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Transactions of the American Mathematical Society

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On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals

Author: László Hatvani
Journal: Trans. Amer. Math. Soc. 354 (2002), 3555-3571
MSC (2000): Primary 34K20
Published electronically: April 30, 2002
MathSciNet review: 1911511
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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-Krasovski{\u{\i}}\kern.15em 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 $\Vert x_t\Vert _2$, 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

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).
Article copyright: © Copyright 2002 American Mathematical Society