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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals
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by László Hatvani PDF
Trans. Amer. Math. Soc. 354 (2002), 3555-3571 Request permission

Abstract:

Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE’s whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov–Krasovskiĭ functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals. The functionals and their derivatives are estimated by either ${x(t)}$, the norm of the instantaneous value of the solutions or $\|x_t\|_2$, the $L_2$-norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two different types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE’s with one delay and with distributed delays.
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Additional Information
  • László Hatvani
  • Affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
  • MR Author ID: 82460
  • Email: hatvani@math.u-szeged.hu
  • Received by editor(s): November 5, 2001
  • Published electronically: April 30, 2002
  • Additional Notes: The author was supported by the Hungarian National Foundation for Scientific Research (OTKA T/029188).
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 3555-3571
  • MSC (2000): Primary 34K20
  • DOI: https://doi.org/10.1090/S0002-9947-02-03029-5
  • MathSciNet review: 1911511