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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Asymptotic stability in functional-differential equations by Liapunov functionals
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by Bo Zhang PDF
Trans. Amer. Math. Soc. 347 (1995), 1375-1382 Request permission


We consider the asymptotic stability in a system of functional differential equations $x’(t) = F(t,{x_t})$ by Liapunov functionals $V$. The work generalizes some well-known results in the literature in that we only require the derivative of $V$ to be negative definite on a sequence of intervals ${I_n} = [{S_n},{t_n}]$. We also show that it is not necessary to require a uniform upper bound on $V$ for nonuniform asymptotic stability.
    T. A. Burton, Uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc. 68 (1978), 195-199. —, Stability and periodic solutions of ordinary and functional differential equations, Academic Press, Orlando, 1985. T. A. Burton and L. Hatvani, Stability theorems for nonautonomous functional differential equations by Liapunov functionals, Tôhoku Math. J. 41 (1989), 65-104. —, On nonuniform asymptotic stability for nonautonomous functional differential equations, Differential and Integral Equations 2 (1990), 285-293. T. A. Burton and G. Makay, Asymptotic stability for functional differential equations, Acta Math. Hungar. 65 (1994), 243-251. J. Hale, Theory of functional differential equations, Springer-Verlag, New York, 1977. J. Kato, A conjecture in Liapunov method for functional differential equations, Preprint. T. Yoshizawa, Stability by Liapunov’s second method, Math. Soc. Japan, Tokyo, 1966.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1375-1382
  • MSC: Primary 34K20
  • DOI:
  • MathSciNet review: 1264834